Unit 12: Algebraic Expression

Exercise 1: (Multiple Choice Questions and Answers 1-16)

In each of the questions 1 to 16, out of the four options, only one is correct. Write the correct answer.

Question: 1

An algebraic expression containing three terms is called a:

a.    Monomial

b.   Binomial

c.    Trinomial

d.   All of these

Solution

(c)

An algebraic expression containing three terms is called trinomial.

Question: 2

Number of terms in the expression 3 x 2 y2 y 2 z z 2 x+5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hadaahaaWcbeqaaiaaikdaaaGccaWG5bGaeyOeI0IaaGOmaiaadMha daahaaWcbeqaaiaaikdaaaGccaWG6bGaeyOeI0IaamOEamaaCaaale qabaGaaGOmaaaakiaadIhacqGHRaWkcaaI1aaaaa@4399@  is

 

a.    2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@3694@

b.   3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3695@

c.    4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaaaa@3696@

d.   5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaaaa@3697@

Solution

(c)

Total number of terms in the expression are 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaaaa@3697@ . They are  3 x 2 y,2 y 2 z, z 2 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hadaahaaWcbeqaaiaaikdaaaGccaWG5bGaaiilaiabgkHiTiaaikda caWG5bWaaWbaaSqabeaacaaIYaaaaOGaamOEaiaacYcacqGHsislca WG6bWaaWbaaSqabeaacaaIYaaaaOGaamiEaaaa@4359@  and 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaaaa@3698@ .

Question: 3

The terms of expression 4 x 2 3xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadI hadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIZaGaamiEaiaadMha aaa@3C2B@  are:

a.    4 x 2 and3xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadI hadaahaaWcbeqaaiaaikdaaaGccaaMc8Uaaeyyaiaab6gacaqGKbGa eyOeI0IaaG4maiaadIhacaWG5baaaa@4072@

b.   4 x 2  and 3xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadI hadaahaaWcbeqaaiaaikdaaaGccaqGGaGaaeyyaiaab6gacaqGKbGa aeiiaiaaiodacaWG4bGaamyEaaaa@3F40@

c.    4 x 2  andxy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadI hadaahaaWcbeqaaiaaikdaaaGccaqGGaGaaeyyaiaab6gacaqGKbGa eyOeI0IaamiEaiaadMhaaaa@3ECD@

d.   x 2  and xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiaabccacaqGHbGaaeOBaiaabsgacaqGGaGa amiEaiaadMhaaaa@3DC5@

Solution

(a)

Terms in the expression 4 x 2 3xy MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadI hadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIZaGaamiEaiaadMha aaa@3C2C@  are 4 x 2 and3xy. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadI hadaahaaWcbeqaaiaaikdaaaGccaaMc8Uaaeyyaiaab6gacaqGKbGa eyOeI0IaaG4maiaadIhacaWG5bGaaiOlaaaa@4124@

Question: 4

Factors of 5 x 2 y 2 z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ynaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bWaaWbaaSqabeaa caaIYaaaaOGaamOEaaaa@3C65@  are

a.    5×x×y×z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG PaVlaaykW7caaI1aGaey41aqRaamiEaiabgEna0kaadMhacqGHxdaT caWG6baaaa@43D9@

b.   5× x 2 ×y×z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG PaVlaaykW7caaI1aGaey41aqRaamiEamaaCaaaleqabaGaaGOmaaaa kiabgEna0kaadMhacqGHxdaTcaWG6baaaa@44CC@

c.    5×x×x×y×y×z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG PaVlaaykW7caaI1aGaey41aqRaamiEaiabgEna0kaadIhacqGHxdaT caWG5bGaey41aqRaamyEaiabgEna0kaadQhaaaa@4A02@

d.    5×x×y× z 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaae iiaiaaiwdacqGHxdaTcaWG4bGaey41aqRaamyEaiabgEna0kaadQha daahaaWcbeqaaiaaikdaaaaaaa@424F@

Solution

(c)

5 x 2 y 2 z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ynaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bWaaWbaaSqabeaa caaIYaaaaOGaamOEaaaa@3C65@  Can be factorized as 5×x×x×y×y×z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ynaiabgEna0kaadIhacqGHxdaTcaWG4bGaey41aqRaamyEaiabgEna 0kaadMhacqGHxdaTcaWG6baaaa@46EC@ .

Question: 5

Coefficient of x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@  in  9x y 2 z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaae iiaiaaiMdacaWG4bGaamyEamaaCaaaleqabaGaaGOmaaaakiaadQha aaa@3C18@  is

a.    9yz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaadM hacaWG6baaaa@3898@

b.   9yz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG yoaiaadMhacaWG6baaaa@3985@

c.    9 y 2 z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaadM hadaahaaWcbeqaaiaaikdaaaGccaWG6baaaa@398B@

d.   9 y 2 z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG yoaiaadMhadaahaaWcbeqaaiaaikdaaaGccaWG6baaaa@3A78@

Solution

(d)

Coefficient of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  in  9x y 2 z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaae iiaiaaiMdacaWG4bGaamyEamaaCaaaleqabaGaaGOmaaaakiaadQha aaa@3C18@  is 9 y 2 z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG yoaiaadMhadaahaaWcbeqaaiaaikdaaaGccaWG6baaaa@3A78@ .

Question: 6

Which of the following is a pair of like terms?

a.    7x y 2 z,7 x 2 yz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG PaVlaaiEdacaWG4bGaamyEamaaCaaaleqabaGaaGOmaaaakiaadQha caGGSaGaaGPaVlaaykW7cqGHsislcaaI3aGaamiEamaaCaaaleqaba GaaGOmaaaakiaadMhacaWG6baaaa@465F@

b.   10ay z 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiaaicdacaWGHbGaamyEaiaadQhadaahaaWcbeqaaiaaikdaaaaa aa@3C07@ , 3ay z 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadg gacaWG5bGaamOEamaaCaaaleqabaGaaGOmaaaaaaa@3A62@

c.    3xyz, 3 x 2 y 2 z 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hacaWG5bGaamOEaiaacYcacaqGGaGaaG4maiaadIhadaahaaWcbeqa aiaaikdaaaGccaWG5bWaaWbaaSqabeaacaaIYaaaaOGaamOEamaaCa aaleqabaGaaGOmaaaaaaa@4168@  

d.   4xy z 2 , 4 x 2 yz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadI hacaWG5bGaamOEamaaCaaaleqabaGaaGOmaaaakiaacYcacaqGGaGa aGinaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bGaamOEaaaa@4081@

Solution

(b)

Like terms are those terms, having same algebraic factor.

Hence, 10ay z 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiaaicdacaWGHbGaamyEaiaadQhadaahaaWcbeqaaiaaikdaaaaa aa@3C07@  and 3ay z 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadg gacaWG5bGaamOEamaaCaaaleqabaGaaGOmaaaaaaa@3A62@  are like terms.

Question: 7

Identify the binomial out of the following:

a.    3x y 2 +5y x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGynaiaadMha cqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEaaaa@3FFD@

b.   x 2 y5y x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiaadMhacqGHsislcaaI1aGaamyEaiabgkHi TiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5baaaa@3F4B@

c.    xy+yz+zx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaadM hacqGHRaWkcaWG5bGaamOEaiabgUcaRiaadQhacaWG4baaaa@3D90@

d.   3x y 2 +5yx y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGynaiaadMha cqGHsislcaWG4bGaamyEamaaCaaaleqabaGaaGOmaaaaaaa@3FF3@

Solution

(d)

Taking option (d),

3x y 2 +5yx y 2 =2 x 2 y+5y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIZa GaamiEaiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI1aGa amyEaiabgkHiTiaadIhacaWG5bWaaWbaaSqabeaacaaIYaaaaaGcba Gaeyypa0JaaGOmaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bGa ey4kaSIaaGynaiaadMhaaaaa@4754@  

As it contains only two terms, hence it is binomial.

Question: 8

The sum of x 4 xy+2 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGinaaaakiabgkHiTiaadIhacaWG5bGaey4kaSIaaGOm aiaadMhadaahaaWcbeqaaiaaikdaaaaaaa@3E37@  and x 4 +xy+2 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaam iEamaaCaaaleqabaGaaGinaaaakiabgUcaRiaadIhacaWG5bGaey4k aSIaaGOmaiaadMhadaahaaWcbeqaaiaaikdaaaaaaa@3F19@  is

a.    Monomial and polynomial in y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@

b.   Binomial and Polynomial

c.    Trinomial and polynomial

d.   Monomial and polynomial in x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@

Solution

(a)

Required sum =( x 4 xy+2 y 2 )+( x 4 +xy+2 y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacaWG4bWaaWbaaSqabeaacaaI0aaaaOGaeyOeI0IaamiEaiaa dMhacqGHRaWkcaaIYaGaamyEamaaCaaaleqabaGaaGOmaaaaaOGaay jkaiaawMcaaiabgUcaRmaabmaabaGaeyOeI0IaamiEamaaCaaaleqa baGaaGinaaaakiabgUcaRiaadIhacaWG5bGaey4kaSIaaGOmaiaadM hadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaaaaa@4C87@

=[ ( x 4 +( x 4 ) ]+( xy+xy )+( 2 y 2 +2 y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG PaVpaadmaabaWaaeqaaeaacaWG4bWaaWbaaSqabeaacaaI0aaaaOGa ey4kaSYaaeWaaeaacqGHsislcaWG4bWaaWbaaSqabeaacaaI0aaaaa GccaGLOaGaayzkaaaacaGLOaaaaiaawUfacaGLDbaacqGHRaWkdaqa daqaaiabgkHiTiaadIhacaWG5bGaey4kaSIaamiEaiaadMhaaiaawI cacaGLPaaacqGHRaWkdaqadaqaaiaaikdacaWG5bWaaWbaaSqabeaa caaIYaaaaOGaey4kaSIaaGOmaiaadMhadaahaaWcbeqaaiaaikdaaa aakiaawIcacaGLPaaaaaa@5334@

=0+0+4 y 2 =4 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG imaiabgUcaRiaaicdacqGHRaWkcaaI0aGaamyEamaaCaaaleqabaGa aGOmaaaakiabg2da9iaaisdacaWG5bWaaWbaaSqabeaacaaIYaaaaa aa@4070@

4 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadM hadaahaaWcbeqaaiaaikdaaaaaaa@387D@  is a monomial and polynomial in y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@ .

Question: 9

The subtraction of 5 times of y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  from x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@  is

a.    5xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaadI hacqGHsislcaWG5baaaa@397F@

b.   y5x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgk HiTiaaiwdacaWG4baaaa@397F@

c.    x5y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgk HiTiaaiwdacaWG5baaaa@397F@

d.   5yx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaadM hacqGHsislcaWG4baaaa@397F@

Solution

(c)

5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaaaa@3697@  times of y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  is 5y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaadM haaaa@3795@ . Now, subtraction of 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaaaa@3697@  times of y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  from x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@  is x5y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgk HiTiaaiwdacaWG5baaaa@397F@

Question: 10

 b  0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaae iiaiaadkgacaqGGaGaeyOeI0Iaaeiiaiaaicdaaaa@3B3C@  is equal to

a.    1×b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiabgEna0kaadkgaaaa@3A7E@  

b.   1b0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiadyc OHsislcaWGIbGaeyOeI0IaaGimaaaa@3B2E@

c.    0( 1 )×b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgk HiTmaabmaabaGaeyOeI0IaaGymaaGaayjkaiaawMcaaiabgEna0kaa dkgaaaa@3DAE@

d.   55 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ynaiaaiwdaaaa@3844@

Solution

(a)

 b  0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaae iiaiaadkgacaqGGaGaeyOeI0Iaaeiiaiaaicdaaaa@3B3C@  is equal to 1×b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiabgEna0kaadkgaaaa@3A7E@

Question: 11

The length of the top of square table is x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@ . The expression for perimeter is:

a.    4+x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabgU caRiaadIhaaaa@3875@

b.   2x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI haaaa@3791@

c.    4x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadI haaaa@3793@

d.   8x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaadI haaaa@3797@

Solution

(c)

Given, length of a square table =x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iEaaaa@37DB@

MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaO qaaiabgsJiCbaa@3B60@  Perimeter of a square

=4×Side =4×x =4x. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaI0aGaey41aqRaaGPaVlaaykW7caqGtbGaaeyAaiaabsgacaqG LbGaaGPaVdqaaiabg2da9iaaisdacqGHxdaTcaWG4baabaGaeyypa0 JaaGinaiaadIhacaGGUaaaaaa@4A37@   

Question: 12

The number of scarfs of length half meter that can be made from y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  meters of cloth is:

a.    2y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadM haaaa@3792@

b.   2y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaamyEaaqaaiaaikdaaaaaaa@385E@

c.    y+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgU caRiaaikdaaaa@3874@

d.   y+ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgU caRmaalaaabaGaaGymaaqaaiaaikdaaaaaaa@393F@

Solution

(a)

We have, length of 1 scarf=1/2m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4Caiaabo gacaqGHbGaaeOCaiaabAgacqGH9aqpcaaIXaGaai4laiaaikdacaaM c8UaamyBaaaa@4023@

So, number of scarfs which can be made from y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  metres

= y 1/2 =2y. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiaadMhaaeaacaaIXaGaai4laiaaikdaaaaabaGaeyyp a0JaaGOmaiaadMhacaGGUaaaaaa@3D8F@

Question: 13

123 x 2 y138 x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaIZaGaamiEamaaCaaaleqabaGaaGOmaaaakiaadMhacqGHsisl caaIXaGaaG4maiaaiIdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaam yEaaaa@410F@  is a like term of:

a. 10xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaWG4bGaamyEaaaa@3948@

b.   15xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiaaiwdacaWG4bGaamyEaaaa@3A3A@

c. 15x y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiaaiwdacaWG4bGaamyEamaaCaaaleqabaGaaGOmaaaaaaa@3B23@

d.   10 x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEaaaa@3A3B@

Solution

(d)

We have, 123 x 2 y138 x 2 y=15 x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaIZaGaamiEamaaCaaaleqabaGaaGOmaaaakiaadMhacqGHsisl caaIXaGaaG4maiaaiIdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaam yEaiabg2da9iabgkHiTiaaigdacaaI1aGaamiEamaaCaaaleqabaGa aGOmaaaakiaadMhaaaa@476A@

Hence, it is like term of 10 x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEaaaa@3A3B@  as both contain x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiaadMhaaaa@38C6@ .

Question: 14

The value of 3 x 2 5x+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI1aGaamiEaiabgUca Riaaiodaaaa@3CCD@  when x=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaaigdaaaa@3896@  is

a.    1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3693@

b.   0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaaaa@3692@

c.    1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaaaa@3780@

d.   11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaig daaaa@374E@

Solution

(a)

Put x=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaaigdaaaa@3896@  in given equation, we get

3 x 2 5x+3 =3 (1) 2 5( 1 )+3 =35+3 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaaIZa GaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaiwdacaWG4bGa ey4kaSIaaG4maaqaaabaaaaaaaaapeGaeyypa0JaaG4maiaacIcaca aIXaGaaiyka8aadaahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0IaaGyn amaabmaapaqaa8qacaaIXaaacaGLOaGaayzkaaGaey4kaSIaaG4maa qaaiabg2da9iaaiodacqGHsislcaaI1aGaey4kaSIaaG4maaqaaiab g2da9iaaigdaaaaa@4E5B@

Question: 15

The expression for the number of diagonals that we can make from one vertex of a n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CB@  sided polygon is:

a.    2n+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaad6 gacqGHRaWkcaaIXaaaaa@3924@

b.   n2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgk HiTiaaikdaaaa@3874@

c.    5n+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaad6 gacqGHRaWkcaaIYaaaaa@3928@

d.   n3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgk HiTiaaiodaaaa@3875@

Solution

(d)

Since, vertex is formed by joining two sides. Diagonal is line segment joining the two opposite vertex. So, number of diagonals formed by one vertex =n3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam OBaiabgkHiTiaaiodaaaa@397B@

Question: 16

The length of a side of square is given as 2x+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI hacqGHRaWkcaaIZaaaaa@3930@ . Which expression represents the perimeter of the square?

a.    2x+16 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI hacqGHRaWkcaaIXaGaaGOnaaaa@39EE@

b.   6x+9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaadI hacqGHRaWkcaaI5aaaaa@393A@

c.    8x+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaadI hacqGHRaWkcaaIZaaaaa@3936@

d.   8x+12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaadI hacqGHRaWkcaaIXaGaaGOmaaaa@39F0@

Solution

(d)

Side of the square =( 2x+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacaaIYaGaamiEaiabgUcaRiaaiodaaiaawIcacaGLPaaaaaa@3BBF@           [Given]

MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaO qaaiabgsJiCbaa@3B60@  Perimeter of square =4×( Side ) =4×( 2x+3 ) =8x+12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaI0aGaey41aq7aaeWaaeaacaqGtbGaaeyAaiaabsgacaqGLbaa caGLOaGaayzkaaaabaGaeyypa0JaaGinaiabgEna0oaabmaabaGaaG OmaiaadIhacqGHRaWkcaaIZaaacaGLOaGaayzkaaaabaGaeyypa0Ja aGioaiaadIhacqGHRaWkcaaIXaGaaGOmaaaaaa@4CAE@

 

In questions 17 to 32, fill in the blanks to make the statements true.

Question: 17

Sum or difference of two like terms is ________.

Solution

Sum or difference of two like terms is a like terms.

Question: 18

In the formula, area of circle =π r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaeq iWdaNaamOCamaaCaaaleqabaGaaGOmaaaaaaa@3A7B@ , the numerical constant of the expression π r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWdaNaam OCamaaCaaaleqabaGaaGOmaaaaaaa@3975@  is ________.

Solution

In π r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWdaNaam OCamaaCaaaleqabaGaaGOmaaaaaaa@3975@ , r is variable, so the numerical constant is π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWdahaaa@3795@ .

Question: 19

3 a 2 b and 7b a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadg gadaahaaWcbeqaaiaaikdaaaGccaWGIbGaaeiiaiaabggacaqGUbGa aeizaiaabccacqGHsislcaaI3aGaamOyaiaadggadaahaaWcbeqaai aaikdaaaaaaa@41BB@  are ________ terms.

Solution

3 a 2 b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadg gadaahaaWcbeqaaiaaikdaaaGccaWGIbaaaa@3955@  and 7b a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4naiaadkgacaWGHbWaaWbaaSqabeaacaaIYaaaaaaa@3A3C@  are like terms as both have same variable factor a 2 b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGOmaaaakiaadkgaaaa@3898@ .

Question: 20

5 a 2 b and 5 b 2 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ynaiaadggadaahaaWcbeqaaiaaikdaaaGccaWGIbGaaeiiaiaabgga caqGUbGaaeizaiaabccacqGHsislcaaI1aGaamOyamaaCaaaleqaba GaaGOmaaaakiaadggaaaa@42B2@  are ________ terms.

Solution

5 a 2 b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ynaiaadggadaahaaWcbeqaaiaaikdaaaGccaWGIbaaaa@3A44@  and 5 b 2 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ynaiaadkgadaahaaWcbeqaaiaaikdaaaGccaWGHbaaaa@3A44@  are unlike terms.

Question: 21

In the expression 2πr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabec 8aWjaadkhaaaa@3948@ , the algebraic variable is ________.

Solution

r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36CF@  is algebraic variable in the expression 2πr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabec 8aWjaadkhaaaa@3948@ .

Question: 22

Number of terms in a monomial is ________.

Solution

Number of terms in a monomial is one.

Question: 23

Like terms in the expression n(n+1)+6(n1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaabI cacaWGUbGaey4kaSIaaGymaiaabMcacqGHRaWkcaaI2aGaaGPaVlaa bIcacaWGUbGaaGPaVlaaykW7cqGHsislcaaIXaGaaeykaaaa@44E7@  are ___________and________.

Solution

We have, n( n+1 )+6( n1 )= n 2 +n+6n6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaabm aabaGaamOBaiabgUcaRiaaigdaaiaawIcacaGLPaaacqGHRaWkcaaI 2aWaaeWaaeaacaWGUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaiabg2 da9iaad6gadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGUbGaey4k aSIaaGOnaiaad6gacqGHsislcaaI2aaaaa@49AD@

Hence, like terms in the expression n( n+1 )+6( n1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaabm aabaGaamOBaiabgUcaRiaaigdaaiaawIcacaGLPaaacqGHRaWkcaaI 2aWaaeWaaeaacaWGUbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaa@40AA@  are n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CB@  and 6n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaad6 gaaaa@378B@ .

Question: 24

The expression 13+90 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaio dacqGHRaWkcaaI5aGaaGimaaaa@39AF@  is a ________.

Solution

13+90=103 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaio dacqGHRaWkcaaI5aGaaGimaiabg2da9iaaigdacaaIWaGaaG4maaaa @3CE7@  is a constant term.

Question: 25

The speed of car is 55 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaaiw daaaa@3756@  km/hrs. The distance covered in y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  hours is________.

Solution

Given, speed of car =55km/hrs MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ynaiaaiwdacaaMc8UaaGPaVlaabUgacaqGTbGaae4laiaabIgacaqG YbGaae4Caaaa@40D8@

We know that, Distance = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DE@  Speed × MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey41aqlaaa@37EF@  Time

MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaO qaaiabgsJiCbaa@3B60@  Distance covered in y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  hours =55xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ynaiaaiwdacaWG4bGaamyEaaaa@3A57@  km

 

Question: 26

x+y+z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgU caRiaadMhacqGHRaWkcaWG6baaaa@3A96@  is an expression which is neither monomial nor ________.

Solution

x+y+z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgU caRiaadMhacqGHRaWkcaWG6baaaa@3A96@  contains three terms, so it is trinomial.

Hence, x+y+z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgU caRiaadMhacqGHRaWkcaWG6baaaa@3A96@  is an expression which is neither monomial nor binomial.

Question: 27

If ( x 2 y+ y 2 +3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEaiabgUcaRiaadMhadaah aaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaaacaGLOaGaayzkaaaaaa@3EC1@  is subtracted from ( 3 x 2 y+2 y 2 +5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaGaamiEamaaCaaaleqabaGaaGOmaaaakiaadMhacqGHRaWkcaaI YaGaamyEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdaaiaawI cacaGLPaaaaaa@403C@ , then coefficient of y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  in the result is ________.

Solution

We have, ( 3 x 2 y+2 y 2 +5 )( x 2 y+ y 2 +3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaGaamiEamaaCaaaleqabaGaaGOmaaaakiaadMhacqGHRaWkcaaI YaGaamyEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdaaiaawI cacaGLPaaacqGHsisldaqadaqaaiaadIhadaahaaWcbeqaaiaaikda aaGccaWG5bGaey4kaSIaamyEamaaCaaaleqabaGaaGOmaaaakiabgU caRiaaiodaaiaawIcacaGLPaaaaaa@4A12@

=3 x 2 y+2 y 2 +5 x 2 y y 2 3 =2 x 2 y+ y 2 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaIZaGaamiEamaaCaaaleqabaGaaGOmaaaakiaadMhacqGHRaWk caaIYaGaamyEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdacq GHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEaiabgkHiTiaa dMhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIZaaabaGaeyypa0 JaaGOmaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bGaey4kaSIa amyEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaikdaaaaa@5143@

MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaO qaaiabgsJiCbaa@3B60@   Coefficient of y=2 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2 da9iaaikdacaWG4bWaaWbaaSqabeaacaaIYaaaaaaa@3A7E@

Question: 28

 abc MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaae iiaiaadggacqGHsislcaWGIbGaeyOeI0Iaam4yaaaa@3BF7@  is same as  a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4eGiaabc cacaWGHbGaai4eGaaa@38CF@  ( ________ ).

Solution

abc=a( b+c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG PaVlaadggacqGHsislcaWGIbGaeyOeI0Iaam4yaiabg2da9iabgkHi TiaadggacqGHsisldaqadaqaaiaadkgacqGHRaWkcaWGJbaacaGLOa Gaayzkaaaaaa@44DF@  

So, abc MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaam yyaiabgkHiTiaadkgacqGHsislcaWGJbaaaa@3B54@  is same as a( b+c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaam yyaiabgkHiTmaabmaabaGaamOyaiabgUcaRiaadogaaiaawIcacaGL Paaaaaa@3CD2@ .

Question: 29

The unlike terms in perimeters of following figures are ________ and ________.

Fig. (i)

Fig. (ii)

Solution

In above fig. (i), Perimeter = Sum of all sides

=2x+y+2x+y=4x+2y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaadIhacqGHRaWkcaWG5bGaey4kaSIaaGOmaiaadIhacqGHRaWk caWG5bGaeyypa0JaaGinaiaadIhacqGHRaWkcaaIYaGaamyEaaaa@444D@

In above fig. (ii), Perimeter = Sum of all sides

=x+ y 2 +x+ y 2 =2x+2 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaaqaaaaa aaaaWdbiabg2da9iaadIhacqGHRaWkcaWG5bWdamaaCaaaleqabaWd biaaikdaaaGccqGHRaWkcaWG4bGaey4kaSIaamyEa8aadaahaaWcbe qaa8qacaaIYaaaaaGcpaqaa8qacqGH9aqpcaaIYaGaamiEaiabgUca RiaaikdacaWG5bWdamaaCaaaleqabaWdbiaaikdaaaaaaaa@4646@

Unlike terms in perimeters are 2y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadM haaaa@3792@  and 2 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadM hadaahaaWcbeqaaiaaikdaaaaaaa@387B@

Question: 30

On adding a monomial _____________ to  2x+4 y 2 +z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaae iiaiaaikdacaWG4bGaey4kaSIaaGinaiaadMhadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaWG6baaaa@3E93@ , the resulting expression becomes a binomial.

Solution

We can add 2x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI haaaa@3791@  to the expression to make it binomial.

Question: 31

3x+23 x 2 +6 y 2 +2x+ y 2 + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hacqGHRaWkcaaIYaGaaG4maiaadIhadaahaaWcbeqaaiaaikdaaaGc cqGHRaWkcaaI2aGaamyEamaaCaaaleqabaGaaGOmaaaakiabgUcaRi aaikdacaWG4bGaey4kaSIaamyEamaaCaaaleqabaGaaGOmaaaakiab gUcaRaaa@45C0@  ____________ =5x+7 y 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ynaiaadIhacqGHRaWkcaaI3aGaamyEamaaCaaaleqabaGaaGOmaaaa kiaac6caaaa@3CE0@

Solution

Assume

( 3x+23 x 2 +6 y 2 +2x+ y 2 )+N=5x+7 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaqadaWdaeaapeGaaG4maiaadIhacqGHRaWkcaaIYaGaaG4maiaa dIhapaWaaWbaaSqabeaapeGaaGOmaaaakiabgUcaRiaaiAdacaWG5b WdamaaCaaaleqabaWdbiaaikdaaaGccqGHRaWkcaaIYaGaamiEaiab gUcaRiaadMhapaWaaWbaaSqabeaapeGaaGOmaaaaaOGaayjkaiaawM caaiabgUcaRiaab6eacqGH9aqpcaaI1aGaamiEaiabgUcaRiaaiEda caWG5bWdamaaCaaaleqabaWdbiaaikdaaaaaaa@4F21@

N=5x+7 y 2 ( 3x+23 x 2 +6 y 2 +2x+ y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHshI3caqGobGaeyypa0JaaGynaiaadIhacqGHRaWkcaaI3aGa amyEa8aadaahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0YaaeWaa8aaba WdbiaaiodacaWG4bGaey4kaSIaaGOmaiaaiodacaWG4bWdamaaCaaa leqabaWdbiaaikdaaaGccqGHRaWkcaaI2aGaamyEa8aadaahaaWcbe qaa8qacaaIYaaaaOGaey4kaSIaaGOmaiaadIhacqGHRaWkcaWG5bWd amaaCaaaleqabaWdbiaaikdaaaaakiaawIcacaGLPaaaaaa@5193@

N=5x+7 y 2 3x23 x 2 6 y 2 2x y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHshI3caqGobGaeyypa0JaaGynaiaadIhacqGHRaWkcaaI3aGa amyEa8aadaahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0IaaG4maiaadI hacqGHsislcaaIYaGaaG4maiaadIhapaWaaWbaaSqabeaapeGaaGOm aaaakiabgkHiTiaaiAdacaWG5bWdamaaCaaaleqabaWdbiaaikdaaa GccqGHsislcaaIYaGaamiEaiabgkHiTiaadMhapaWaaWbaaSqabeaa peGaaGOmaaaaaaa@500D@

N=23 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGHshI3caqGobGaeyypa0JaeyOeI0IaaGOmaiaaiodacaWG4bWd amaaCaaaleqabaWdbiaaikdaaaaaaa@3E97@

Question: 32

If Rohit has 5xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaadI hacaWG5baaaa@3892@  toffees and Shantanu has 20xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaic dacaWG4bGaamyEaaaa@3949@  toffees, then Shantanu has ___________ more toffees.

Solution

We have, Rohit’s toffees =5xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ynaiaadIhacaWG5baaaa@3998@

Shantanu’s toffees =20xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaaicdacaWG4bGaamyEaaaa@3A4F@

Difference:

=20xy5xy =15xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIYaGaaGimaiaadIhacaWG5bGaeyOeI0IaaGynaiaadIhacaWG 5baabaGaeyypa0JaaGymaiaaiwdacaWG4bGaamyEaaaaaa@4278@

Hence, Shantanu had 15xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiw dacaWG4bGaamyEaaaa@394D@  more toffees.

--

In questions 33 to 52, state whether the statements given are True or False.

Question: 33

1+ x 2 + x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgU caRmaalaaabaGaamiEaaqaaiaaikdaaaGaey4kaSIaamiEamaaCaaa leqabaGaaG4maaaaaaa@3C07@  is a polynomial.

Solution

True

Expression with three or more than three terms is called a polynomial.

Question: 34

( 3ab+3 )( ab ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaGaamyyaiabgkHiTiaadkgacqGHRaWkcaaIZaaacaGLOaGaayzk aaGaeyOeI0YaaeWaaeaacaWGHbGaaGPaVlaabUcacaqGGaGaamOyaa GaayjkaiaawMcaaaaa@4396@  is a binomial.

Solution

False

We have, ( 3ab+3 )( ab ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaGaamyyaiabgkHiTiaadkgacqGHRaWkcaaIZaaacaGLOaGaayzk aaGaeyOeI0YaaeWaaeaacaWGHbGaaGPaVlaabUcacaqGGaGaamOyaa GaayjkaiaawMcaaaaa@4396@

=3ab+3ab MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4maiaadggacqGHsislcaWGIbGaey4kaSIaaG4maiabgkHiTiaadgga cqGHsislcaWGIbaaaa@3F9B@

=3aabb+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4maiaadggacqGHsislcaWGHbGaeyOeI0IaamOyaiabgkHiTiaadkga cqGHRaWkcaaIZaaaaa@3F9B@

=2a2b+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaadggacqGHsislcaaIYaGaamOyaiabgUcaRiaaiodaaaa@3CAF@

The expression has three terms, it is a trinomial.

Question: 35

A trinomial can be a polynomial.

Solution

True

Trinomial is a polynomial.

Question: 36

A polynomial with more than two terms is a trinomial.

Solution

False

A trinomial have exact three terms.

Question: 37

Sum of x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@  and y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  is x+y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgU caRiaadMhaaaa@38B6@ .

Solution

True

Sum of xandyisx+y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaayk W7caaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlaaykW7caWG5bGaaGPa VlaaykW7caqGPbGaae4CaiaaykW7caaMc8UaamiEaiabgUcaRiaadM haaaa@4BA6@

Question: 38

Sum of 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@3695@  and p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36CE@  is 2p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadc haaaa@378A@ .

Solution

False

Sum of 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@3694@  and p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36CD@  is 2+p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgU caRiaadchaaaa@386B@

Question: 39

A binomial has more than two terms.

Solution

False

Binomial has exactly two terms.

Question: 40

A trinomial has exactly three terms.

Solution

True

A trinomial has exactly three unlike terms.

Question: 41

In like terms, variables and their powers are the same.

Solution

True

In like terms, variables and their powers are the same.

Question: 42

The expression x+y+5x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgU caRiaadMhacqGHRaWkcaaI1aGaamiEaaaa@3B53@  is a trinomial.

Solution

False

x+y+5x=6x+y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWLaam iEaiabgUcaRiaadMhacqGHRaWkcaaI1aGaamiEaiabg2da9iaaiAda caWG4bGaey4kaSIaamyEaaaa@4134@  it is a binomial.

Question: 43

4p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadc haaaa@378B@  is the numerical coefficient of q 2  in4p q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaCa aaleqabaGaaGOmaaaakiaabccacaqGPbGaaeOBaiabgkHiTiaaisda caWGWbGaamyCamaaCaaaleqabaGaaGOmaaaaaaa@3EC0@ .

Solution

False

Numerical coefficient of q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaCa aaleqabaGaaGOmaaaaaaa@37B7@  in 4p q 2 =4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG inaiaadchacaWGXbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaeyOe I0IaaGinaaaa@3D12@ .

Question: 44

5a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaadg gaaaa@377D@  and 5b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaadk gaaaa@377E@  are unlike terms.

Solution

True

Both the terms have different algebraic factors.

Question: 45

Sum of x 2 +x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaadIhaaaa@39A7@  and y+ y 2  is 2 x 2 +2 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgU caRiaadMhadaahaaWcbeqaaiaaikdaaaGccaqGGaGaaeyAaiaaboha caqGGaGaaGOmaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkca aIYaGaamyEamaaCaaaleqabaGaaGOmaaaaaaa@4302@ .

Solution

False

Sum =( x 2 +x )+( y+ y 2 )= x 2 +x+y+ y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamiEaaGa ayjkaiaawMcaaiabgUcaRmaabmaabaGaamyEaiabgUcaRiaadMhada ahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaacqGH9aqpcaWG4bWa aWbaaSqabeaacaaIYaaaaOGaey4kaSIaamiEaiabgUcaRiaadMhacq GHRaWkcaWG5bWaaWbaaSqabeaacaaIYaaaaaaa@4BF0@

= x 2 + y 2 +x+y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadMhadaahaaWcbeqa aiaaikdaaaGccqGHRaWkcaWG4bGaey4kaSIaamyEaaaa@3F60@

Question: 46

Subtracting a term from a given expression is the same as adding its additive inverse to the given expression.

Solution

True

Additive inverse is the negation of a number or expression.

Question: 47

The total number of planets of Sun can be denoted by the variable n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CB@ .

Solution

False

As, Sun has infinite planets around it.

Question: 48

In like terms, the numerical coefficients should also be the same.

Solution

False

e.g. 3 x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4maiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5baaaa@3A70@  and 4 x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadI hadaahaaWcbeqaaiaaikdaaaGccaWG5baaaa@3984@  are like terms as they have same algebraic factor x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiaadMhaaaa@38C6@  but have different numerical coefficients.

Question: 49

If we add a monomial and binomial, then answer can never be a monomial.

Solution

False

If we add a monomial and a binomial, then answer can be a monomial, e.g.

Add x 2 & x 2 + y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiaacAcacqGHsislcaWG4bWaaWbaaSqabeaa caaIYaaaaOGaey4kaSIaamyEamaaCaaaleqabaGaaGOmaaaaaaa@3E18@

= x 2 +( x 2 + y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iEamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaabmaabaGaeyOeI0Ia amiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadMhadaahaaWcbe qaaiaaikdaaaaakiaawIcacaGLPaaaaaa@40E9@

= x 2 x 2 + y 2 = y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamiEamaaCaaa leqabaGaaGOmaaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaikdaaa aakeaacqGH9aqpcaWG5bWaaWbaaSqabeaacaaIYaaaaaaaaa@4171@

The answer is monomial.

Question: 50

If we subtract a monomial from a binomial, then answer is at least a binomial.

Solution

False

If we subtract a monomial from a binomial, then answer is at least a monomial.

Question: 51

When we subtract a monomial from a trinomial, then answer can be a polynomial.

Solution

True

When we subtract a monomial from a trinomial, then answer can be binomial or polynomial.

Question: 52

When we add a monomial and a trinomial, then answer can be a monomial.

Solution

False

When we add a monomial and a trinomial, then it can be binomial or trinomial.

Question: 53

Write the following statements in the form of algebraic expression and write whether it is monomial, binomial or trinomial.

a.    x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  is multiplied by itself and then added to the product of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  and y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D7@ .

b.   Three times of 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaaaa@3698@  and two times of q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaaaa@36CF@  are multiplied and then subtracted from r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36D0@ .

c.    Product of p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36CE@ , twice of q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaaaa@36CF@  and thrice of r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@36D0@ .

d.   Sum of the products of a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36BF@  and b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36C0@ , b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36C0@  and c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36C1@  and c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36C1@  and a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36BF@ .

e.    Perimeter of an equilateral triangle of
side x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@ .

f.Perimeter of a rectangle with length p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36CE@  and breadth q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaaaa@36CF@ .

g.   Area of a triangle with base m and
height n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36CC@ .

h.   Area of a square with side x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@ .

i.  Cube of s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@36D1@  subtracted from cube of t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaaaa@36D2@ .

j.  Quotient of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  and 15 multiplied by x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@ .

k.   The sum of square of x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  and cube of z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36D8@ .

l.  Two times q subtracted from cube of q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaaaa@36CF@ .

Solution

a.    x 2 +xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaadIhacaWG5baaaa@3AA5@

b.   r( 3p×2q ) =r6pq MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaWGYb GaeyOeI0YaaeWaaeaacaaIZaGaamiCaiabgEna0kaaikdacaWGXbaa caGLOaGaayzkaaaabaGaeyypa0JaamOCaiabgkHiTiaaiAdacaWGWb GaamyCaaaaaa@445B@

c.    p×2q×3r =6pqr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaWGWb Gaey41aqRaaGOmaiaadghacqGHxdaTcaaIZaGaamOCaaqaaiabg2da 9iaaiAdacaWGWbGaamyCaiaadkhaaaaa@430F@

d.   ab+bc+ca MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadk gacqGHRaWkcaWGIbGaam4yaiabgUcaRiaadogacaWGHbaaaa@3D06@

e.    3x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI haaaa@3792@

f. 2( p+q ) =2p+2q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaaIYa WaaeWaaeaacaWGWbGaey4kaSIaamyCaaGaayjkaiaawMcaaaqaaiab g2da9iaaikdacaWGWbGaey4kaSIaaGOmaiaadghaaaaa@403B@

g.   1 2 mn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOmaaaacaWGTbGaamOBaaaa@3944@

h.   x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaaaaa@37BE@

i.    t 3 s 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaCa aaleqabaGaaG4maaaakiabgkHiTiaadohadaahaaWcbeqaaiaaioda aaaaaa@3A94@

j.   x 15 ×x or x 3 15 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaadIhaaeaacaaIXaGaaGynaaaacqGHxdaTcaWG4bGaaGPaVlaa ykW7aeaacaqGVbGaaeOCaiaaykW7daWcaaqaaiaadIhadaahaaWcbe qaaiaaiodaaaaakeaacaaIXaGaaGynaaaaaaaa@457C@

k.   x 2 + z 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaadQhadaahaaWcbeqaaiaaioda aaaaaa@3A93@

l.   q 3 2q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCamaaCa aaleqabaGaaG4maaaakiabgkHiTiaaikdacaWGXbaaaa@3A61@

Question: 54

Write the coefficient of x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaaaaa@37BF@  in the following:

(i)             x 2 x+4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiabgkHiTiaadIhacqGHRaWkcaaI0aaaaa@3B52@

(ii)         x 3 2 x 2 +3x+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgkHiTiaaikdacaWG4bWaaWbaaSqabeaa caaIYaaaaOGaey4kaSIaaG4maiaadIhacqGHRaWkcaaIXaaaaa@3F9B@

(iii)      1+2x+3 x 2 +4 x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgU caRiaaikdacaWG4bGaey4kaSIaaG4maiaadIhadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaI0aGaamiEamaaCaaaleqabaGaaG4maaaaaa a@4044@

(iv)       y+ y 2 x+ y 3 x 2 + y 4 x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabgU caRiaadMhadaahaaWcbeqaaiaaikdaaaGccaWG4bGaey4kaSIaamyE amaaCaaaleqabaGaaG4maaaakiaadIhadaahaaWcbeqaaiaaikdaaa GccqGHRaWkcaWG5bWaaWbaaSqabeaacaaI0aaaaOGaamiEamaaCaaa leqabaGaaG4maaaaaaa@4426@      

Solution

(i)             1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3693@

(ii)         2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG Omaaaa@3781@

(iii)      3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3695@

(iv)       y 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa aaleqabaGaaG4maaaaaaa@37C0@  

Question: 55

Find the numerical coefficient of each of the terms:

(i)             x 3 y 3 z,x y 2 z 2 ,3x y 2 z 3 ,5 x 3 y 3 z,7 x 2 y 2 z 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiaadMhadaahaaWcbeqaaiaaiodaaaGccaWG 6bGaaiilaiaaykW7caaMc8UaamiEaiaadMhadaahaaWcbeqaaiaaik daaaGccaWG6bWaaWbaaSqabeaacaaIYaaaaOGaaiilaiabgkHiTiaa iodacaWG4bGaamyEamaaCaaaleqabaGaaGOmaaaakiaadQhadaahaa WcbeqaaiaaiodaaaGccaGGSaGaaGPaVlaaiwdacaWG4bWaaWbaaSqa beaacaaIZaaaaOGaamyEamaaCaaaleqabaGaaG4maaaakiaadQhaca GGSaGaeyOeI0IaaG4naiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG 5bWaaWbaaSqabeaacaaIYaaaaOGaamOEamaaCaaaleqabaGaaGOmaa aaaaa@5A9F@

(ii)         10xyz,7x y 2 z,9xyz,2x y 2 z,2 x 2 y 2 z 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaWG4bGaamyEaiaadQhacaGGSaGaeyOeI0IaaG4naiaadIhacaWG 5bWaaWbaaSqabeaacaaIYaaaaOGaamOEaiaacYcacqGHsislcaaI5a GaamiEaiaadMhacaWG6bGaaiilaiaaikdacaWG4bGaamyEamaaCaaa leqabaGaaGOmaaaakiaadQhacaGGSaGaaGOmaiaadIhadaahaaWcbe qaaiaaikdaaaGccaWG5bWaaWbaaSqabeaacqGHYaGmaaGccaWG6bWa aWbaaSqabeaacaaIYaaaaaaa@52AF@

Solution

(i)             Coefficient of:

x 3 y 2 z=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiaadMhadaahaaWcbeqaaiaaikdaaaGccaWG 6bGaeyypa0JaaGymaaaa@3C7A@

(ii)         Coefficient of:

10xyz=10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaaMc8UaamiEaiaadMhacaWG6bGaeyypa0JaaGymaiaaicdaaaa@3E4D@

7x y 2 z=7 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4naiaadIhacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaamOEaiabg2da 9iabgkHiTiaaiEdaaaa@3E27@

9xyz=9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG yoaiaadIhacaWG5bGaamOEaiabg2da9iabgkHiTiaaiMdaaaa@3D38@

2x y 2 z=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI hacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaamOEaiabg2da9iaaikda aaa@3C43@

2 x 2 y 2 z 2 =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI hadaahaaWcbeqaaiaaikdaaaGccaWG5bWaaWbaaSqabeaacaaIYaaa aOGaamOEamaaCaaaleqabaGaaGOmaaaakiabg2da9iaaikdaaaa@3E29@

Question: 56

Simplify the following by combining the like terms and then write whether the expression is a monomial, a binomial or a trinomial.

a.    3 x 2 y z 2 3x y 2 z+ x 2 y z 2 +7x y 2 z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hadaahaaWcbeqaaiaaikdaaaGccaWG5bGaamOEamaaCaaaleqabaGa aGOmaaaakiabgkHiTiaaiodacaWG4bGaamyEamaaCaaaleqabaGaaG OmaaaakiaadQhacqGHRaWkcaWG4bWaaWbaaSqabeaacaaIYaaaaOGa amyEaiaadQhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI3aGaam iEaiaadMhadaahaaWcbeqaaiaaikdaaaGccaWG6baaaa@4C5E@

b.   x 4 +3 x 3 y+3 x 2 y 2 3 x 3 y3x y 3   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGinaaaakiabgUcaRiaaiodacaWG4bWaaWbaaSqabeaa caaIZaaaaOGaamyEaiabgUcaRiaaiodacaWG4bWaaWbaaSqabeaaca aIYaaaaOGaamyEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaioda caWG4bWaaWbaaSqabeaacaaIZaaaaOGaamyEaiabgkHiTiaaiodaca WG4bGaamyEamaaCaaaleqabaGaaG4maaaakiaabccaaaa@4BAD@

        +  y 4 3 x 2 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaae iiaiaadMhadaahaaWcbeqaaiaaisdaaaGccqGHsislcaaIZaGaamiE amaaCaaaleqabaGaaGOmaaaakiaadMhadaahaaWcbeqaaiaaikdaaa aaaa@3ED1@

c.    p 3 q 2 r+p q 2 r 2 +3 p 2 q r 2 9 p 2 q r 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaG4maaaakiaadghadaahaaWcbeqaaiaaikdaaaGccaWG YbGaey4kaSIaamiCaiaadghadaahaaWcbeqaaiaaikdaaaGccaWGYb WaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG4maiaadchadaahaaWc beqaaiaaikdaaaGccaWGXbGaamOCamaaCaaaleqabaGaaGOmaaaaki abgkHiTiaaiMdacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaamyCaiaa dkhadaahaaWcbeqaaiaaikdaaaaaaa@4D21@

d.   2a+2b+2c2a2b2c2b  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadg gacqGHRaWkcaaIYaGaamOyaiabgUcaRiaaikdacaWGJbGaeyOeI0Ia aGOmaiaadggacqGHsislcaaIYaGaamOyaiabgkHiTiaaikdacaWGJb GaeyOeI0IaaGOmaiaadkgacaqGGaaaaa@4768@

        + 2c+2a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaae iiaiaaikdacaWGJbGaey4kaSIaaGOmaiaadggaaaa@3B85@

e.    50 x 3 21x+107+41 x 3 x+1  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaaic dacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGOmaiaaigda caWG4bGaey4kaSIaaGymaiaaicdacaaI3aGaey4kaSIaaGinaiaaig dacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamiEaiabgUca RiaaigdacaqGGaaaaa@4831@

       

Solution

a.    3 x 2 y z 2 3x y 2 z+ x 2 y z 2 +7x y 2 z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hadaahaaWcbeqaaiaaikdaaaGccaWG5bGaamOEamaaCaaaleqabaGa aGOmaaaakiabgkHiTiaaiodacaWG4bGaamyEamaaCaaaleqabaGaaG OmaaaakiaadQhacqGHRaWkcaWG4bWaaWbaaSqabeaacaaIYaaaaOGa amyEaiaadQhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI3aGaam iEaiaadMhadaahaaWcbeqaaiaaikdaaaGccaWG6baaaa@4C5E@

By combining the like terms

          =3 x 2 y z 2 + x 2 y z 2 3x y 2 z+7x y 2 z =4 x 2 y z 2 +4x y 2 z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIZaGaamiEamaaCaaaleqabaGaaGOmaaaakiaadMhacaWG6bWa aWbaaSqabeaacaaIYaaaaOGaey4kaSIaamiEamaaCaaaleqabaGaaG OmaaaakiaadMhacaWG6bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Ia aG4maiaadIhacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaamOEaiabgU caRiaaiEdacaWG4bGaamyEamaaCaaaleqabaGaaGOmaaaakiaadQha aeaacqGH9aqpcaaI0aGaamiEamaaCaaaleqabaGaaGOmaaaakiaadM hacaWG6bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGinaiaadIha caWG5bWaaWbaaSqabeaacaaIYaaaaOGaamOEaaaaaa@599C@

The expression contains 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3695@  terms.
So, it is trinomial.

b.   x 4 +3 x 3 y+3 x 2 y 2 3 x 3 y3x y 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGinaaaakiabgUcaRiaaiodacaWG4bWaaWbaaSqabeaa caaIZaaaaOGaamyEaiabgUcaRiaaiodacaWG4bWaaWbaaSqabeaaca aIYaaaaOGaamyEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaioda caWG4bWaaWbaaSqabeaacaaIZaaaaOGaamyEaiabgkHiTiaaiodaca WG4bGaamyEamaaCaaaleqabaGaaG4maaaaaaa@4B00@

          + y 4 3 x 2 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaam yEamaaCaaaleqabaGaaGinaaaakiabgkHiTiaaiodacaWG4bWaaWba aSqabeaacaaIYaaaaOGaamyEamaaCaaaleqabaGaaGOmaaaaaaa@3E2E@

         By combining the like terms

          = x 4 +3 x 3 y3 x 3 y+3 x 2 y 2 3 x 2 y 2 3 x 3 y+ y 3 = x 4 0+03 x 3 y+ y 4 = x 4 + y 4 3 x 3 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWG4bWaaWbaaSqabeaacaaI0aaaaOGaey4kaSIaaG4maiaadIha daahaaWcbeqaaiaaiodaaaGccaWG5bGaeyOeI0IaaG4maiaadIhada ahaaWcbeqaaiaaiodaaaGccaWG5bGaey4kaSIaaG4maiaadIhadaah aaWcbeqaaiaaikdaaaGccaWG5bWaaWbaaSqabeaacaaIYaaaaaGcba GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHsislcaaIZaGaamiE amaaCaaaleqabaGaaGOmaaaakiaadMhadaahaaWcbeqaaiaaikdaaa GccqGHsislcaaIZaGaamiEamaaCaaaleqabaGaaG4maaaakiaadMha cqGHRaWkcaWG5bWaaWbaaSqabeaacaaIZaaaaaGcbaGaeyypa0Jaam iEamaaCaaaleqabaGaaGinaaaakiabgkHiTiaaicdacqGHRaWkcaaI WaGaeyOeI0IaaG4maiaadIhadaahaaWcbeqaaiaaiodaaaGccaWG5b Gaey4kaSIaamyEamaaCaaaleqabaGaaGinaaaaaOqaaiabg2da9iaa dIhadaahaaWcbeqaaiaaisdaaaGccqGHRaWkcaWG5bWaaWbaaSqabe aacaaI0aaaaOGaeyOeI0IaaG4maiaadIhadaahaaWcbeqaaiaaioda aaGccaWG5baaaaa@743E@

The expression contains 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3695@  terms.
So, it is trinomial.

c.    p 3 q 2 r+p q 2 r 3 +3 p 2 q r 2 9 p 2 q r 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaG4maaaakiaadghadaahaaWcbeqaaiaaikdaaaGccaWG YbGaey4kaSIaamiCaiaadghadaahaaWcbeqaaiaaikdaaaGccaWGYb WaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaG4maiaadchadaahaaWc beqaaiaaikdaaaGccaWGXbGaamOCamaaCaaaleqabaGaaGOmaaaaki abgkHiTiaaiMdacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaamyCaiaa dkhadaahaaWcbeqaaiaaiodaaaaaaa@4D22@

By combining the like terms

= p 3 q 2 r+p q 2 r 3 +3 p 2 q r 2 9 p 2 q r 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWGWbWaaWbaaSqabeaacaaIZaaaaOGaamyCamaaCaaaleqabaGa aGOmaaaakiaadkhacqGHRaWkcaWGWbGaamyCamaaCaaaleqabaGaaG OmaaaakiaadkhadaahaaWcbeqaaiaaiodaaaGccqGHRaWkcaaIZaGa amiCamaaCaaaleqabaGaaGOmaaaakiaadghacaWGYbWaaWbaaSqabe aacaaIYaaaaaGcbaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGH sislcaaI5aGaamiCamaaCaaaleqabaGaaGOmaaaakiaadghacaWGYb WaaWbaaSqabeaacaaIYaaaaaaaaa@55E6@

= p 3 q 2 r+p q 2 r 3 6 p 2 q r 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iCamaaCaaaleqabaGaaG4maaaakiaadghadaahaaWcbeqaaiaaikda aaGccaWGYbGaey4kaSIaamiCaiaadghadaahaaWcbeqaaiaaikdaaa GccaWGYbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGOnaiaadcha daahaaWcbeqaaiaaikdaaaGccaWGXbGaamOCamaaCaaaleqabaGaaG Omaaaaaaa@47BE@

The expression contains terms.
So, it is trinomial.

d.   2a+2b+2c2a2b2c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadg gacqGHRaWkcaaIYaGaamOyaiabgUcaRiaaikdacaWGJbGaeyOeI0Ia aGOmaiaadggacqGHsislcaaIYaGaamOyaiabgkHiTiaaikdacaWGJb aaaa@4436@

2b+2c+2a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiaadkgacqGHRaWkcaaIYaGaam4yaiabgUcaRiaaikdacaWGHbaa aa@3D73@

By combining the like terms

=2a2a+2a+2b2b2b +2c2c+2c =2a2b+2c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIYaGaamyyaiabgkHiTiaaikdacaWGHbGaey4kaSIaaGOmaiaa dggacqGHRaWkcaaIYaGaamOyaiabgkHiTiaaikdacaWGIbGaeyOeI0 IaaGOmaiaadkgaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlab gUcaRiaaikdacaWGJbGaeyOeI0IaaGOmaiaadogacqGHRaWkcaaIYa Gaam4yaaqaaiabg2da9iaaikdacaWGHbGaeyOeI0IaaGOmaiaadkga cqGHRaWkcaaIYaGaam4yaaaaaa@5C53@

The expression contains 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3696@  terms.
So, it is trinomial.

e.    50 x 3 21x+107+41 x 3 x+193 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaaic dacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGOmaiaaigda caWG4bGaey4kaSIaaGymaiaaicdacaaI3aGaey4kaSIaaGinaiaaig dacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamiEaiabgUca RiaaigdacqGHsislcaaI5aGaaG4maaaa@49FC@

+71x31 x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaaG PaVlaaiEdacaaIXaGaamiEaiabgkHiTiaaiodacaaIXaGaamiEamaa CaaaleqabaGaaG4maaaaaaa@3F0B@

By combining the like terms

=50 x 3 +41 x 3 31 x 3 21xx +71x+107+193 =60 x 3 +49x+15 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaI1aGaaGimaiaadIhadaahaaWcbeqaaiaaiodaaaGccqGHRaWk caaI0aGaaGymaiaadIhadaahaaWcbeqaaiaaiodaaaGccqGHsislca aIZaGaaGymaiaadIhadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaI YaGaaGymaiaadIhacqGHsislcaWG4baabaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7cqGHRaWkcaaMc8UaaG4naiaaigdacaWG4bGaey4k aSIaaGymaiaaicdacaaI3aGaey4kaSIaaGymaiabgkHiTiaaiMdaca aIZaaabaGaeyypa0JaaGOnaiaaicdacaWG4bWaaWbaaSqabeaacaaI ZaaaaOGaey4kaSIaaGinaiaaiMdacaWG4bGaey4kaSIaaGymaiaaiw daaaaa@662A@

The expression contains 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3696@  terms.
So, it is trinomial.

Question: 57

Add the following expressions:

a.    p 2 7pq q 2  and3 p 2 2pq+7 q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaGOmaaaakiadycOHsislcaaI3aGaamiCaiaadghacqGH sislcaWGXbWaaWbaaSqabeaacaaIYaaaaOGaaeiiaiaabggacaqGUb GaaeizaiabgkHiTiaaiodacaWGWbWaaWbaaSqabeaacaaIYaaaaOGa eyOeI0IaaGOmaiaadchacaWGXbGaey4kaSIaaG4naiaadghadaahaa Wcbeqaaiaaikdaaaaaaa@4D56@

b.   x 3 x 2 yx y 2 y 3  and  x 3 2 x 2 y+ 3x y 2 +4y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgkHiTiaadIhadaahaaWcbeqaaiaaikda aaGccaWG5bGaeyOeI0IaamiEaiaadMhadaahaaWcbeqaaiaaikdaaa GccqGHsislcaWG5bWaaWbaaSqabeaacaaIZaaaaOGaaeiiaiGacgga caGGUbGaaiizaiaabccacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaey OeI0IaaGOmaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bGaey4k aSIaaeiiaiaaiodacaWG4bGaamyEamaaCaaaleqabaGaaGOmaaaaki abgUcaRiaaisdacaWG5baaaa@54BB@

c.    ab+bc+ca andbccaab MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadk gacqGHRaWkcaWGIbGaam4yaiabgUcaRiaadogacaWGHbGaaeiiaiGa cggacaGGUbGaaiizaiaaykW7caaMc8UaeyOeI0IaamOyaiaadogacq GHsislcaWGJbGaamyyaiabgkHiTiaadggacaWGIbaaaa@4BB1@

d.   p 2 q+r,  q 2 r+p and  r 2 p+q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaiGjGikdaaaGccqGHsislcaaMc8UaaGPaVlaadghacqGH RaWkcaWGYbGaaiilaiaabccacaWGXbWaaWbaaSqabeaacaaIYaaaaO GaeyOeI0IaamOCaiabgUcaRiaadchacaqGGaGaciyyaiaac6gacaGG KbGaaeiiaiaadkhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGWb Gaey4kaSIaamyCaaaa@5054@

e.    x 3 y 2 + x 2 y 3 +3 y 4  and  x 4 +3 x 2 y 3 +4 y 4   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiaadMhadaahaaWcbeqaaiaaikdaaaGccqGH RaWkcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEamaaCaaaleqaba GaaG4maaaakiabgUcaRiaaiodacaWG5bWaaWbaaSqabeaacaaI0aaa aOGaaeiiaiGacggacaGGUbGaaiizaiaabccacaWG4bWaaWbaaSqabe aacaaI0aaaaOGaey4kaSIaaG4maiaadIhadaahaaWcbeqaaiaaikda aaGccaWG5bWaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaGinaiaadM hadaahaaWcbeqaaiaaisdaaaGccaqGGaaaaa@51C0@

f. p 2 qr+p q 2 r+pq r 2  and 3p q 2 r2pq r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaGOmaaaakiaadghacaWGYbGaey4kaSIaamiCaiaadgha daahaaWcbeqaaiacyciIYaaaaOGaamOCaiabgUcaRiaadchacaWGXb GaamOCamaaCaaaleqabaGaiGjGikdaaaGccaqGGaGaciyyaiaac6ga caGGKbGaaGPaVlabgkHiTiaabccacaaIZaGaamiCaiaadghadaahaa WcbeqaaiaaikdaaaGccaWGYbGaeyOeI0IaaGOmaiaadchacaWGXbGa amOCamaaCaaaleqabaGaaGOmaaaaaaa@55E0@

g.   uvvw, vwwu and wuuv MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiaadA hacqGHsislcaWG2bGaam4DaiaacYcacaqGGaGaamODaiaadEhacqGH sislcaWG3bGaamyDaiaabccaciGGHbGaaiOBaiaacsgacaqGGaGaam 4DaiaadwhacqGHsislcaWG1bGaamODaaaa@49BD@

h.   a 2 +3abbc,  b 2 +3bcca and MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaaiodacaWGHbGaamOyaiabgkHi TiaadkgacaWGJbGaaiilaiaabccacaWGIbWaaWbaaSqabeaacaaIYa aaaOGaey4kaSIaaG4maiaadkgacaWGJbGaeyOeI0Iaam4yaiaadgga caqGGaGaaeyyaiaab6gacaqGKbaaaa@4A8E@   c 2 +3caab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaaiodacaWGJbGaamyyaiabgkHi TiaadggacaWGIbaaaa@3DDB@

i.   5 8 p 4 +2 p 2 + 5 8 ; 1 8 17p+ 9 8 p 2 and MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaaGioaaaacaWGWbWaaWbaaSqabeaacaaI0aaaaOGaey4k aSIaaGOmaiaadchadaahaaWcbeqaaiaaikdaaaGccqGHRaWkdaWcaa qaaiaaiwdaaeaacaaI4aaaaiaacUdadaWcaaqaaiaaigdaaeaacaaI 4aaaaiabgkHiTiaaigdacaaI3aGaamiCaiabgUcaRmaalaaabaGaaG yoaaqaaiaaiIdaaaGaamiCamaaCaaaleqabaGaaGOmaaaakiaaykW7 ciGGHbGaaiOBaiaacsgaaaa@4DA1@   p 5 p 3 +7a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaGynaaaakiabgkHiTiaadchadaahaaWcbeqaaiaaioda aaGccqGHRaWkcaaI3aGaamyyaaaa@3D22@

j.        t t 2 t 3 14; 15 t 3 +13+9t8 t 2 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk HiTiaadshadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG0bWaaWba aSqabeaacaaIZaaaaOGaeyOeI0IaaGymaiaaisdacaGG7aGaaeiiai aaigdacaaI1aGaamiDamaaCaaaleqabaGaaG4maaaakiabgUcaRiaa igdacaaIZaGaey4kaSIaaGyoaiaadshacqGHsislcaaI4aGaamiDam aaCaaaleqabaGaaGOmaaaakiaacUdacaaMc8oaaa@4E90@

Solution

a.    p 2 7pq q 2 and MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaGOmaaaakiabgkHiTiaaiEdacaWGWbGaamyCaiabgkHi TiaadghadaahaaWcbeqaaiaaikdaaaGccaaMc8Uaaeyyaiaab6gaca qGKbGaaGPaVdaa@4402@

3 p 2 2pq+7 q 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4maiaadchadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIYaGaamiC aiaadghacqGHRaWkcaaI3aGaamyCamaaCaaaleqabaGaaGOmaaaaaa a@4081@

= p 2 7pq q 2 3 p 2 2pq+7 q 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iCamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaiEdacaWGWbGaamyC aiabgkHiTiaadghadaahaaWcbeqaaiaaikdaaaGccaaMc8UaaGPaVl abgkHiTiaaiodacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Ia aGOmaiaadchacaWGXbGaey4kaSIaaG4naiaadghadaahaaWcbeqaai aaikdaaaaaaa@4CF4@

By combining the like terms, we get

  = p 2 3 p 2 7pq2pq q 2 +7 q 2 =2 p 2 9pq+6 q 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWGWbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaG4maiaadcha daahaaWcbeqaaiaaikdaaaGccqGHsislcaaI3aGaamiCaiaadghacq GHsislcaaIYaGaamiCaiaadghacqGHsislcaWGXbWaaWbaaSqabeaa caaIYaaaaOGaey4kaSIaaG4naiaadghadaahaaWcbeqaaiaaikdaaa aakeaacqGH9aqpcqGHsislcaaIYaGaamiCamaaCaaaleqabaGaaGOm aaaakiabgkHiTiaaiMdacaWGWbGaamyCaiabgUcaRiaaiAdacaWGXb WaaWbaaSqabeaacaaIYaaaaaaaaa@55A1@

b.   x 3 x 2 yx y 2 y 3 and MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgkHiTiaadIhadaahaaWcbeqaaiaaikda aaGccaWG5bGaeyOeI0IaamiEaiaadMhadaahaaWcbeqaaiaaikdaaa GccqGHsislcaWG5bWaaWbaaSqabeaacaaIZaaaaOGaaGPaVlaaykW7 caqGHbGaaeOBaiaabsgacaaMc8oaaa@49BB@

  x 3 2 x 2 y+3x y 2 +4y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgkHiTiaaikdacaWG4bWaaWbaaSqabeaa caaIYaaaaOGaamyEaiabgUcaRiaaiodacaWG4bGaamyEamaaCaaale qabaGaaGOmaaaakiabgUcaRiaaisdacaWG5baaaa@438C@

By combining the like terms, we get

  = x 3 + x 3 x 2 y2 x 2 yx y 2 +3x y 2 y 3 +4y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iEamaaCaaaleqabaGaaG4maaaakiabgUcaRiaadIhadaahaaWcbeqa aiaaiodaaaGccqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaam yEaiabgkHiTiaaikdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyE aiabgkHiTiaadIhacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaey4kaS IaaG4maiaadIhacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Ia amyEamaaCaaaleqabaGaaG4maaaakiabgUcaRiaaisdacaWG5baaaa@51FA@

  =2 x 3 3 x 2 y+2x y 2 y 3 +4y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaadIhadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaIZaGaamiE amaaCaaaleqabaGaaGOmaaaakiaadMhacqGHRaWkcaaIYaGaamiEai aadMhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG5bWaaWbaaSqa beaacaaIZaaaaOGaey4kaSIaaGinaiaadMhaaaa@482D@

c.    ab+bc+caandbccaab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadk gacqGHRaWkcaWGIbGaam4yaiabgUcaRiaadogacaWGHbGaaGPaVlaa bggacaqGUbGaaeizaiaaykW7cqGHsislcaWGIbGaam4yaiabgkHiTi aadogacaWGHbGaeyOeI0Iaamyyaiaadkgaaaa@4B0A@

  =ab+ac+bcbccaab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam yyaiaadkgacqGHRaWkcaWGHbGaam4yaiabgUcaRiaadkgacaWGJbGa eyOeI0IaamOyaiaadogacqGHsislcaWGJbGaamyyaiabgkHiTiaadg gacaWGIbaaaa@463E@

By combining the like terms,

  =ababbc+bc+caca =0+0+0=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWGHbGaamOyaiabgkHiTiaadggacaWGIbGaeyOeI0IaamOyaiaa dogacqGHRaWkcaWGIbGaam4yaiabgUcaRiaadogacaWGHbGaeyOeI0 Iaam4yaiaadggaaeaacqGH9aqpcaaIWaGaey4kaSIaaGimaiabgUca RiaaicdacqGH9aqpcaaIWaaaaaa@4CFD@

d.   p 2 q+r, q 2 r+pand r 2 p+q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaGOmaaaakiabgkHiTiaadghacqGHRaWkcaWGYbGaaiil aiaadghadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGYbGaey4kaS IaamiCaiaaykW7caaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlaaykW7 caWGYbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamiCaiabgUcaRi aadghaaaa@505D@

  = p 2 q+r+ q 2 r+p+ r 2 p+q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iCamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadghacqGHRaWkcaWG YbGaey4kaSIaamyCamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadk hacqGHRaWkcaWGWbGaey4kaSIaamOCamaaCaaaleqabaGaaGOmaaaa kiabgkHiTiaadchacqGHRaWkcaWGXbaaaa@498F@

By combining the like terms,

  = p 2 + q 2 + r 2 q+q+rr+pp = p 2 + q 2 + r 2 +0+0+0= p 2 + q 2 + r 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWGWbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyCamaaCaaa leqabaGaaGOmaaaakiabgUcaRiaadkhadaahaaWcbeqaaiaaikdaaa GccqGHsislcaWGXbGaey4kaSIaamyCaiabgUcaRiaadkhacqGHsisl caWGYbGaey4kaSIaamiCaiabgkHiTiaadchaaeaacqGH9aqpcaWGWb WaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyCamaaCaaaleqabaGa aGOmaaaakiabgUcaRiaadkhadaahaaWcbeqaaiaaikdaaaGccqGHRa WkcaaIWaGaey4kaSIaaGimaiabgUcaRiaaicdacqGH9aqpcaWGWbWa aWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyCamaaCaaaleqabaGaaG OmaaaakiabgUcaRiaadkhadaahaaWcbeqaaiaaikdaaaaaaaa@5F6A@

e.    x 3 y 2 + x 2 y 3 +3 y 4 and x 4 +3 x 2 y 3 +4 y 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiaadMhadaahaaWcbeqaaiaaikdaaaGccqGH RaWkcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEamaaCaaaleqaba GaaG4maaaakiabgUcaRiaaiodacaWG5bWaaWbaaSqabeaacaaI0aaa aOGaaGPaVlGacggacaGGUbGaaiizaiaaykW7caWG4bWaaWbaaSqabe aacaaI0aaaaOGaey4kaSIaaG4maiaadIhadaahaaWcbeqaaiaaikda aaGccaWG5bWaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaGinaiaadM hadaahaaWcbeqaaiaaisdaaaaaaa@52E4@

  = x 3 y 2 + x 2 y 3 +3 y 4 + x 4 +3 x 2 y 3 +4 y 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iEamaaCaaaleqabaGaaG4maaaakiaadMhadaahaaWcbeqaaiaaikda aaGccqGHRaWkcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEamaaCa aaleqabaGaaG4maaaakiabgUcaRiaaiodacaWG5bWaaWbaaSqabeaa caaI0aaaaOGaey4kaSIaamiEamaaCaaaleqabaGaaGinaaaakiabgU caRiaaiodacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEamaaCaaa leqabaGaaG4maaaakiabgUcaRiaaisdacaWG5bWaaWbaaSqabeaaca aI0aaaaaaa@4EF5@

By combining the like terms,

= x 3 y 2 + x 2 y 3 +3 y 4 + x 4 +3 x 2 y 3 +4 y 4 = x 4 +7 y 4 + x 3 y 2 +4 x 2 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWG4bWaaWbaaSqabeaacaaIZaaaaOGaamyEamaaCaaaleqabaGa aGOmaaaakiabgUcaRiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5b WaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaG4maiaadMhadaahaaWc beqaaiaaisdaaaGccqGHRaWkcaWG4bWaaWbaaSqabeaacaaI0aaaaO Gaey4kaSIaaG4maiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bWa aWbaaSqabeaacaaIZaaaaOGaey4kaSIaaGinaiaadMhadaahaaWcbe qaaiaaisdaaaaakeaacqGH9aqpcaWG4bWaaWbaaSqabeaacaaI0aaa aOGaey4kaSIaaG4naiaadMhadaahaaWcbeqaaiaaisdaaaGccqGHRa WkcaWG4bWaaWbaaSqabeaacaaIZaaaaOGaamyEamaaCaaaleqabaGa aGOmaaaakiabgUcaRiaaisdacaWG4bWaaWbaaSqabeaacaaIYaaaaO GaamyEamaaCaaaleqabaGaaG4maaaaaaaa@5FD0@

f.     p 2 qr+p q 2 r+pq r 2 and3p q 2 r2pq r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaGOmaaaakiaadghacaWGYbGaey4kaSIaamiCaiaadgha daahaaWcbeqaaiaaikdaaaGccaWGYbGaey4kaSIaamiCaiaadghaca WGYbWaaWbaaSqabeaacaaIYaaaaOGaaGPaVlaabggacaqGUbGaaeiz aiaaykW7cqGHsislcaaIZaGaamiCaiaadghadaahaaWcbeqaaiaabk daaaGccaWGYbGaeyOeI0IaaGOmaiaadchacaWGXbGaamOCamaaCaaa leqabaGaaGOmaaaaaaa@53D9@

  = p 2 qr+p q 2 r+pq r 2 3p q 2 r2pq r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iCamaaCaaaleqabaGaaGOmaaaakiaadghacaWGYbGaey4kaSIaamiC aiaadghadaahaaWcbeqaaiaaikdaaaGccaWGYbGaey4kaSIaamiCai aadghacaWGYbWaaWbaaSqabeaacaaIYaaaaOGaaGPaVlabgkHiTiaa iodacaWGWbGaamyCamaaCaaaleqabaGaaGOmaaaakiaadkhacqGHsi slcaaIYaGaamiCaiaadghacaWGYbWaaWbaaSqabeaacaaIYaaaaaaa @509F@

By combing the like terms,

  = p 2 qr+p q 2 r3p q 2 r+pq r 2 2pq r 2 = p 2 qr2p q 2 rpq r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWGWbWaaWbaaSqabeaacaaIYaaaaOGaamyCaiaadkhacqGHRaWk caWGWbGaamyCamaaCaaaleqabaGaaGOmaaaakiaadkhacqGHsislca aIZaGaamiCaiaadghadaahaaWcbeqaaiaaikdaaaGccaWGYbGaey4k aSIaamiCaiaadghacaWGYbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0 IaaGOmaiaadchacaWGXbGaamOCamaaCaaaleqabaGaaGOmaaaaaOqa aiabg2da9iaadchadaahaaWcbeqaaiaaikdaaaGccaWGXbGaamOCai abgkHiTiaaikdacaWGWbGaamyCamaaCaaaleqabaGaaGOmaaaakiaa dkhacqGHsislcaWGWbGaamyCaiaadkhadaahaaWcbeqaaiaaikdaaa aaaaa@5E36@

g.   uvvw,vwwu andwuuv MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaWG1b GaamODaiabgkHiTiaadAhacaWG3bGaaiilaiaadAhacaWG3bGaeyOe I0Iaam4DaiaadwhacaaMc8UaaGPaVdqaaiaabggacaqGUbGaaeizai aaykW7caaMc8Uaam4DaiaadwhacqGHsislcaWG1bGaamODaaaaaa@4E02@

  =uvvw+vwwu+wuuv MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam yDaiaadAhacqGHsislcaWG2bGaam4DaiabgUcaRiaadAhacaWG3bGa eyOeI0Iaam4DaiaadwhacqGHRaWkcaWG3bGaamyDaiabgkHiTiaadw hacaWG2baaaa@472E@

By combining like terms,

= 0 

h.      a 2 +3abbc, b 2 +3bcca and c 2 +3caab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaWGHb WaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG4maiaadggacaWGIbGa eyOeI0IaamOyaiaadogacaGGSaGaaGPaVlaadkgadaahaaWcbeqaai aaikdaaaGccqGHRaWkcaaIZaGaamOyaiaadogacqGHsislcaWGJbGa amyyaiaaykW7caaMc8oabaGaaeyyaiaab6gacaqGKbGaaGPaVlaayk W7caWGJbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG4maiaadoga caWGHbGaeyOeI0Iaamyyaiaadkgaaaaa@5908@

( a 2 +3abbc )+( b 2 +3bcca )+( c 2 +3caab ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG4maiaadggacaWG IbGaeyOeI0IaamOyaiaadogaaiaawIcacaGLPaaacqGHRaWkdaqada qaaiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaamOy aiaadogacqGHsislcaWGJbGaamyyaaGaayjkaiaawMcaaiabgUcaRm aabmaabaGaam4yamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaioda caWGJbGaamyyaiabgkHiTiaadggacaWGIbaacaGLOaGaayzkaaaaaa@543E@

a 2 + b 2 + c 2 +2ab+2bc+2ca MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaadkgadaahaaWcbeqaaiaaikda aaGccqGHRaWkcaWGJbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG OmaiaadggacaWGIbGaey4kaSIaaGOmaiaadkgacaWGJbGaey4kaSIa aGOmaiaadogacaWGHbaaaa@476F@

i.         5 8 p 4 +2 p 2 + 5 8 ; 1 8 17p+ 9 8 p 2 and p 5 p 3 +7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaWcaa qaaiaaiwdaaeaacaaI4aaaaiaadchadaahaaWcbeqaaiaaisdaaaGc cqGHRaWkcaaIYaGaamiCamaaCaaaleqabaGaaGOmaaaakiabgUcaRm aalaaabaGaaGynaaqaaiaaiIdaaaGaai4oamaalaaabaGaaGymaaqa aiaaiIdaaaGaeyOeI0IaaGymaiaaiEdacaWGWbGaey4kaSYaaSaaae aacaaI5aaabaGaaGioaaaacaWGWbWaaWbaaSqabeaacaaIYaaaaOGa aGPaVlaaykW7aeaacaqGHbGaaeOBaiaabsgacaaMc8UaaGPaVlaadc hadaahaaWcbeqaaiaaiwdaaaGccqGHsislcaWGWbWaaWbaaSqabeaa caaIZaaaaOGaey4kaSIaaG4naaaaaa@58A8@

  = 5 8 p 4 +2 p 2 + 5 8 + 1 8 17p+ 9 8 p 2 + p 5 p 3 +7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaacaaI1aaabaGaaGioaaaacaWGWbWaaWbaaSqabeaacaaI0aaa aOGaey4kaSIaaGOmaiaadchadaahaaWcbeqaaiaaikdaaaGccqGHRa WkdaWcaaqaaiaaiwdaaeaacaaI4aaaaiabgUcaRmaalaaabaGaaGym aaqaaiaaiIdaaaGaeyOeI0IaaGymaiaaiEdacaWGWbGaey4kaSYaaS aaaeaacaaI5aaabaGaaGioaaaacaWGWbWaaWbaaSqabeaacaaIYaaa aOGaey4kaSIaamiCamaaCaaaleqabaGaaGynaaaakiabgkHiTiaadc hadaahaaWcbeqaaiaaiodaaaGccqGHRaWkcaaI3aaaaa@51C5@

By combining like terms,

  = p 5 + 5 8 p 4 p 3 +( 2+ 9 8 ) p 2 17p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iCamaaCaaaleqabaGaaGynaaaakiabgUcaRmaalaaabaGaaGynaaqa aiaaiIdaaaGaamiCamaaCaaaleqabaGaaGinaaaakiabgkHiTiaadc hadaahaaWcbeqaaiaaiodaaaGccqGHRaWkdaqadaqaaiaaikdacqGH RaWkdaWcaaqaaiaaiMdaaeaacaaI4aaaaaGaayjkaiaawMcaaiaadc hadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIXaGaaG4naiaadcha aaa@4AE1@

     +( 5 8 + 1 8 +7 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSYaae WaaeaadaWcaaqaaiaaiwdaaeaacaaI4aaaaiabgUcaRmaalaaabaGa aGymaaqaaiaaiIdaaaGaey4kaSIaaG4naaGaayjkaiaawMcaaaaa@3DE7@

  = p 5 + 5 8 p 4 p 3 +( 16+9 8 ) p 2 17p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iCamaaCaaaleqabaGaaGynaaaakiabgUcaRmaalaaabaGaaGynaaqa aiaaiIdaaaGaamiCamaaCaaaleqabaGaaGinaaaakiabgkHiTiaadc hadaahaaWcbeqaaiaaiodaaaGccqGHRaWkdaqadaqaamaalaaabaGa aGymaiaaiAdacqGHRaWkcaaI5aaabaGaaGioaaaaaiaawIcacaGLPa aacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGymaiaaiEda caWGWbaaaa@4BA0@

      +( 5+1+56 8 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSYaae WaaeaadaWcaaqaaiaaiwdacqGHRaWkcaaIXaGaey4kaSIaaGynaiaa iAdaaeaacaaI4aaaaaGaayjkaiaawMcaaaaa@3DD3@

  = p 5 + 5 8 p 4 p 3 + 25 8 p 2 17p+ 62 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iCamaaCaaaleqabaGaaGynaaaakiabgUcaRmaalaaabaGaaGynaaqa aiaaiIdaaaGaamiCamaaCaaaleqabaGaaGinaaaakiabgkHiTiaadc hadaahaaWcbeqaaiaaiodaaaGccqGHRaWkdaWcaaqaaiaaikdacaaI 1aaabaGaaGioaaaacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0 IaaGymaiaaiEdacaWGWbGaey4kaSYaaSaaaeaacaaI2aGaaGOmaaqa aiaaiIdaaaaaaa@4BA2@

  = p 5 + 5 8 p 4 p 3 + 25 8 p 2 17p+ 31 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iCamaaCaaaleqabaGaaGynaaaakiabgUcaRmaalaaabaGaaGynaaqa aiaaiIdaaaGaamiCamaaCaaaleqabaGaaGinaaaakiabgkHiTiaadc hadaahaaWcbeqaaiaaiodaaaGccqGHRaWkdaWcaaqaaiaaikdacaaI 1aaabaGaaGioaaaacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0 IaaGymaiaaiEdacaWGWbGaey4kaSYaaSaaaeaacaaIZaGaaGymaaqa aiaaisdaaaaaaa@4B9A@

j.         t t 2 t 3 14;15 t 3 +13+9t8 t 2 ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabgk HiTiaadshadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG0bWaaWba aSqabeaacaaIZaaaaOGaeyOeI0IaaGymaiaaisdacaGG7aGaaGPaVl aaykW7caaIXaGaaGynaiaadshadaahaaWcbeqaaiaaiodaaaGccqGH RaWkcaaIXaGaaG4maiabgUcaRiaaiMdacaWG0bGaeyOeI0IaaGioai aadshadaahaaWcbeqaaiaaikdaaaGccaGG7aaaaa@4F79@

  12 t 2 1924tand4t9 t 2 +19 t 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaWG0bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGymaiaaiMda cqGHsislcaaIYaGaaGinaiGacshacaaMc8UaaGPaVlaabggacaqGUb GaaeizaiaaykW7caaMc8UaaGinaiaadshacqGHsislcaaI5aGaamiD amaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdacaaI5aGaamiDam aaCaaaleqabaGaaG4maaaaaaa@5186@

  =t t 2 t 3 14+15 t 3 +13+9t8 t 2 +12 t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iDaiabgkHiTiaadshadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG 0bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGymaiaaisdacqGHRa WkcaaIXaGaaGynaiaadshadaahaaWcbeqaaiaaiodaaaGccqGHRaWk caaIXaGaaG4maiabgUcaRiaaiMdacaWG0bGaeyOeI0IaaGioaiaads hadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaGOmaiaadsha daahaaWcbeqaaiaaikdaaaaaaa@5108@

  1924t+4t9 t 2 +19 t 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiaaiMdacqGHsislcaaIYaGaaGinaiaadshacqGHRaWkcaaI0aGa amiDaiabgkHiTiaaiMdacaWG0bWaaWbaaSqabeaacaaIYaaaaOGaey 4kaSIaaGymaiaaiMdacaWG0bWaaWbaaSqabeaacaaIZaaaaaaa@461B@

By combining like terms,

  =t+9t24t+4t t 2 8 t 2 +12 t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iDaiabgUcaRiaaiMdacaWG0bGaeyOeI0IaaGOmaiaaisdacaWG0bGa ey4kaSIaaGinaiaadshacqGHsislcaWG0bWaaWbaaSqabeaacaaIYa aaaOGaeyOeI0IaaGioaiaadshadaahaaWcbeqaaiaaikdaaaGccqGH RaWkcaaIXaGaaGOmaiaadshadaahaaWcbeqaaiaaikdaaaaaaa@4B1E@

     9 t 2 t 3 +15 t 3 +19 t 3 14+1319 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG yoaiaadshadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG0bWaaWba aSqabeaacaaIZaaaaOGaey4kaSIaaGymaiaaiwdacaWG0bWaaWbaaS qabeaacaaIZaaaaOGaey4kaSIaaGymaiaaiMdacaWG0bWaaWbaaSqa beaacaaIZaaaaOGaeyOeI0IaaGymaiaaisdacqGHRaWkcaaIXaGaaG 4maiabgkHiTiaaigdacaaI5aaaaa@4C10@

  =10t6 t 2 +33 t 3 20 =33 t 3 6 t 2 10t20 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcqGHsislcaaIXaGaaGimaiaadshacqGHsislcaaI2aGaamiDamaa CaaaleqabaGaaGOmaaaakiabgUcaRiaaiodacaaIZaGaamiDamaaCa aaleqabaGaaG4maaaakiabgkHiTiaaikdacaaIWaaabaGaeyypa0Ja aG4maiaaiodacaWG0bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaG OnaiaadshadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIXaGaaGim aiaadshacqGHsislcaaIYaGaaGimaaaaaa@524A@

Question: 58

Subtract

a.     7 p 2 qr from3 p 2 qr. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaae iiaiaaiEdacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaamyCaiaadkha caqGGaGaaeOzaiaabkhacaqGVbGaaeyBaiabgkHiTiaaiodacaWGWb WaaWbaaSqabeaacaaIYaaaaOGaamyCaiaadkhacaGGUaaaaa@4692@

b.   a 2 ab from  b 2 +ab. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaam yyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadggacaWGIbGaaeii aiaabAgacaqGYbGaae4Baiaab2gacaqGGaGaamOyamaaCaaaleqaba GaaGOmaaaakiabgUcaRiaadggacaWGIbGaaiOlaaaa@4599@

c.    4 x 2 y y 3  from  x 3 +3x y 2 x 2 y. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG inaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bGaeyOeI0IaamyE amaaCaaaleqabaGaaG4maaaakiaabccacaqGMbGaaeOCaiaab+gaca qGTbGaaeiiaiaadIhadaahaaWcbeqaaiaaiodaaaGccqGHRaWkcaaI ZaGaamiEaiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG4b WaaWbaaSqabeaacaaIYaaaaOGaamyEaiaac6caaaa@4D61@

d.   x 4 +3 x 3 y 3 +5 y 4  from 2 x 4 x 3 y 3 +7 y 4 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGinaaaakiabgUcaRiaaiodacaWG4bWaaWbaaSqabeaa caaIZaaaaOGaamyEamaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiw dacaWG5bWaaWbaaSqabeaacaaI0aaaaOGaaeiiaiaabAgacaqGYbGa ae4Baiaab2gacaqGGaGaaGOmaiaadIhadaahaaWcbeqaaiaaisdaaa GccqGHsislcaWG4bWaaWbaaSqabeaacaaIZaaaaOGaamyEamaaCaaa leqabaGaaG4maaaakiabgUcaRiaaiEdacaWG5bWaaWbaaSqabeaaca aI0aaaaOGaaiOlaaaa@51AC@

e.    abbcca fromab+bc+ca. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadk gacqGHsislcaWGIbGaam4yaiabgkHiTiaadogacaWGHbGaaeiiaiaa bAgacaqGYbGaae4Baiaab2gacqGHsislcaWGHbGaamOyaiabgUcaRi aadkgacaWGJbGaey4kaSIaam4yaiaadggacaGGUaaaaa@4A4C@

f. 2 a 2 2 b 2  from  a 2 b 2 +2ab. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIYaGaamOy amaaCaaaleqabaGaaGOmaaaakiaabccacaqGMbGaaeOCaiaab+gaca qGTbGaaeiiaiabgkHiTiaadggadaahaaWcbeqaaiaaikdaaaGccqGH sislcaWGIbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOmaiaadg gacaWGIbGaaiOlaaaa@4B8D@

g.   x 3 y 2 +3 x 2 y 2 7x y 3  from  x 4 + y 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiaadMhadaahaaWcbeqaaiaaikdaaaGccqGH RaWkcaaIZaGaamiEamaaCaaaleqabaGaiGjGikdaaaGccaWG5bWaaW baaSqabeaacaaIYaaaaOGaeyOeI0IaaG4naiaadIhacaWG5bWaaWba aSqabeaacaaIZaaaaOGaaeiiaiaabAgacaqGYbGaae4Baiaab2gaca qGGaGaamiEamaaCaaaleqabaGaaGinaaaakiabgUcaRiaadMhadaah aaWcbeqaaiaaisdaaaaaaa@4EBA@

        + 3 x 2 y 2 x y 3 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaae iiaiaaiodacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEamaaCaaa leqabaGaaGOmaaaakiabgkHiTiaadIhacaWG5bWaaWbaaSqabeaaca aIZaaaaOGaaiOlaaaa@4089@

h.   2 (ab+bc+ca) fromabbcca. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaabc cacaGGOaGaamyyaiaadkgacqGHRaWkcaWGIbGaam4yaiabgUcaRiaa dogacaWGHbGaaiykaiaabccacaqGMbGaaeOCaiaab+gacaqGTbGaey OeI0IaamyyaiaadkgacqGHsislcaWGIbGaam4yaiabgkHiTiaadoga caWGHbGaaiOlaaaa@4D04@

i.    4.5 x 5 3.4 x 2 +5.7 from 5 x 4    MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaac6 cacaaI1aGaamiEamaaCaaaleqabaGaaGynaaaakiabgkHiTiaaioda caGGUaGaaGinaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkca aI1aGaaiOlaiaaiEdacaqGGaGaaeOzaiaabkhacaqGVbGaaeyBaiaa bccacaaI1aGaamiEamaaCaaaleqabaGaaGinaaaakiaabccacaqGGa aaaa@4B15@

       

j.   1115 y 2  from  y 3 15 y 2 y11. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaig dacqGHsislcaaIXaGaaGynaiaadMhadaahaaWcbeqaaiaaikdaaaGc caqGGaGaaeOzaiaabkhacaqGVbGaaeyBaiaabccacaWG5bWaaWbaaS qabeaacaaIZaaaaOGaeyOeI0IaaGymaiaaiwdacaWG5bWaaWbaaSqa beaacaaIYaaaaOGaeyOeI0IaamyEaiabgkHiTiaaigdacaaIXaGaai Olaaaa@4BF6@

Solution

a.    We have,

        3 p 2 qr( 7 p 2 qr ) =3 p 2 qr+7 p 2 qr = p 2 qr( 3+7 ) =4 p 2 qr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGHsi slcaaIZaGaamiCamaaCaaaleqabaGaaGOmaaaakiaadghacaWGYbGa eyOeI0YaaeWaaeaacqGHsislcaaI3aGaamiCamaaCaaaleqabaGaaG OmaaaakiaadghacaWGYbaacaGLOaGaayzkaaaabaGaeyypa0JaeyOe I0IaaG4maiaadchadaahaaWcbeqaaiaaikdaaaGccaWGXbGaamOCai abgUcaRiaaiEdacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaamyCaiaa dkhaaeaacqGH9aqpcaWGWbWaaWbaaSqabeaacaaIYaaaaOGaamyCai aadkhadaqadaqaaiabgkHiTiaaiodacqGHRaWkcaaI3aaacaGLOaGa ayzkaaaabaGaeyypa0JaaGinaiaadchadaahaaWcbeqaaiaaikdaaa GccaWGXbGaamOCaaaaaa@5EA0@

b.   We have,

b 2 +ab( a 2 ab ) = b 2 +ab+ a 2 +ab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaWGIb WaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyyaiaadkgacqGHsisl daqadaqaaiabgkHiTiaadggadaahaaWcbeqaaiaaikdaaaGccqGHsi slcaWGHbGaamOyaaGaayjkaiaawMcaaaqaaiabg2da9iaadkgadaah aaWcbeqaaiaaikdaaaGccqGHRaWkcaWGHbGaamOyaiabgUcaRiaadg gadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGHbGaamOyaaaaaa@4D57@

By combining like terms,

= b 2 + a 2 +ab+ab = a 2 +  b 2 +2ab MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWGIbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyyamaaCaaa leqabaGaaGOmaaaakiabgUcaRiaadggacaWGIbGaey4kaSIaamyyai aadkgaaeaacqGH9aqpcaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaaeiiaiaadkgadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYa Gaamyyaiaadkgaaaaa@4A81@

c.    We have,

x 3 +3x y 2 x 2 y( 4 x 2 y y 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgUcaRiaaiodacaWG4bGaamyEamaaCaaa leqabaGaaGOmaaaakiabgkHiTiaadIhadaahaaWcbeqaaiaaikdaaa GccaWG5bGaeyOeI0YaaeWaaeaacqGHsislcaaI0aGaamiEamaaCaaa leqabaGaaGOmaaaakiaadMhacqGHsislcaWG5bWaaWbaaSqabeaaca aIZaaaaaGccaGLOaGaayzkaaaaaa@4A20@

= x 3 +3x y 2 x 2 y+4 x 2 y+ y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iEamaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiodacaWG4bGaamyE amaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadIhadaahaaWcbeqaai aaikdaaaGccaWG5bGaey4kaSIaaGinaiaadIhadaahaaWcbeqaaiaa ikdaaaGccaWG5bGaey4kaSIaamyEamaaCaaaleqabaGaaG4maaaaaa a@4890@

= x 3 + y 3 +3 x 2 y+3x y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iEamaaCaaaleqabaGaaG4maaaakiabgUcaRiaadMhadaahaaWcbeqa aiaaiodaaaGccqGHRaWkcaaIZaGaamiEamaaCaaaleqabaGaaGOmaa aakiaadMhacqGHRaWkcaaIZaGaamiEaiaadMhadaahaaWcbeqaaiaa ikdaaaaaaa@44B4@

d.   We have,

2 x 4 x 3 y 3 +7 y 4 ( x 4 +3 x 3 y 3 +5 y 4 ) =2 x 4 x 3 y 3 +7 y 4 x 4 3 x 3 y 3 5 y 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIYa GaamiEamaaCaaaleqabaGaaGinaaaakiabgkHiTiaadIhadaahaaWc beqaaiaaiodaaaGccaWG5bWaaWbaaSqabeaacaaIZaaaaOGaey4kaS IaaG4naiaadMhadaahaaWcbeqaaiaaisdaaaGccqGHsisldaqadaqa aiaadIhadaahaaWcbeqaaiaaisdaaaGccqGHRaWkcaaIZaGaamiEam aaCaaaleqabaGaaG4maaaakiaadMhadaahaaWcbeqaaiaaiodaaaGc cqGHRaWkcaaI1aGaamyEamaaCaaaleqabaGaaGinaaaaaOGaayjkai aawMcaaaqaaiabg2da9iaaikdacaWG4bWaaWbaaSqabeaacaaI0aaa aOGaeyOeI0IaamiEamaaCaaaleqabaGaaG4maaaakiaadMhadaahaa WcbeqaaiaaiodaaaGccqGHRaWkcaaI3aGaamyEamaaCaaaleqabaGa aGinaaaakiabgkHiTiaadIhadaahaaWcbeqaaiaaisdaaaGccqGHsi slcaaIZaGaamiEamaaCaaaleqabaGaaG4maaaakiaadMhadaahaaWc beqaaiaaiodaaaGccqGHsislcaaI1aGaamyEamaaCaaaleqabaGaaG inaaaaaaaa@668D@

By combining like terms,

=2 x 4 x 4 x 3 y 3 3 x 3 y 3 +7 y 4 5 y 4 = x 4 4 x 3 y 3 +2 y 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIYaGaamiEamaaCaaaleqabaGaaGinaaaakiabgkHiTiaadIha daahaaWcbeqaaiaaisdaaaGccqGHsislcaWG4bWaaWbaaSqabeaaca aIZaaaaOGaamyEamaaCaaaleqabaGaaG4maaaakiabgkHiTiaaioda caWG4bWaaWbaaSqabeaacaaIZaaaaOGaamyEamaaCaaaleqabaGaaG 4maaaakiabgUcaRiaaiEdacaWG5bWaaWbaaSqabeaacaaI0aaaaOGa eyOeI0IaaGynaiaadMhadaahaaWcbeqaaiaaisdaaaaakeaacqGH9a qpcaWG4bWaaWbaaSqabeaacaaI0aaaaOGaeyOeI0IaaGinaiaadIha daahaaWcbeqaaiaaiodaaaGccaWG5bWaaWbaaSqabeaacaaIZaaaaO Gaey4kaSIaaGOmaiaadMhadaahaaWcbeqaaiaaisdaaaaaaaa@5A12@

e.    We have,

ab+bc+ca( abbcca ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaam yyaiaadkgacqGHRaWkcaWGIbGaam4yaiabgUcaRiaadogacaWGHbGa eyOeI0YaaeWaaeaacaWGHbGaamOyaiabgkHiTiaadkgacaWGJbGaey OeI0Iaam4yaiaadggaaiaawIcacaGLPaaaaaa@47AE@

=ab+bc+caab+bc+ca MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaey OeI0IaamyyaiaadkgacqGHRaWkcaWGIbGaam4yaiabgUcaRiaadoga caWGHbGaeyOeI0IaamyyaiaadkgacqGHRaWkcaWGIbGaam4yaiabgU caRiaadogacaWGHbaaaa@4715@

By combining like terms,

=abab+bc+bc+ca+ca =2ab+2bc+2ca MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcqGHsislcaWGHbGaamOyaiabgkHiTiaadggacaWGIbGaey4kaSIa amOyaiaadogacqGHRaWkcaWGIbGaam4yaiabgUcaRiaadogacaWGHb Gaey4kaSIaam4yaiaadggaaeaacqGH9aqpcqGHsislcaaIYaGaamyy aiaadkgacqGHRaWkcaaIYaGaamOyaiaadogacqGHRaWkcaaIYaGaam 4yaiaadggaaaaa@5271@

f.We have,

( a 2 b 2 +2ab )( 2 a 2 2 b 2 ) = a 2 b 2 +2ab+2 a 2 +2 b 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqada qaaiabgkHiTiaadggadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG IbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOmaiaadggacaWGIb aacaGLOaGaayzkaaGaeyOeI0YaaeWaaeaacqGHsislcaaIYaGaamyy amaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaikdacaWGIbWaaWbaaS qabeaacaaIYaaaaaGccaGLOaGaayzkaaaabaGaeyypa0JaeyOeI0Ia amyyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadkgadaahaaWcbe qaaiaaikdaaaGccqGHRaWkcaaIYaGaamyyaiaadkgacqGHRaWkcaaI YaGaamyyamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWGIb WaaWbaaSqabeaacaaIYaaaaaaaaa@5ABF@

By combining like terms,

= a 2 +2 a 2 b 2 +2 a 2 +2ab = a 2 b 2 +2ab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcqGHsislcaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOm aiaadggadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGIbWaaWbaaS qabeaacaaIYaaaaOGaey4kaSIaaGOmaiaadggadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaIYaGaamyyaiaadkgaaeaacqGH9aqpcaWGHb WaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamOyamaaCaaaleqabaGa aGOmaaaakiabgUcaRiaaikdacaWGHbGaamOyaaaaaa@4FDD@

g.   We have,

  x 4 + y 4 +3 x 2 y 2 x y 3 ( x 3 y 2 +3 x 2 y 2 7x y 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGinaaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaisda aaGccqGHRaWkcaaIZaGaamiEamaaCaaaleqabaGaaGOmaaaakiaadM hadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG4bGaamyEamaaCaaa leqabaGaaG4maaaakiabgkHiTmaabmaabaGaamiEamaaCaaaleqaba GaaG4maaaakiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI ZaGaamiEamaaCaaaleqabaGaaGOmaaaakiaadMhadaahaaWcbeqaai aaikdaaaGccqGHsislcaaI3aGaamiEaiaadMhadaahaaWcbeqaaiaa iodaaaaakiaawIcacaGLPaaaaaa@5471@

  = x 4 + y 4 +3 x 2 y 2 x y 3 x 3 y 2 3 x 2 y 2 +7x y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iEamaaCaaaleqabaGaaGinaaaakiabgUcaRiaadMhadaahaaWcbeqa aiaaisdaaaGccqGHRaWkcaaIZaGaamiEamaaCaaaleqabaGaaGOmaa aakiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG4bGaamyE amaaCaaaleqabaGaaG4maaaakiabgkHiTiaadIhadaahaaWcbeqaai aaiodaaaGccaWG5bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaG4m aiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bWaaWbaaSqabeaaca aIYaaaaOGaey4kaSIaaG4naiaadIhacaWG5bWaaWbaaSqabeaacaaI Zaaaaaaa@53E4@

By combining the like terms,

  = x 4 + y 4 +3 x 2 y 2 3 x 2 y 2 x y 3 +7x y 3 x 3 y 2 = x 4 + y 4 +6x y 3 x 3 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWG4bWaaWbaaSqabeaacaaI0aaaaOGaey4kaSIaamyEamaaCaaa leqabaGaaGinaaaakiabgUcaRiaaiodacaWG4bWaaWbaaSqabeaaca aIYaaaaOGaamyEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaioda caWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEamaaCaaaleqabaGaaG OmaaaakiabgkHiTiaadIhacaWG5bWaaWbaaSqabeaacaaIZaaaaOGa ey4kaSIaaG4naiaadIhacaWG5bWaaWbaaSqabeaacaaIZaaaaOGaey OeI0IaamiEamaaCaaaleqabaGaaG4maaaakiaadMhadaahaaWcbeqa aiaaikdaaaaakeaacqGH9aqpcaWG4bWaaWbaaSqabeaacaaI0aaaaO Gaey4kaSIaamyEamaaCaaaleqabaGaaGinaaaakiabgUcaRiaaiAda caWG4bGaamyEamaaCaaaleqabaGaaG4maaaakiabgkHiTiaadIhada ahaaWcbeqaaiaaiodaaaGccaWG5bWaaWbaaSqabeaacaaIYaaaaaaa aa@6318@

h.   We have,

  abbcca2( ab+bc+ca ) =abbcca2ab2bc2ca MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGHsi slcaWGHbGaamOyaiabgkHiTiaadkgacaWGJbGaeyOeI0Iaam4yaiaa dggacqGHsislcaaIYaWaaeWaaeaacaWGHbGaamOyaiabgUcaRiaadk gacaWGJbGaey4kaSIaam4yaiaadggaaiaawIcacaGLPaaaaeaacqGH 9aqpcqGHsislcaWGHbGaamOyaiabgkHiTiaadkgacaWGJbGaeyOeI0 Iaam4yaiaadggacqGHsislcaaIYaGaamyyaiaadkgacqGHsislcaaI YaGaamOyaiaadogacqGHsislcaaIYaGaam4yaiaadggaaaaa@5C0D@

By combining the like terms,

  =ab2abbc2bcca2ca =3ab3bc3ca MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcqGHsislcaWGHbGaamOyaiabgkHiTiaaikdacaWGHbGaamOyaiab gkHiTiaadkgacaWGJbGaeyOeI0IaaGOmaiaadkgacaWGJbGaeyOeI0 Iaam4yaiaadggacqGHsislcaaIYaGaam4yaiaadggaaeaacqGH9aqp cqGHsislcaaIZaGaamyyaiaadkgacqGHsislcaaIZaGaamOyaiaado gacqGHsislcaaIZaGaam4yaiaadggaaaaa@54EA@

i.      We have,

  5 x 4 32 x 2 7.3x( 4.5 x 5 3.4 x 2 +5.7 ) =5 x 4 3.2 x 2 7.3x4.5 x 5 +3.4 x 2 5.7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaI1a GaamiEamaaCaaaleqabaGaaGinaaaakiabgkHiTiaaiodacaaIYaGa amiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaiEdacaGGUaGaaG 4maiaadIhacqGHsisldaqadaqaaiaaisdacaGGUaGaaGynaiaadIha daahaaWcbeqaaiaaiwdaaaGccqGHsislcaaIZaGaaiOlaiaaisdaca WG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGynaiaac6cacaaI 3aaacaGLOaGaayzkaaaabaGaeyypa0JaaGynaiaadIhadaahaaWcbe qaaiaaisdaaaGccqGHsislcaaIZaGaaiOlaiaaikdacaWG4bWaaWba aSqabeaacaaIYaaaaOGaeyOeI0IaaG4naiaac6cacaaIZaGaamiEai abgkHiTiaaisdacaGGUaGaaGynaiaadIhadaahaaWcbeqaaiaaiwda aaGccqGHRaWkcaaIZaGaaiOlaiaaisdacaWG4bWaaWbaaSqabeaaca aIYaaaaOGaeyOeI0IaaGynaiaac6cacaaI3aaaaaa@69BD@

By combining the like terms,

  =4.5 x 5 +5 x 4 3.2 x 2 +3.4 x 2 7.3x5.7 =4.5 x 5 +5 x 4 +0.2 x 2 7.3x5.7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcqGHsislcaaI0aGaaiOlaiaaiwdacaWG4bWaaWbaaSqabeaacaaI 1aaaaOGaey4kaSIaaGynaiaadIhadaahaaWcbeqaaiaaisdaaaGccq GHsislcaaIZaGaaiOlaiaaikdacaWG4bWaaWbaaSqabeaacaaIYaaa aOGaey4kaSIaaG4maiaac6cacaaI0aGaamiEamaaCaaaleqabaGaaG OmaaaakiabgkHiTiaaiEdacaGGUaGaaG4maiaadIhacqGHsislcaaI 1aGaaiOlaiaaiEdaaeaacqGH9aqpcqGHsislcaaI0aGaaiOlaiaaiw dacaWG4bWaaWbaaSqabeaacaaI1aaaaOGaey4kaSIaaGynaiaadIha daahaaWcbeqaaiaaisdaaaGccqGHRaWkcaaIWaGaaiOlaiaaikdaca WG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaG4naiaac6cacaaI ZaGaamiEaiabgkHiTiaaiwdacaGGUaGaaG4naaaaaa@66A3@

j.      We have,

  y 3 15 y 2 y11( 1115 y 2 ) = y 3 15 y 2 y1111+15 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG5b WaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGymaiaaiwdacaWG5bWa aWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyEaiabgkHiTiaaigdaca aIXaGaeyOeI0YaaeWaaeaacaaIXaGaaGymaiabgkHiTiaaigdacaaI 1aGaamyEamaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaqaai abg2da9iaadMhadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaIXaGa aGynaiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG5bGaey OeI0IaaGymaiaaigdacqGHsislcaaIXaGaaGymaiabgUcaRiaaigda caaI1aGaamyEamaaCaaaleqabaGaaGOmaaaaaaaa@5B00@

By combining the like terms,

  = y 3 15 y 2 +15 y 2 y1111 = y 3 y22 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWG5bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGymaiaaiwda caWG5bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaiaaiwdaca WG5bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyEaiabgkHiTiaa igdacaaIXaGaeyOeI0IaaGymaiaaigdaaeaacqGH9aqpcaWG5bWaaW baaSqabeaacaaIZaaaaOGaeyOeI0IaamyEaiabgkHiTiaaikdacaaI Yaaaaaa@4F76@

Question: 59

a.    What should be added to

       

b.   What should be added to

        3pq+5 p 2 q 2 + p 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadc hacaWGXbGaey4kaSIaaGynaiaadchadaahaaWcbeqaaiaaikdaaaGc caWGXbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamiCamaaCaaale qabaGaaG4maaaaaaa@40B4@  to get p 3 +2 p 2 q 2 +4pq MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaG4maaaakiabgUcaRiaaikdacaWGWbWaaWbaaSqabeaa caaIYaaaaOGaamyCamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaais dacaWGWbGaamyCaaaa@40BC@

Solution

a.    Subtract: x 3 +3 x 2 y+3x y 2 + y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgUcaRiaaiodacaWG4bWaaWbaaSqabeaa caaIYaaaaOGaamyEaiabgUcaRiaaiodacaWG4bGaamyEamaaCaaale qabaGaaGOmaaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaiodaaaaa aa@43AE@  from x 3 + y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaioda aaaaaa@3A94@

        Required expression is

        x 3 + y 3 ( x 3 +3 x 2 y+3x y 2 + y 3 ) = x 3 + y 3 x 3 3 x 2 y3x y 2 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4b WaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaamyEamaaCaaaleqabaGa aG4maaaakiabgkHiTmaabmaabaGaamiEamaaCaaaleqabaGaaG4maa aakiabgUcaRiaaiodacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyE aiabgUcaRiaaiodacaWG4bGaamyEamaaCaaaleqabaGaaGOmaaaaki abgUcaRiaadMhadaahaaWcbeqaaiaaiodaaaaakiaawIcacaGLPaaa aeaacqGH9aqpcaWG4bWaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaam yEamaaCaaaleqabaGaaG4maaaakiabgkHiTiaadIhadaahaaWcbeqa aiaaiodaaaGccqGHsislcaaIZaGaamiEamaaCaaaleqabaGaaGOmaa aakiaadMhacqGHsislcaaIZaGaamiEaiaadMhadaahaaWcbeqaaiaa ikdaaaGccqGHsislcaWG5bWaaWbaaSqabeaacaaIZaaaaaaaaa@5FA8@

By combining like terms,

        = x 3 x 3 + y 3 y 3 3 x 2 y3x y 2 =3 x 2 y3x y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWG4bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamiEamaaCaaa leqabaGaaG4maaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaiodaaa GccqGHsislcaWG5bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaG4m aiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bGaeyOeI0IaaG4mai aadIhacaWG5bWaaWbaaSqabeaacaaIYaaaaaGcbaGaeyypa0JaeyOe I0IaaG4maiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bGaeyOeI0 IaaG4maiaadIhacaWG5bWaaWbaaSqabeaacaaIYaaaaaaaaa@54C4@

  So, if we add 3 x 2 y3x y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4maiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bGaeyOeI0IaaG4m aiaadIhacaWG5bWaaWbaaSqabeaacaaIYaaaaaaa@3EFF@  in    

  x 3 +3 x 2 y+3x y 2 + y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgUcaRiaaiodacaWG4bWaaWbaaSqabeaa caaIYaaaaOGaamyEaiabgUcaRiaaiodacaWG4bGaamyEamaaCaaale qabaGaaGOmaaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaiodaaaaa aa@43AE@  we get x 3 + y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaioda aaaaaa@3A94@

b.   Subtract: 3pq+5 p 2 q 2 + p 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadc hacaWGXbGaey4kaSIaaGynaiaadchadaahaaWcbeqaaiaaikdaaaGc caWGXbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamiCamaaCaaale qabaGaaG4maaaaaaa@40B4@  from p 3 +2 p 2 q 2 +4pq MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaG4maaaakiabgUcaRiaaikdacaWGWbWaaWbaaSqabeaa caaIYaaaaOGaamyCamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaais dacaWGWbGaamyCaaaa@40BC@

        Required expression is,

        p 3 +2 p 2 q 2 +4pq( 3pq+5 p 2 q 2 + p 3 ) = p 3 +2 p 2 q 2 +4pq3pq5 p 2 q 2 p 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGWb WaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaGOmaiaadchadaahaaWc beqaaiaaikdaaaGccaWGXbWaaWbaaSqabeaacaaIYaaaaOGaey4kaS IaaGinaiaadchacaWGXbGaeyOeI0YaaeWaaeaacaaIZaGaamiCaiaa dghacqGHRaWkcaaI1aGaamiCamaaCaaaleqabaGaaGOmaaaakiaadg hadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGWbWaaWbaaSqabeaa caaIZaaaaaGccaGLOaGaayzkaaaabaGaeyypa0JaamiCamaaCaaale qabaGaaG4maaaakiabgUcaRiaaikdacaWGWbWaaWbaaSqabeaacaaI YaaaaOGaamyCamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaisdaca WGWbGaamyCaiabgkHiTiaaiodacaWGWbGaamyCaiabgkHiTiaaiwda caWGWbWaaWbaaSqabeaacaaIYaaaaOGaamyCamaaCaaaleqabaGaaG OmaaaakiabgkHiTiaadchadaahaaWcbeqaaiaaiodaaaaaaaa@65E4@

        By combining like terms,

        = p 3 p 3 +2 p 2 q 2 5 p 2 q 2 +4pq3pq =3 p 2 q 2 +pq MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaWGWbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamiCamaaCaaa leqabaGaaG4maaaakiabgUcaRiaaikdacaWGWbWaaWbaaSqabeaaca aIYaaaaOGaamyCamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaiwda caWGWbWaaWbaaSqabeaacaaIYaaaaOGaamyCamaaCaaaleqabaGaaG OmaaaakiabgUcaRiaaisdacaWGWbGaamyCaiabgkHiTiaaiodacaWG WbGaamyCaaqaaiabg2da9iaaykW7caaMc8UaaGPaVlaaykW7cqGHsi slcaaIZaGaamiCamaaCaaaleqabaGaaGOmaaaakiaadghadaahaaWc beqaaiaaikdaaaGccqGHRaWkcaWGWbGaamyCaaaaaa@5D2A@

        So, if we add 3 p 2 q 2 +pq MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4maiaadchadaahaaWcbeqaaiaaikdaaaGccaWGXbWaaWbaaSqabeaa caaIYaaaaOGaey4kaSIaamiCaiaadghaaaa@3E20@  in 3p q 2 +5 p 2 q 2 + p 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadc hacaWGXbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGynaiaadcha daahaaWcbeqaaiaaikdaaaGccaWGXbWaaWbaaSqabeaacaaIYaaaaO Gaey4kaSIaamiCamaaCaaaleqabaGaaG4maaaaaaa@41A6@ ,

        we get p 3 +2 p 2 q 2 +4pq MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaG4maaaakiabgUcaRiaaikdacaWGWbWaaWbaaSqabeaa caaIYaaaaOGaamyCamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaais dacaWGWbGaamyCaaaa@40BB@

Question: 60

a.    What should be subtracted from

        2 x 3 3 x 2 y+2x y 2 +3 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI hadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaIZaGaamiEamaaCaaa leqabaGaaGOmaaaakiaadMhacqGHRaWkcaaIYaGaamiEaiaadMhada ahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaamyEamaaCaaaleqa baGaaG4maaaaaaa@4531@  to get

       

b.   What should be subtracted from

        7mn+2 m 2 +3 n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4naiaad2gacaWGUbGaey4kaSIaaGOmaiaad2gadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaIZaGaamOBamaaCaaaleqabaGaaGOmaaaaaa a@406A@  to get

       

Solution

a.    Subtract: x 3 2 x 2 y+3x y 2 +4 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgkHiTiaaikdacaWG4bWaaWbaaSqabeaa caaIYaaaaOGaamyEaiabgUcaRiaaiodacaWG4bGaamyEamaaCaaale qabaGaaGOmaaaakiabgUcaRiaaisdacaWG5bWaaWbaaSqabeaacaaI Zaaaaaaa@4476@  from

        2 x 3 3 x 2 y+2x y 2 +3 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI hadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaIZaGaamiEamaaCaaa leqabaGaaGOmaaaakiaadMhacqGHRaWkcaaIYaGaamiEaiaadMhada ahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaamyEamaaCaaaleqa baGaaG4maaaaaaa@4531@

  Required expression is

  2 x 3 3 x 2 y+2x y 2 +3 y 3  ( x 3 2 x 2 y+3x y 2 +4 y 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIYa GaamiEamaaCaaaleqabaGaaG4maaaakiabgkHiTiaaiodacaWG4bWa aWbaaSqabeaacaaIYaaaaOGaamyEaiabgUcaRiaaikdacaWG4bGaam yEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiodacaWG5bWaaWba aSqabeaacaaIZaaaaaGcbaGaeyOeI0IaaeiiamaabmaabaGaamiEam aaCaaaleqabaGaaG4maaaakiabgkHiTiaaikdacaWG4bWaaWbaaSqa beaacaaIYaaaaOGaamyEaiabgUcaRiaaiodacaWG4bGaamyEamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaaisdacaWG5bWaaWbaaSqabeaa caaIZaaaaaGccaGLOaGaayzkaaaaaaa@5701@    

  =2 x 3 3 x 2 y+2x y 2 +3 y 3 x 3 +2 x 2 y3x y 2 4 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIYaGaamiEamaaCaaaleqabaGaaG4maaaakiabgkHiTiaaioda caWG4bWaaWbaaSqabeaacaaIYaaaaOGaamyEaiabgUcaRiaaikdaca WG4bGaamyEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiodacaWG 5bWaaWbaaSqabeaacaaIZaaaaaGcbaGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7cqGHsislcaWG4bWaaWbaaSqabeaacaaIZaaaaOGaey4k aSIaaGOmaiaadIhadaahaaWcbeqaaiaaikdaaaGccaWG5bGaeyOeI0 IaaG4maiaadIhacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Ia aGinaiaadMhadaahaaWcbeqaaiaaiodaaaaaaaa@5D94@

By combining like terms,

  =2 x 3 x 3 3 x 2 y+2 x 2 y+2x y 2 3x y 2 +3 y 3 4 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIYaGaamiEamaaCaaaleqabaGaaG4maaaakiabgkHiTiaadIha daahaaWcbeqaaiaaiodaaaGccqGHsislcaaIZaGaamiEamaaCaaale qabaGaaGOmaaaakiaadMhacqGHRaWkcaaIYaGaamiEamaaCaaaleqa baGaaGOmaaaakiaadMhacqGHRaWkcaaIYaGaamiEaiaadMhadaahaa WcbeqaaiaaikdaaaaakeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlabgkHiTiaaiodacaWG4bGaamyEamaaCaaaleqabaGaaGOmaaaaki abgUcaRiaaiodacaWG5bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0Ia aGinaiaadMhadaahaaWcbeqaaiaaiodaaaaaaaa@5D94@

  = x 3 x 2 yx y 2 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iEamaaCaaaleqabaGaaG4maaaakiabgkHiTiaadIhadaahaaWcbeqa aiaaikdaaaGccaWG5bGaeyOeI0IaamiEaiaadMhadaahaaWcbeqaai aaikdaaaGccqGHsislcaWG5bWaaWbaaSqabeaacaaIZaaaaaaa@435B@

  So, if we subtract x 3 x 2 yx y 2 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgkHiTiaadIhadaahaaWcbeqaaiaaikda aaGccaWG5bGaeyOeI0IaamiEaiaadMhadaahaaWcbeqaaiaaikdaaa GccqGHsislcaWG5bWaaWbaaSqabeaacaaIZaaaaaaa@4255@  from  

  2 x 3 3 x 2 y+2x y 2 +3 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI hadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaIZaGaamiEamaaCaaa leqabaGaaGOmaaaakiaadMhacqGHRaWkcaaIYaGaamiEaiaadMhada ahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaamyEamaaCaaaleqa baGaaG4maaaaaaa@4531@ , then we get

  x 3 2 x 2 y+3x y 2 +4 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgkHiTiaaikdacaWG4bWaaWbaaSqabeaa caaIYaaaaOGaamyEaiabgUcaRiaaiodacaWG4bGaamyEamaaCaaale qabaGaaGOmaaaakiabgUcaRiaaisdacaWG5bWaaWbaaSqabeaacaaI Zaaaaaaa@4476@

b.   Subtract: m 2 +2mn+ n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWGTbGaamOBaiabgUca Riaad6gadaahaaWcbeqaaiaaikdaaaaaaa@3DFF@  from 7mn+2 m 2 +3 n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4naiaad2gacaWGUbGaey4kaSIaaGOmaiaad2gadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaIZaGaamOBamaaCaaaleqabaGaaGOmaaaaaa a@406A@

  Required expression is,

  7mn+2 m 2 +3 n 2 ( m 2 +2mn+ n 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4naiaad2gacaWGUbGaey4kaSIaaGOmaiaad2gadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaIZaGaamOBamaaCaaaleqabaGaaGOmaaaaki abgkHiTmaabmaabaGaamyBamaaCaaaleqabaGaaGOmaaaakiabgUca RiaaikdacaWGTbGaamOBaiabgUcaRiaad6gadaahaaWcbeqaaiaaik daaaaakiaawIcacaGLPaaaaaa@4B1A@

  =7mn+2 m 2 +3 n 2 m 2 2mn n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaey OeI0IaaG4naiaad2gacaWGUbGaey4kaSIaaGOmaiaad2gadaahaaWc beqaaiaaikdaaaGccqGHRaWkcaaIZaGaamOBamaaCaaaleqabaGaaG OmaaaakiabgkHiTiaad2gadaahaaWcbeqaaiaaikdaaaGccqGHsisl caaIYaGaamyBaiaad6gacqGHsislcaWGUbWaaWbaaSqabeaacaaIYa aaaaaa@4AA3@

  By combining like terms,

  =7mn2mn+2 m 2 m 2 +3 n 2 n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaey OeI0IaaG4naiaad2gacaWGUbGaeyOeI0IaaGOmaiaad2gacaWGUbGa ey4kaSIaaGOmaiaad2gadaahaaWcbeqaaiaaikdaaaGccqGHsislca WGTbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG4maiaad6gadaah aaWcbeqaaiaaikdaaaGccqGHsislcaWGUbWaaWbaaSqabeaacaaIYa aaaaaa@4AA3@

  =9mn+ m 2 +2 n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaey OeI0IaaGyoaiaad2gacaWGUbGaey4kaSIaamyBamaaCaaaleqabaGa aGOmaaaakiabgUcaRiaaikdacaWGUbWaaWbaaSqabeaacaaIYaaaaa aa@40B5@

  So, if we subtract m 2 +2 n 2 9mn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWGUbWaaWbaaSqabeaa caaIYaaaaOGaeyOeI0IaaGyoaiaad2gacaWGUbaaaa@3ED7@  from

  7mn+2 m 2 +3 n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG 4naiaad2gacaWGUbGaey4kaSIaaGOmaiaad2gadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaIZaGaamOBamaaCaaaleqabaGaaGOmaaaaaa a@406A@ , then we get

  m 2 +2mn+ n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWGTbGaamOBaiabgUca Riaad6gadaahaaWcbeqaaiaaikdaaaaaaa@3DFF@

Question: 61

How much is 21 a 3 17 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaig dacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGymaiaaiEda caWGHbWaaWbaaSqabeaacaaIYaaaaaaa@3D61@  less than 89 a 3 64 a 2 +6a+16? MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaaiM dacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGOnaiaaisda caWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOnaiaadggacq GHRaWkcaaIXaGaaGOnaiaac+daaaa@4323@

Solution

Required expression is

89 a 3 64 a 2 +6a+16( 21 a 3 17 a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaaiM dacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGOnaiaaisda caWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOnaiaadggacq GHRaWkcaaIXaGaaGOnaiabgkHiTmaabmaabaGaaGOmaiaaigdacaWG HbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGymaiaaiEdacaWGHb WaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaaaaa@4C6A@

=89 a 3 64 a 2 +6a+1621 a 3 +17 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiaaiMdacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGOn aiaaisdacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOnai aadggacqGHRaWkcaaIXaGaaGOnaiabgkHiTiaaikdacaaIXaGaamyy amaaCaaaleqabaGaaG4maaaakiabgUcaRiaaigdacaaI3aGaamyyam aaCaaaleqabaGaaGOmaaaaaaa@4BD2@

By combining the like terms,

=89 a 3 21 a 3 64 a 2 +17 a 2 +6a+16 =68 a 3 47 a 2 +6a+16 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaI4aGaaGyoaiaadggadaahaaWcbeqaaiaaiodaaaGccqGHsisl caaIYaGaaGymaiaadggadaahaaWcbeqaaiaaiodaaaGccqGHsislca aI2aGaaGinaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI XaGaaG4naiaadggadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI2a GaamyyaiabgUcaRiaaigdacaaI2aaabaGaeyypa0JaaGOnaiaaiIda caWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGinaiaaiEdaca WGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOnaiaadggacqGH RaWkcaaIXaGaaGOnaaaaaa@596F@

So, 21 a 3 17 a 2 is68 a 3 47 a 2 +6a+16 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaig dacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGymaiaaiEda caWGHbWaaWbaaSqabeaacaaIYaaaaOGaaGPaVlaaykW7caqGPbGaae 4CaiaaykW7caaMc8UaaGOnaiaaiIdacaWGHbWaaWbaaSqabeaacaaI ZaaaaOGaeyOeI0IaaGinaiaaiEdacaWGHbWaaWbaaSqabeaacaaIYa aaaOGaey4kaSIaaGOnaiaadggacqGHRaWkcaaIXaGaaGOnaaaa@5200@  less than

89 a 3 64 a 2 +6a+16 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaaiM dacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGOnaiaaisda caWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOnaiaadggacq GHRaWkcaaIXaGaaGOnaaaa@4261@ .

Question: 62

How much is y 4 12 y 2 +y+14 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa aaleqabaGaiGjGisdaaaGccqGHsislcaaIXaGaaGOmaiaadMhadaah aaWcbeqaaiaaikdaaaGccqGHRaWkcaWG5bGaey4kaSIaaGymaiaais daaaa@417B@  greater than 17 y 3 +34 y 2 51y+68? MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiE dacaWG5bWaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaG4maiaaisda caWG5bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGynaiaaigdaca WG5bGaey4kaSIaaGOnaiaaiIdacaGG=aaaaa@4420@

Solution

Required expression is

y 4 12 y 2 +y+14( 17 y 3 +34 y 2 51y+68 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa aaleqabaGaaGinaaaakiabgkHiTiaaigdacaaIYaGaamyEamaaCaaa leqabaGaaGOmaaaakiabgUcaRiaadMhacqGHRaWkcaaIXaGaaGinai abgkHiTmaabmaabaGaaGymaiaaiEdacaWG5bWaaWbaaSqabeaacaaI ZaaaaOGaey4kaSIaaG4maiaaisdacaWG5bWaaWbaaSqabeaacaaIYa aaaOGaeyOeI0IaaGynaiaaigdacaWG5bGaey4kaSIaaGOnaiaaiIda aiaawIcacaGLPaaaaaa@5057@

= y 4 12 y 2 +y+1417 y 3 34 y 2 +51y68 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam yEamaaCaaaleqabaGaaGinaaaakiabgkHiTiaaigdacaaIYaGaamyE amaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadMhacqGHRaWkcaaIXa GaaGinaiabgkHiTiaaigdacaaI3aGaamyEamaaCaaaleqabaGaaG4m aaaakiabgkHiTiaaiodacaaI0aGaamyEamaaCaaaleqabaGaaGOmaa aakiabgUcaRiaaiwdacaaIXaGaamyEaiabgkHiTiaaiAdacaaI4aaa aa@4FDF@

By combining like terms,

= y 4 12 y 2 34 y 2 +y+51y+146817 y 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam yEamaaCaaaleqabaGaaGinaaaakiabgkHiTiaaigdacaaIYaGaamyE amaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaiodacaaI0aGaamyEam aaCaaaleqabaGaaGOmaaaakiabgUcaRiaadMhacqGHRaWkcaaI1aGa aGymaiaadMhacqGHRaWkcaaIXaGaaGinaiabgkHiTiaaiAdacaaI4a GaeyOeI0IaaGymaiaaiEdacaWG5bWaaWbaaSqabeaacaaIZaaaaaaa @4FD5@

= y 4 46 y 2 +52y17 y 3 54 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam yEamaaCaaaleqabaGaaGinaaaakiabgkHiTiaaisdacaaI2aGaamyE amaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdacaaIYaGaamyEai abgkHiTiaaigdacaaI3aGaamyEamaaCaaaleqabaGaaG4maaaakiab gkHiTiaaiwdacaaI0aaaaa@474E@

= y 4 17 y 3 46 y 2 +52y54 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam yEamaaCaaaleqabaGaaGinaaaakiabgkHiTiaaigdacaaI3aGaamyE amaaCaaaleqabaGaaG4maaaakiabgkHiTiaaisdacaaI2aGaamyEam aaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdacaaIYaGaamyEaiab gkHiTiaaiwdacaaI0aaaaa@474E@

So, y 4 12 y 2 +y+14 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa aaleqabaGaaGinaaaakiabgkHiTiaaigdacaaIYaGaamyEamaaCaaa leqabaGaaGOmaaaakiabgUcaRiaadMhacqGHRaWkcaaIXaGaaGinaa aa@405C@  is y 4 17 y 3 46 y 2 +52y54 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa aaleqabaGaaGinaaaakiabgkHiTiaaigdacaaI3aGaamyEamaaCaaa leqabaGaaG4maaaakiabgkHiTiaaisdacaaI2aGaamyEamaaCaaale qabaGaaGOmaaaakiabgUcaRiaaiwdacaaIYaGaamyEaiabgkHiTiaa iwdacaaI0aaaaa@4648@  greater than 17 y 3 +34 y 2 51y+68 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiE dacaWG5bWaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaG4maiaaisda caWG5bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGynaiaaigdaca WG5bGaey4kaSIaaGOnaiaaiIdaaaa@435E@

Question: 63

How much does 93 p 2 55p+4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaaio dacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGynaiaaiwda caWGWbGaey4kaSIaaGinaaaa@3E40@  exceed 13 p 3 5 p 2 +17p90? MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaio dacaWGWbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGynaiaadcha daahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaG4naiaadchacq GHsislcaaMc8UaaGPaVlaaiMdacaaIWaGaai4paaaa@4663@

Solution

Required expression is

93 p 2 55p+4( 13 p 3 5 p 2 +17p90 ) =93 p 2 55p+413 p 3 +5 p 2 17p+90 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaI5a GaaG4maiaadchadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI1aGa aGynaiaadchacqGHRaWkcaaI0aGaeyOeI0YaaeWaaeaacaaIXaGaaG 4maiaadchadaahaaWcbeqaaiaaiodaaaGccqGHsislcaaI1aGaamiC amaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdacaaI3aGaamiCai abgkHiTiaaiMdacaaIWaaacaGLOaGaayzkaaaabaGaeyypa0JaaGyo aiaaiodacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGynai aaiwdacaWGWbGaey4kaSIaaGinaiabgkHiTiaaigdacaaIZaGaamiC amaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiwdacaWGWbWaaWbaaS qabeaacaaIYaaaaOGaeyOeI0IaaGymaiaaiEdacaWGWbGaey4kaSIa aGyoaiaaicdaaaaa@6472@

By combining the like terms,

=93 p 2 +5 p 2 55p17p+4+9013 p 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG yoaiaaiodacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGyn aiaadchadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI1aGaaGynai aadchacqGHsislcaaIXaGaaG4naiaadchacqGHRaWkcaaI0aGaey4k aSIaaGyoaiaaicdacqGHsislcaaIXaGaaG4maiaadchadaahaaWcbe qaaiaaiodaaaaaaa@4CD1@

=98 p 2 72p+9413 p 3 =13 p 3 +98 p 2 72p+94 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaI5aGaaGioaiaadchadaahaaWcbeqaaiaaikdaaaGccqGHsisl caaI3aGaaGOmaiaadchacqGHRaWkcaaI5aGaaGinaiabgkHiTiaaig dacaaIZaGaamiCamaaCaaaleqabaGaaG4maaaaaOqaaiabg2da9iab gkHiTiaaigdacaaIZaGaamiCamaaCaaaleqabaGaaG4maaaakiabgU caRiaaiMdacaaI4aGaamiCamaaCaaaleqabaGaaGOmaaaakiabgkHi TiaaiEdacaaIYaGaamiCaiabgUcaRiaaiMdacaaI0aaaaaa@53C8@

So, 93 p 2 55p+4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaaio dacaWGWbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGynaiaaiwda caWGWbGaey4kaSIaaGinaaaa@3E41@  is 13 p 3 +9 p 2 72p+94 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiaaiodacaWGWbWaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaGyo aiaadchadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI3aGaaGOmai aadchacqGHRaWkcaaI5aGaaGinaaaa@4376@

more from 13 p 3 5 p 2 +17p90. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaio dacaWGWbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaGynaiaadcha daahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaG4naiaadchacq GHsislcaaI5aGaaGimaiaac6caaaa@433D@

Question: 64

To what expression must 99 x 3 33 x 2 13x41 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaaiM dacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaG4maiaaioda caWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGymaiaaiodaca WG4bGaeyOeI0IaaGinaiaaigdaaaa@436E@  be added to make the sum zero?

Solution

In order to find the solution, we will subtract 99 x 3 33 x 2 13x41 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaaiM dacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaG4maiaaioda caWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGymaiaaiodaca WG4bGaeyOeI0IaaGinaiaaigdaaaa@436F@  from 0.

Required expression is

0( 99 x 3 33 x 2 13x41 ) =099 x 3 +33 x 2 +13x+41 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaIWa GaeyOeI0YaaeWaaeaacaaI5aGaaGyoaiaadIhadaahaaWcbeqaaiaa iodaaaGccqGHsislcaaIZaGaaG4maiaadIhadaahaaWcbeqaaiaaik daaaGccqGHsislcaaIXaGaaG4maiaadIhacqGHsislcaaI0aGaaGym aaGaayjkaiaawMcaaaqaaiabg2da9iaaicdacqGHsislcaaI5aGaaG yoaiaadIhadaahaaWcbeqaaiaaiodaaaGccqGHRaWkcaaIZaGaaG4m aiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaG4mai aadIhacqGHRaWkcaaI0aGaaGymaaaaaa@56C8@

So, if we add 99 x 3 +33 x 2 +13x+41 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG yoaiaaiMdacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaG4m aiaaiodacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymai aaiodacaWG4bGaey4kaSIaaGinaiaaigdaaaa@443B@  to 99 x 3 33 x 2 13x41 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaaiM dacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaG4maiaaioda caWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGymaiaaiodaca WG4bGaeyOeI0IaaGinaiaaigdaaaa@436F@ , then the sum is zero.

Question: 65

Subtract 9 a 2 15a+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaadg gadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIXaGaaGynaiaadgga cqGHRaWkcaaIZaaaaa@3D60@  from unity.

Solution

In order to find the solution, we will subtract 9 a 2 15a+3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaadg gadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIXaGaaGynaiaadgga cqGHRaWkcaaIZaaaaa@3D61@  from unity, i.e. 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@ .

Required expression is 1( 9 a 2 15a+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk HiTmaabmaabaGaaGyoaiaadggadaahaaWcbeqaaiaaikdaaaGccqGH sislcaaIXaGaaGynaiaadggacqGHRaWkcaaIZaaacaGLOaGaayzkaa aaaa@4092@

=19 a 2 +15a3 =9 a 2 +15a2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaIXaGaeyOeI0IaaGyoaiaadggadaahaaWcbeqaaiaaikdaaaGc cqGHRaWkcaaIXaGaaGynaiaadggacqGHsislcaaIZaaabaGaeyypa0 JaeyOeI0IaaGyoaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHRaWk caaIXaGaaGynaiaadggacqGHsislcaaIYaaaaaa@498F@

Question: 66

Find the values of the following polynomials at a= 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 da9iabgkHiTiaabccacaaIYaaaaa@3A10@  and b=3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg2 da9iaaiodaaaa@3882@ :

a.    a 2 +2ab+ b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWGHbGaamOyaiabgUca RiaadkgadaahaaWcbeqaaiaaikdaaaaaaa@3DCE@

b.   a 2 2ab+ b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaiGjGikdaaaGccqGHsislcaaIYaGaamyyaiaadkgacqGH RaWkcaWGIbWaaWbaaSqabeaacaaIYaaaaaaa@3EF9@

c.    a 3 +3 a 2 b+3a b 2 + b 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaiGjGiodaaaGccqGHRaWkcaaIZaGaamyyamaaCaaaleqa baGaiGjGikdaaaGccaWGIbGaey4kaSIaaG4maiaadggacaWGIbWaaW baaSqabeaacGaMaIOmaaaakiabgUcaRiaadkgadaahaaWcbeqaaiaa iodaaaaaaa@4683@

d.   a 3 3 a 2 b+3a b 2 b 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaG4maaaakiabgkHiTiaaiodacaWGHbWaaWbaaSqabeaa cGaMaIOmaaaakiaadkgacqGHRaWkcaaIZaGaamyyaiaadkgadaahaa WcbeqaaiaaikdaaaGccqGHsislcaWGIbWaaWbaaSqabeaacaaIZaaa aaaa@4459@

e.    a 2 + b 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqa baGaaGOmaaaaaOqaaiaaiodaaaaaaa@3B3A@

f. a 2 b 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamOyamaaCaaaleqa baGaaGOmaaaaaOqaaiaaiodaaaaaaa@3B45@

g.   a b + b a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbaabaGaamOyaaaacqGHRaWkdaWcaaqaaiaadkgaaeaacaWGHbaa aaaa@3A74@

h.   a 2 + b 2 ab b 2 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaadkgadaahaaWcbeqaaiacyciI YaaaaOGaeyOeI0IaamyyaiaadkgacqGHsislcaWGIbWaaWbaaSqabe aacaaIYaaaaOGaeyOeI0IaamyyamaaCaaaleqabaGaaGOmaaaaaaa@43CA@

Solution

Given, a=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 da9iabgkHiTiaaikdaaaa@396E@  and MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35D8@ b=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg2 da9iaaiodaaaa@3883@

Put a=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 da9iabgkHiTiaaikdaaaa@396E@  and b=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg2 da9iaaiodaaaa@3883@  in the given expressions, we get

a.    a 2 +2ab+ b 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWGHbGaamOyaiabgUca RiaadkgadaahaaWcbeqaaiaaikdaaaaaaa@3DCF@

         = ( 2 ) 2 +2( 2 )( 3 )+ ( 3 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacqGHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI YaaaaOGaey4kaSIaaGOmamaabmaabaGaeyOeI0IaaGOmaaGaayjkai aawMcaamaabmaabaGaaG4maaGaayjkaiaawMcaaiabgUcaRmaabmaa baGaaG4maaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaa@462B@

        =412+9 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaI0aGaeyOeI0IaaGymaiaaikdacqGHRaWkcaaI5aaabaGaeyyp a0JaaGymaaaaaa@3D6D@

b.   a 2 2ab+ b 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGOmaaaakiabgkHiTiaaikdacaWGHbGaamOyaiabgUca RiaadkgadaahaaWcbeqaaiaaikdaaaaaaa@3DDA@

        = ( 2 ) 2 2( 2 )( 3 )+ ( 3 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacqGHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI YaaaaOGaeyOeI0IaaGOmamaabmaabaGaeyOeI0IaaGOmaaGaayjkai aawMcaamaabmaabaGaaG4maaGaayjkaiaawMcaaiabgUcaRmaabmaa baGaaG4maaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaa@4636@

        =4+12+9 =25 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaI0aGaey4kaSIaaGymaiaaikdacqGHRaWkcaaI5aaabaGaeyyp a0JaaGOmaiaaiwdaaaaa@3E23@

c.    a 3 +3 a 2 b+3a b 2 + b 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaG4maaaakiabgUcaRiaaiodacaWGHbWaaWbaaSqabeaa caaIYaaaaOGaamOyaiabgUcaRiaaiodacaWGHbGaamOyamaaCaaale qabaGaaGOmaaaakiabgUcaRiaadkgadaahaaWcbeqaaiaaiodaaaaa aa@4324@

        = ( 2 ) 3 +3 ( 2 ) 2 ( 3 )+3( 2 ) ( 3 ) 2 + ( 3 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacqGHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI ZaaaaOGaey4kaSIaaG4mamaabmaabaGaeyOeI0IaaGOmaaGaayjkai aawMcaamaaCaaaleqabaGaaGOmaaaakmaabmaabaGaaG4maaGaayjk aiaawMcaaiabgUcaRiaaiodadaqadaqaaiabgkHiTiaaikdaaiaawI cacaGLPaaadaqadaqaaiaaiodaaiaawIcacaGLPaaadaahaaWcbeqa aiaaikdaaaGccqGHRaWkdaqadaqaaiaaiodaaiaawIcacaGLPaaada ahaaWcbeqaaiaaiodaaaaaaa@4F2B@

         8+3654+27 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGHsi slcaqGGaGaaGioaiabgUcaRiaaiodacaaI2aGaeyOeI0IaaGynaiaa isdacqGHRaWkcaaIYaGaaG4naaqaaiabg2da9iaaigdaaaaa@411B@

d.   a 3 3 a 2 b+3a b 2 b 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaG4maaaakiabgkHiTiaaiodacaWGHbWaaWbaaSqabeaa caaIYaaaaOGaamOyaiabgUcaRiaaiodacaWGHbGaamOyamaaCaaale qabaGaaGOmaaaakiabgkHiTiaadkgadaahaaWcbeqaaiaaiodaaaaa aa@433A@

        = ( 2 ) 3 3 ( 2 ) 2 ( 3 )+3( 2 ) ( 3 ) 2 ( 3 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacqGHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI ZaaaaOGaeyOeI0IaaG4mamaabmaabaGaeyOeI0IaaGOmaaGaayjkai aawMcaamaaCaaaleqabaGaaGOmaaaakmaabmaabaGaaG4maaGaayjk aiaawMcaaiabgUcaRiaaiodadaqadaqaaiabgkHiTiaaikdaaiaawI cacaGLPaaadaqadaqaaiaaiodaaiaawIcacaGLPaaadaahaaWcbeqa aiaaikdaaaGccqGHsisldaqadaqaaiaaiodaaiaawIcacaGLPaaada ahaaWcbeqaaiaaiodaaaaaaa@4F41@

        =8365427 =125 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcqGHsislcaaI4aGaeyOeI0IaaG4maiaaiAdacqGHsislcaaI1aGa aGinaiabgkHiTiaaikdacaaI3aaabaGaeyypa0JaeyOeI0IaaGymai aaikdacaaI1aaaaaa@43FC@

e.    a 2 + b 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaaleqa baGaaGOmaaaaaOqaaiaaiodaaaaaaa@3B3B@

        = ( 2 ) 2 + ( 3 ) 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaadaqadaqaaiabgkHiTiaaikdaaiaawIcacaGLPaaadaahaaWc beqaaiaaikdaaaGccqGHRaWkdaqadaqaaiaaiodaaiaawIcacaGLPa aadaahaaWcbeqaaiaaikdaaaaakeaacaaIZaaaaaaa@3FEC@

        = 4+9 3 = 13 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaaisdacqGHRaWkcaaI5aaabaGaaG4maaaaaeaacqGH 9aqpdaWcaaqaaiaaigdacaaIZaaabaGaaG4maaaaaaaa@3D60@

f. a 2 b 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamOyamaaCaaaleqa baGaaGOmaaaaaOqaaiaaiodaaaaaaa@3B46@

= ( 2 ) 2 ( 3 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaS aaaeaadaqadaqaaiabgkHiTiaaikdaaiaawIcacaGLPaaadaahaaWc beqaaiaaikdaaaGccqGHsisldaqadaqaaiaaiodaaiaawIcacaGLPa aaaeaacaaIZaaaaaaa@3F04@

= 49 3 = 5 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaaisdacqGHsislcaaI5aaabaGaaG4maaaaaeaacqGH 9aqpcqGHsisldaWcaaqaaiaaiwdaaeaacaaIZaaaaaaaaa@3D9F@

g.   a b + b a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbaabaGaamOyaaaacqGHRaWkdaWcaaqaaiaadkgaaeaacaWGHbaa aaaa@3A75@

= 2 3 + 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaey OeI0YaaSaaaeaacaaIYaaabaGaaG4maaaacqGHRaWkdaWcaaqaaiaa iodaaeaacqGHsislcaaIYaaaaaaa@3CAD@

= 2 3 3 2 = 49 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcqGHsisldaWcaaqaaiaaikdaaeaacaaIZaaaaiabgkHiTmaalaaa baGaaG4maaqaaiaaikdaaaaabaGaeyypa0ZaaSaaaeaacqGHsislca aI0aGaeyOeI0IaaGyoaaqaaiaaiAdaaaaaaaa@4103@

= 13 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaey OeI0YaaSaaaeaacaaIXaGaaG4maaqaaiaaiAdaaaaaaa@3A14@

h.   a 2 + b 2 ab b 2 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaadkgadaahaaWcbeqaaiaaikda aaGccqGHsislcaWGHbGaamOyaiabgkHiTiaadkgadaahaaWcbeqaai aaikdaaaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIYaaaaaaa@42AB@

= ( 2 ) 2 + ( 3 ) 2 ( 2 )( 3 ) ( 2 ) 2 ( 3 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacqGHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI YaaaaOGaey4kaSYaaeWaaeaacaaIZaaacaGLOaGaayzkaaWaaWbaaS qabeaacaaIYaaaaOGaeyOeI0YaaeWaaeaacqGHsislcaaIYaaacaGL OaGaayzkaaWaaeWaaeaacaaIZaaacaGLOaGaayzkaaGaeyOeI0Yaae WaaeaacqGHsislcaaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaI YaaaaOGaeyOeI0YaaeWaaeaacaaIZaaacaGLOaGaayzkaaWaaWbaaS qabeaacaaIYaaaaaaa@4EB2@

=4+9+694 =6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaI0aGaey4kaSIaaGyoaiabgUcaRiaaiAdacqGHsislcaaI5aGa eyOeI0IaaGinaaqaaiabg2da9iaaiAdaaaaa@400C@

Question: 67

Find the values of following polynomials at m=1, n=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2 da9iaaigdacaGGSaGaaeiiaiaad6gacqGH9aqpcqGHsislcaaIXaaa aa@3D7F@  and p=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9iaaikdaaaa@388F@ :

a.    m+n+p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabgU caRiaad6gacqGHRaWkcaWGWbaaaa@3A76@

b.   m 2 + n 2 + p 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa aaleqabaGaiGjGikdaaaGccqGHRaWkcaWGUbWaaWbaaSqabeaacGaM aIOmaaaakiabgUcaRiaadchadaahaaWcbeqaaiaaikdaaaaaaa@3F85@

c.    m 3 + n 3 + p 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa aaleqabaGaaG4maaaakiabgUcaRiaad6gadaahaaWcbeqaaiacyciI ZaaaaOGaey4kaSIaamiCamaaCaaaleqabaGaaG4maaaaaaa@3E68@

d.   mn+np+pm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaad6 gacqGHRaWkcaWGUbGaamiCaiabgUcaRiaadchacaWGTbaaaa@3D50@

e.    m 3 + n 3 + p 3 3mnp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa aaleqabaGaiGjGiodaaaGccqGHRaWkcaWGUbWaaWbaaSqabeaacGaM aI4maaaakiabgUcaRiaadchadaahaaWcbeqaaiaaiodaaaGccqGHsi slcaaIZaGaamyBaiaad6gacaWGWbaaaa@4416@

f. m 2 n 2 + n 2 p 2 + p 2 m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa aaleqabaGaaGOmaaaakiaad6gadaahaaWcbeqaaiaaikdaaaGccqGH RaWkcaWGUbWaaWbaaSqabeaacaaIYaaaaOGaamiCamaaCaaaleqaba GaaGOmaaaakiabgUcaRiaadchadaahaaWcbeqaaiaaikdaaaGccaWG TbWaaWbaaSqabeaacaaIYaaaaaaa@42F8@

Solution

Given, m=1,n=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2 da9iaaigdacaGGSaGaaGPaVlaad6gacqGH9aqpcqGHsislcaaIXaaa aa@3E68@  

and p=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9iaaikdaaaa@3890@

Put m=1,n=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2 da9iaaigdacaGGSaGaaGPaVlaaykW7caWGUbGaeyypa0JaeyOeI0Ia aGymaaaa@3FF3@  

and p=2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9iaaikdaaaa@3890@  in the expression, we get

a.    m+n+p =11+2 =2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaWGTb Gaey4kaSIaamOBaiabgUcaRiaadchaaeaacqGH9aqpcaaIXaGaeyOe I0IaaGymaiabgUcaRiaaikdaaeaacqGH9aqpcaaIYaaaaaa@4147@

b.   m 2 + n 2 + p 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaad6gadaahaaWcbeqaaiaaikda aaGccqGHRaWkcaWGWbWaaWbaaSqabeaacaaIYaaaaaaa@3D46@

= ( 1 ) 2 + ( 1 ) 2 +( 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa ey4kaSYaaeWaaeaacqGHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaS qabeaacaaIYaaaaOGaey4kaSYaaeWaaeaacaaIYaaacaGLOaGaayzk aaaaaa@4243@

=1+1+4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiabgUcaRiaaigdacqGHRaWkcaaI0aaaaa@3AD7@

c.    m 3 + n 3 + p 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa aaleqabaGaaG4maaaakiabgUcaRiaad6gadaahaaWcbeqaaiaaioda aaGccqGHRaWkcaWGWbWaaWbaaSqabeaacaaIZaaaaaaa@3D49@

= ( 1 ) 3 + ( 1 ) 3 + ( 2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaOGa ey4kaSYaaeWaaeaacqGHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaS qabeaacaaIZaaaaOGaey4kaSYaaeWaaeaacaaIYaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaIZaaaaaaa@432F@

=11+8=8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiabgkHiTiaaigdacqGHRaWkcaaI4aGaeyypa0JaaGioaaaa@3CAE@

d.   mn+np+pm MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaad6 gacqGHRaWkcaWGUbGaamiCaiabgUcaRiaadchacaWGTbaaaa@3D51@

=( 1 )( 1 )+( 1 )( 2 )+( 2 )( 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacaaIXaaacaGLOaGaayzkaaWaaeWaaeaacqGHsislcaaIXaaa caGLOaGaayzkaaGaey4kaSYaaeWaaeaacqGHsislcaaIXaaacaGLOa GaayzkaaWaaeWaaeaacaaIYaaacaGLOaGaayzkaaGaey4kaSYaaeWa aeaacaaIYaaacaGLOaGaayzkaaWaaeWaaeaacaaIXaaacaGLOaGaay zkaaaaaa@4817@

=12+2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcqGHsislcaaIXaGaeyOeI0IaaGOmaiabgUcaRiaaikdaaeaacqGH 9aqpcqGHsislcaaIXaaaaaa@3E82@

e.    m 3 + n 3 + p 3 3mnp MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa aaleqabaGaaG4maaaakiabgUcaRiaad6gadaahaaWcbeqaaiaaioda aaGccqGHRaWkcaWGWbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaG 4maiaad2gacaWGUbGaamiCaaaa@41D7@

= ( 1 ) 3 + ( 1 ) 3 + ( 2 ) 3 3( 1 )( 1 )( 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaOGa ey4kaSYaaeWaaeaacqGHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaS qabeaacaaIZaaaaOGaey4kaSYaaeWaaeaacaaIYaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaaG4mamaabmaabaGaaG ymaaGaayjkaiaawMcaamaabmaabaGaeyOeI0IaaGymaaGaayjkaiaa wMcaamaabmaabaGaaGOmaaGaayjkaiaawMcaaaaa@4C9D@

=11+8+6 =14 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaIXaGaeyOeI0IaaGymaiabgUcaRiaaiIdacqGHRaWkcaaI2aaa baGaeyypa0JaaGymaiaaisdaaaaa@3F0D@

f. m 2 n 2 + n 2 p 2 + p 2 n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaCa aaleqabaGaaGOmaaaakiaad6gadaahaaWcbeqaaiaaikdaaaGccqGH RaWkcaWGUbWaaWbaaSqabeaacaaIYaaaaOGaamiCamaaCaaaleqaba GaaGOmaaaakiabgUcaRiaadchadaahaaWcbeqaaiaaikdaaaGccaWG UbWaaWbaaSqabeaacaaIYaaaaaaa@42FA@

= ( 1 ) 2 ( 1 ) 2 + ( 1 ) 2 ( 2 ) 2 + ( 2 ) 2 ( 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOWa aeWaaeaacqGHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaaca aIYaaaaOGaey4kaSYaaeWaaeaacqGHsislcaaIXaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIYaaacaGLOaGaay zkaaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaeWaaeaacaaIYaaa caGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIXa aacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@4DBF@

=1+4+4 =9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaIXaGaey4kaSIaaGinaiabgUcaRiaaisdaaeaacqGH9aqpcaaI 5aaaaaa@3CA9@

Question: 68

If A=3 x 2 4x+1,  B=5 x 2 +3x8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaqGbb Gaeyypa0JaaG4maiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHsisl caaI0aGaamiEaiabgUcaRiaaigdacaGGSaGaaeiiaaqaaiaabkeacq GH9aqpcaaI1aGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaa iodacaWG4bGaeyOeI0IaaGioaaaaaa@48B2@  

and C=4 x 2 7x+3, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4qaiabg2 da9iaaisdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaG4n aiaadIhacqGHRaWkcaaIZaGaaiilaaaa@3F4C@  then find:

1.   (A + B) - C

2.   B + C - A

3.   A + B + C

Solution

Given, A=3 x 2 4x+1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiabg2 da9iaaiodacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaai4eGiaaisda caWG4bGaey4kaSIaaGymaiaacYcaaaa@3F0E@

B=5 x 2 +3x8  and C=4 x 2 7x+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceiqaaeaaCiqaai aabkeacqGH9aqpcaaI1aGaamiEamaaCaaaleqabaGaaGOmaaaakiab gUcaRiaaiodacaWG4bGaai4eGiaaiIdacaqGGaaabaGaaeyyaiaab6 gacaqGKbGaaeiiaiaaboeacqGH9aqpcaaI0aGaamiEamaaCaaaleqa baGaaGOmaaaakiaacobicaaI3aGaamiEaiabgUcaRiaaiodaaaaa@4B1E@

1.   (A+B)C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeikaiaabg eacqGHRaWkcaqGcbGaaeykaiabgkHiTiaaboeaaaa@3B4E@

=(3 x 2 4x+1+5 x 2 +3x8)(4 x 2 7x+3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaiodacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGin aiaadIhacqGHRaWkcaaIXaGaey4kaSIaaGynaiaadIhadaahaaWcbe qaaiaaikdaaaGccqGHRaWkcaaIZaGaamiEaiabgkHiTiaaiIdacaGG PaGaeyOeI0IaaiikaiaaisdacaWG4bWaaWbaaSqabeaacaaIYaaaaO GaeyOeI0IaaG4naiaadIhacqGHRaWkcaaIZaGaaiykaaaa@5043@

By combining the like terms,

=( 3 x 2 +5 x 2 4x+3x+18 )( 4 x 2 7x+3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Zaae WaaeaacaaIZaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaa iwdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaai4eGiaaisdacaWG4b Gaey4kaSIaaG4maiaadIhacqGHRaWkcaaIXaGaai4eGiaaiIdaaiaa wIcacaGLPaaacaGGtaYaaeWaaeaacaaI0aGaamiEamaaCaaaleqaba GaaGOmaaaakiaacobicaaI3aGaamiEaiabgUcaRiaaiodaaiaawIca caGLPaaaaaa@4FCB@

 =(8 x 2 x7)(4 x 2 7x+3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiOaiabg2 da9iaacIcacaaI4aGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHi TiaadIhacqGHsislcaaI3aGaaiykaiabgkHiTiaacIcacaaI0aGaam iEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaiEdacaWG4bGaey4k aSIaaG4maiaacMcaaaa@48E3@

=8 x 2 x74 x 2 +7x3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG4bGaeyOe I0IaaG4naiabgkHiTiaaisdacaWG4bWaaWbaaSqabeaacaaIYaaaaO Gaey4kaSIaaG4naiaadIhacqGHsislcaaIZaaaaa@450D@

=8 x 2 4 x 2 x+7x73  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI0aGaamiE amaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadIhacqGHRaWkcaaI3a GaamiEaiabgkHiTiaaiEdacqGHsislcaaIZaGaaiiOaaaa@4631@

=4 x 2 +6x10 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG inaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI2aGaamiE aiabgkHiTiaaigdacaaIWaaaaa@3E8D@

2.   B+CA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOqaiabgU caRiaaboeacqGHsislcaqGbbaaaa@39F6@

=5 x 2 +3x8+4 x 2 7x+3(3 x 2 4x+1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ynaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaamiE aiabgkHiTiaaiIdacqGHRaWkcaaI0aGaamiEamaaCaaaleqabaGaaG OmaaaakiabgkHiTiaaiEdacaWG4bGaey4kaSIaaG4maiabgkHiTiaa cIcacaaIZaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaais dacaWG4bGaey4kaSIaaGymaiaacMcaaaa@4EEA@

By combining the like terms, 

=(5 x 2 +4 x 2 +3x7x8+3)( 3 x 2 4x+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaiwdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGin aiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaamiEai abgkHiTiaaiEdacaWG4bGaeyOeI0IaaGioaiabgUcaRiaaiodacaGG PaGaeyOeI0YaaeWaaeaacaaIZaGaamiEamaaCaaaleqabaGaaGOmaa aakiaacobicaaI0aGaamiEaiabgUcaRiaaigdaaiaawIcacaGLPaaa aaa@503D@

=(9 x 2 4x5)( 3 x 2 4x+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaiMdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGin aiaadIhacqGHsislcaaI1aGaaiykaiabgkHiTmaabmaabaGaaG4mai aadIhadaahaaWcbeqaaiaaikdaaaGccaGGtaIaaGinaiaadIhacqGH RaWkcaaIXaaacaGLOaGaayzkaaaaaa@4870@

=9 x 2 4x53 x 2 +4x1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG yoaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI0aGaamiE aiabgkHiTiaaiwdacqGHsislcaaIZaGaamiEamaaCaaaleqabaGaaG OmaaaakiabgUcaRiaaisdacaWG4bGaeyOeI0IaaGymaaaa@45C4@

=9 x 2 3 x 2 4x+4x51 =6 x 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaI5aGaamiEamaaCaaaleqabaGaaGOmaaaakiaacobicaaIZaGa amiEamaaCaaaleqabaGaiWiGikdaaaGccaGGtaIaaGinaiaadIhacq GHRaWkcaaI0aGaamiEaiaacobicaaI1aGaeyOeI0IaaGymaaqaaiab g2da9iaaiAdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaG Onaaaaaa@4B9F@

3.   A+B+C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiabgU caRiaabkeacqGHRaWkcaqGdbaaaa@39EB@

=3 x 2 4x+1+5 x 2 +3x8+4 x 2 7x+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4maiaadIhadaahaaWcbeqaaiacmciIYaaaaOGaeyOeI0IaaGinaiaa dIhacqGHRaWkcaaIXaGaey4kaSIaaGynaiaadIhadaahaaWcbeqaai aaikdaaaGccqGHRaWkcaaIZaGaamiEaiabgkHiTiaaiIdacqGHRaWk caaI0aGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaiEdaca WG4bGaey4kaSIaaG4maaaa@4E9A@

By combining the like terms, 

=3 x 2 +5 x 2 +4 x 2 4x+3x7x+18+3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4maiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI1aGaamiE amaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaisdacaWG4bWaaWbaaS qabeaacaaIYaaaaOGaeyOeI0IaaGinaiaadIhacqGHRaWkcaaIZaGa amiEaiabgkHiTiaaiEdacaWG4bGaey4kaSIaaGymaiaacobicaaI4a Gaey4kaSIaaG4maaaa@4D50@

=12 x 2 8x4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiaaikdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGio aiaadIhacqGHsislcaaI0aaaaa@3E9C@

Question: 69

If =(x2), Q =2(y+1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuaiaabc cacqGH9aqpcqGHsislcaGGOaGaamiEaiabgkHiTiaaikdacaGGPaGa aiilaiaabccacaqGrbGaaeiiaiabg2da9iabgkHiTiaaikdacaGGOa GaamyEaiabgUcaRiaaigdacaGGPaaaaa@46AD@  and

=x+2y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOuaiaabc cacqGH9aqpcqGHsislcaWG4bGaey4kaSIaaGOmaiaadMhaaaa@3CDC@ , find a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36BE@ , when P+Q+R=ax. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuaiabgU caRiaabgfacqGHRaWkcaqGsbGaeyypa0JaamyyaiaadIhacaGGUaaa aa@3DB3@

Solution

Given, P=(x2),  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuaiabg2 da9iabgkHiTiaacIcacaWG4bGaeyOeI0IaaGOmaiaacMcacaGGSaGa aeiiaaaa@3DF0@

Q=2( y+1 )   and R=x+2y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGrb Gaeyypa0JaeyOeI0IaaGOmamaabmaabaGaamyEaiabgUcaRiaaigda aiaawIcacaGLPaaacaqGGaGaaeiiaaqaaiaabggacaqGUbGaaeizai aabccacaqGsbGaeyypa0JaeyOeI0IaamiEaiabgUcaRiaaikdacaWG 5bGaaGjbVdaaaa@4A19@

Also given, P+Q+R=ax MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiuaiabgU caRiaabgfacqGHRaWkcaqGsbGaeyypa0JaamyyaiaadIhaaaa@3D01@   

Put the values of P, Q and R on LHS, we get 

(x2)+[2( y+1 )]+(x+2y)=ax MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaai ikaiaadIhacqGHsislcaaIYaGaaiykaiabgUcaRiaacUfacqGHsisl caaIYaWaaeWaaeaacaWG5bGaey4kaSIaaGymaaGaayjkaiaawMcaai aac2facqGHRaWkcaGGOaGaeyOeI0IaamiEaiabgUcaRiaaikdacaWG 5bGaaiykaiabg2da9iaadggacaWG4baaaa@4CDD@

x+2+(2y2)x+2y=ax MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaam iEaiabgUcaRiaaikdacqGHRaWkcaGGOaGaeyOeI0IaaGOmaiaadMha cqGHsislcaaIYaGaaiykaiabgkHiTiaadIhacqGHRaWkcaaIYaGaam yEaiabg2da9iaadggacaWG4baaaa@475A@

x+22y2x+2y=ax  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaam iEaiabgUcaRiaaikdacaGGtaIaaGOmaiaadMhacaGGtaIaaGOmaiaa cobicaWG4bGaey4kaSIaaGOmaiaadMhacqGH9aqpcaWGHbGaamiEai aacckaaaa@45A1@

By combining the like terms, 

xx2y+2y+22=ax 2x=ax MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGHsi slcaWG4bGaai4eGiaadIhacaGGtaIaaGOmaiaadMhacqGHRaWkcaaI YaGaamyEaiabgUcaRiaaikdacaGGtaIaaGOmaiabg2da9iaadggaca WG4baabaGaeyOeI0IaaGOmaiaadIhacqGH9aqpcaWGHbGaamiEaaaa aa@4A13@

By comparing LHS and RHS, we get a=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 da9iabgkHiTiaaikdaaaa@396D@   

Question: 70

From the sum of x 2 y 2 1,  y 2 x 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiabgkHiTiaadMhadaahaaWcbeqaaiaaikda aaGccqGHsislcaaIXaGaaiilaiaabccacaWG5bWaaWbaaSqabeaaca aIYaaaaOGaeyOeI0IaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHi Tiaaigdaaaa@4417@  and 1 x 2 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk HiTiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG5bWaaWba aSqabeaacaaIYaaaaaaa@3C44@  subtract  (1+ y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaae iiaiaacIcacaaIXaGaey4kaSIaamyEamaaCaaaleqabaGaaGOmaaaa kiaacMcaaaa@3C4F@ .

Solution

Sum of x 2 y 2 1, y 2 x 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGOmaaaakiabgkHiTiaadMhadaahaaWcbeqaaiaaikda aaGccqGHsislcaaIXaGaaiilaiaadMhadaahaaWcbeqaaiaaikdaaa GccqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGym aaaa@4374@  and  1 x 2 y 2 = x 2 y 2 1+ y 2 x 2 1+1 x 2 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacaaIXa GaeyOeI0IaamiEamaaCaaaleqabaGaiWiGikdaaaGccqGHsislcaWG 5bWaaWbaaSqabeaacaaIYaaaaaGcbaGaeyypa0JaamiEamaaCaaale qabaGaiWiGikdaaaGccqGHsislcaWG5bWaaWbaaSqabeaacaaIYaaa aOGaeyOeI0IaaGymaiabgUcaRiaadMhadaahaaWcbeqaaiaaikdaaa GccqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGym aiabgUcaRiaaigdacqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaO GaeyOeI0IaamyEamaaCaaaleqabaGaaGOmaaaaaaaa@549E@

By combining like terms,

= x 2 x 2 x 2 y 2 + y 2 y 2 11+1 = x 2 y 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaai4eGiaadIhadaahaaWc beqaaiaaikdaaaGccaGGtaIaamiEamaaCaaaleqabaGaaGOmaaaaki aacobicaWG5bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyEamaa CaaaleqabaGaaGOmaaaakiaacobicaWG5bWaaWbaaSqabeaacaaIYa aaaOGaai4eGiaaigdacaGGtaIaaGymaiabgUcaRiaaigdaaeaacqGH 9aqpcqGHsislcaWG4bWaaWbaaSqabeaacaaIYaaaaOGaai4eGiaadM hadaahaaWcbeqaaiaaikdaaaGccaGGtaIaaGymaaaaaa@52C3@

Now, subtract ( 1+ y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4eGmaabm aabaGaaGymaiabgUcaRiaadMhadaahaaWcbeqaaiaaikdaaaaakiaa wIcacaGLPaaaaaa@3BA6@  from x 2 y 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4eGiaadI hadaahaaWcbeqaaiaaikdaaaGccaGGtaIaamyEamaaCaaaleqabaGa iWiGikdaaaGccqGHsislcaaIXaaaaa@3DE3@

= x 2 y 2 1[ ( 1+ y 2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaey OeI0IaamiEamaaCaaaleqabaGaiWiGikdaaaGccaGGtaIaamyEamaa CaaaleqabaGaaGOmaaaakiabgkHiTiaaigdacaGGtaYaamWaaeaacq GHsisldaqadaqaaiaaigdacqGHRaWkcaWG5bWaaWbaaSqabeaacaaI YaaaaaGccaGLOaGaayzkaaaacaGLBbGaayzxaaaaaa@47CC@

= x 2 y 2 1+1+ y 2 = x 2 y 2 + y 2 1+1 = x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaGGtaIaamiEamaaCaaaleqabaGaaGOmaaaakiaacobicaWG5bWa aWbaaSqabeaacaaIYaaaaOGaai4eGiaaigdacqGHRaWkcaaIXaGaey 4kaSIaamyEamaaCaaaleqabaGaaGOmaaaaaOqaaiabg2da9iabgkHi TiaadIhadaahaaWcbeqaaiaaikdaaaGccaGGtaIaamyEamaaCaaale qabaGaaGOmaaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaikdaaaGc caGGtaIaaGymaiabgUcaRiaaigdaaeaacqGH9aqpcqGHsislcaWG4b WaaWbaaSqabeaacaaIYaaaaaaaaa@525C@

Question: 71

Subtract the sum of 12ab10 b 2 18 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaWGHbGaamOyaiabgkHiTiaaigdacaaIWaGaamOyamaaCaaaleqa baGaaGOmaaaakiabgkHiTiaaigdacaaI4aGaamyyamaaCaaaleqaba GaaGOmaaaaaaa@4191@  and 9ab+12 b 2 +14 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaadg gacaWGIbGaey4kaSIaaGymaiaaikdacaWGIbWaaWbaaSqabeaacaaI YaaaaOGaey4kaSIaaGymaiaaisdacaWGHbWaaWbaaSqabeaacaaIYa aaaaaa@40C5@  from the sum of ab+2 b 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadk gacqGHRaWkcaaIYaGaamOyamaaCaaaleqabaGaaGOmaaaaaaa@3B13@  and 3 b 2 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadk gadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGHbWaaWbaaSqabeaa caaIYaaaaaaa@3B2B@ .

Solution

Sum of 12ab10 b 2 18 a 2  and  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaWGHbGaamOyaiabgkHiTiaaigdacaaIWaGaamOyamaaCaaaleqa baGaaGOmaaaakiabgkHiTiaaigdacaaI4aGaamyyamaaCaaaleqaba GaaGOmaaaakiaabccacaqGHbGaaeOBaiaabsgacaqGGaaaaa@459D@

9ab+12 b 2 +14 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaadg gacaWGIbGaey4kaSIaaGymaiaaikdacaWGIbWaaWbaaSqabeaacaaI YaaaaOGaey4kaSIaaGymaiaaisdacaWGHbWaaWbaaSqabeaacaaIYa aaaaaa@40C5@

=12ab10 b 2 18 a 2 +9ab+12 b 2 +14 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiaaikdacaWGHbGaamOyaiaacobicaaIXaGaaGimaiaadkgadaah aaWcbeqaaiaaikdaaaGccaGGtaIaaGymaiaaiIdacaWGHbWaaWbaaS qabeaacaaIYaaaaOGaey4kaSIaaGyoaiaadggacaWGIbGaey4kaSIa aGymaiaaikdacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG ymaiaaisdacaWGHbWaaWbaaSqabeaacaaIYaaaaaaa@4E04@

By combining the like terms,

=12ab+9ab10 b 2 +12 b 2 18 a 2 +14 a 2 =21ab+2 b 2 4 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaIXaGaaGOmaiaadggacaWGIbGaey4kaSIaaGyoaiaadggacaWG IbGaai4eGiaaigdacaaIWaGaamOyamaaCaaaleqabaGaaGOmaaaaki abgUcaRiaaigdacaaIYaGaamOyamaaCaaaleqabaGaaGOmaaaakiaa cobicaaIXaGaaGioaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHRa WkcaaIXaGaaGinaiaadggadaahaaWcbeqaaiaaikdaaaaakeaacqGH 9aqpcaaIYaGaaGymaiaadggacaWGIbGaey4kaSIaaGOmaiaadkgada ahaaWcbeqaaiaaikdaaaGccaGGtaIaaGinaiaadggadaahaaWcbeqa aiaaikdaaaaaaaa@591A@

Sum of ab+2 b 2  and 3 b 2 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadk gacqGHRaWkcaaIYaGaamOyamaaCaaaleqabaGaaGOmaaaakiaabcca caqGHbGaaeOBaiaabsgacaqGGaGaaG4maiaadkgadaahaaWcbeqaai aaikdaaaGccaGGtaIaamyyamaaCaaaleqabaGaaGOmaaaaaaa@443C@

=ab+2 b 2 +3 b 2 a 2 =ab+5 b 2 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaWGHbGaamOyaiabgUcaRiaaikdacaWGIbWaaWbaaSqabeaacaaI YaaaaOGaey4kaSIaaG4maiaadkgadaahaaWcbeqaaiaaikdaaaGcca GGtaIaamyyamaaCaaaleqabaGaaGOmaaaaaOqaaiabg2da9iaadgga caWGIbGaey4kaSIaaGynaiaadkgadaahaaWcbeqaaiaaikdaaaGcca GGtaIaamyyamaaCaaaleqabaGaaGOmaaaaaaaa@4B06@  

Now, subtracting 21ab+2 b 2 4 a 2   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaig dacaWGHbGaamOyaiabgUcaRiaaikdacaWGIbWaaWbaaSqabeaacaaI YaaaaOGaai4eGiaaisdacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaae iiaaaa@4085@  
from ab+5 b 2 a 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOzaiaabk hacaqGVbGaaeyBaiaabccacaWGHbGaamOyaiabgUcaRiaaiwdacaWG IbWaaWbaaSqabeaacaaIYaaaaOGaai4eGiaadggadaahaaWcbeqaai aaikdaaaGccaGGSaaaaa@42C3@  we get,

=(ab+5 b 2 a 2 )(21 ab+2 b 2 4 a 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaadggacaWGIbGaey4kaSIaaGynaiaadkgadaahaaWcbeqaaiaa ikdaaaGccaGGtaIaamyyamaaCaaaleqabaGaaGOmaaaakiaacMcaca GGtaIaaiikaiaaikdacaaIXaGaaeiiaiaadggacaWGIbGaey4kaSIa aGOmaiaadkgadaahaaWcbeqaaiaaikdaaaGccaGGtaIaaGinaiaadg gadaahaaWcbeqaaiaaikdaaaGccaGGPaaaaa@4CCC@

=ab+5 b 2 a 2 21ab2 b 2 +4 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam yyaiaadkgacqGHRaWkcaaI1aGaamOyamaaCaaaleqabaGaaGOmaaaa kiaacobicaWGHbWaaWbaaSqabeaacaaIYaaaaOGaai4eGiaaikdaca aIXaGaamyyaiaadkgacaGGtaIaaGOmaiaadkgadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaI0aGaamyyamaaCaaaleqabaGaaGOmaaaaaa a@496D@

By combining the like terms,

=ab21ab+5 b 2 2 b 2 a 2 +4 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam yyaiaadkgacaGGtaIaaGOmaiaaigdacaWGHbGaamOyaiabgUcaRiaa iwdacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaai4eGiaaikdacaWGIb WaaWbaaSqabeaacaaIYaaaaOGaai4eGiaadggadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaI0aGaamyyamaaCaaaleqabaGaiqhGikdaaa aaaa@4A69@

=20ab+3 b 2 +3 a 2 =3 a 2 +3 b 2 20ab MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaGGtaIaaGOmaiaaicdacaWGHbGaamOyaiabgUcaRiaaiodacaWG IbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG4maiaadggadaahaa WcbeqaaiaaikdaaaaakeaacqGH9aqpcaaIZaGaamyyamaaCaaaleqa baGaaGOmaaaakiabgUcaRiaaiodacaWGIbWaaWbaaSqabeaacaaIYa aaaOGaai4eGiaaikdacaaIWaGaamyyaiaadkgaaaaa@4CDF@

Question: 72

Each symbol given below represents an algebraic expression:

CH10 EX Q72a_self  =2 x 2 +3y, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaamyE aiaacYcaaaa@3CD7@   

CH10 EX Q72b_self  =5 x 2 +3x, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ynaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaamiE aiaacYcaaaa@3CD9@   

CH10 EX Q72c_self  =8 y 2 3 x 2 +2x+3y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIZaGaamiE amaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWG4bGaey4kaS IaaG4maiaadMhaaaa@4263@

 

The symbols are then represented in the expression:

CH10 EX Q72a_self        + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaScaaa@36BA@           CH10 EX Q72b_self          MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0caaa@36C5@            CH10 EX Q72c_self 

 

Find the expression which is represented by the above symbols.

Solution

Given,

CH10 EX Q72a_self  =2 x 2 +3y, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaamyE aiaacYcaaaa@3CD7@   

CH10 EX Q72b_self  =5 x 2 +3x, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ynaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaamiE aiaacYcaaaa@3CD9@   

CH10 EX Q72c_self  =8 y 2 3 x 2 +2x+3y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaIZaGaamiE amaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWG4bGaey4kaS IaaG4maiaadMhaaaa@4263@

MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3716@      CH10 EX Q72a_self      + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaScaaa@36BA@       CH10 EX Q72b_self          MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0caaa@36C5@           CH10 EX Q72c_self 

 

=(2 x 2 +3y)+(5 x 2 +3x)(8 y 2 3 x 2 +2x+3y) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaikdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG4m aiaadMhacaGGPaGaey4kaSIaaiikaiaaiwdacaWG4bWaaWbaaSqabe aacaaIYaaaaOGaey4kaSIaaG4maiaadIhacaGGPaGaeyOeI0Iaaiik aiaaiIdacaWG5bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaG4mai aadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaamiEaiab gUcaRiaaiodacaWG5bGaaiykaaaa@52D1@

=2 x 2 +3y+5 x 2 +3x8 y 2 +3 x 2 2x3y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaamyE aiabgUcaRiaaiwdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaS IaaG4maiaadIhacqGHsislcaaI4aGaamyEamaaCaaaleqabaGaaGOm aaaakiabgUcaRiaaiodacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey OeI0IaaGOmaiaadIhacqGHsislcaaIZaGaamyEaaaa@4ED1@

By combining the like terms,

=2 x 2 +5 x 2 +3 x 2 +3y3y+3x2x8 y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI1aGaamiE amaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiodacaWG4bWaaWbaaS qabeaacaaIYaaaaOGaey4kaSIaaG4maiaadMhacqGHsislcaaIZaGa amyEaiabgUcaRiaaiodacaWG4bGaeyOeI0IaaGOmaiaadIhacqGHsi slcaaI4aGaamyEamaaCaaaleqabaGaaGOmaaaaaaa@4EC7@

=10 x 2 8 y 2 +x  MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ymaiaaicdacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGio aiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG4bGaaeiiaa aa@4065@

Question: 73

Observe the following nutritional chart carefully:

Food Item
(Per Unit
=100 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBbXgBd9 gzLbvyNv2CaeHbnfgBNvNBGC0B0HwAJbacfeGaa8xpaiaa=fdacaWF WaGaa8hmaaaa@4376@  g)

Carbohydrates

Rajma

60 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaic daaaa@3752@  g

Cabbage

5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaaaa@3697@  g

Potato

22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaik daaaa@3750@  g

Carrot

11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaig daaaa@374E@  g

Tomato

4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaaaa@3696@  g

Apples

14 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaais daaaa@3751@  g

 

Write an algebraic expression for the amount of carbohydrates in ‘g’ for

a. y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  units of potatoes and 2 units of rajma

b. 2x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI haaaa@3791@  units tomatoes and y units apples.

Solution

(a)

By unitary method,

MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3716@   1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@  unit of potatoes contain carbohydrates

=22g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaaikdacaaMe8Uaae4zaaaa@3ACD@

y units MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaabc cacaqG1bGaaeOBaiaabMgacaqG0bGaae4Caaaa@3C3C@  of potatoes contain carbohydrates

=22×y =22y g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaIYaGaaGOmaiabgEna0kaadMhaaeaacqGH9aqpcaaIYaGaaGOm aiaadMhacaqGGaGaae4zaaaaaa@407A@  

Similarly,

MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3716@   1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@  unit of rajma contain carbohydrates

=60 g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OnaiaaicdacaqGGaGaae4zaaaa@39E5@

MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3716@   2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@3695@  units of rajma contain carbohydrates

=60×2 =120 g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaI2aGaaGimaiabgEna0kaaikdaaeaacqGH9aqpcaaIXaGaaGOm aiaaicdacaqGGaGaae4zaaaaaa@3FF5@

Hence, required expression is 22y+120. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaik dacaWG5bGaey4kaSIaaGymaiaaikdacaaIWaGaaiOlaaaa@3C13@

(b)

By unitary method,

MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3716@   1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@  unit of tomatoes contain carbohydrates

=4 g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG inaiaabccacaqGNbaaaa@3929@

MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3716@   2x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI haaaa@3792@  units of tomatoes contain carbohydrates

=2x x4 =8x g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIYaGaamiEaiaabccacaWG4bGaaGinaaqaaiabg2da9iaaiIda caWG4bGaaeiiaiaabEgaaaaa@3F4E@  

Similarly,

MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3716@   1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@  unit apples contain carbohydrates

 =14 g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiiOaiabg2 da9iaaigdacaaI0aGaaeiiaiaabEgaaaa@3B08@

y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D7@  units apples contain carbohydrates

=14×y =14y g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaIXaGaaGinaiabgEna0kaadMhaaeaacqGH9aqpcaaIXaGaaGin aiaadMhacaqGGaGaae4zaaaaaa@407D@  

Hence, the required expression is 8x+14y. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaadI hacqGHRaWkcaaIXaGaaGinaiaadMhacaGGUaaaaa@3BA2@

Question: 74

Arjun bought a rectangular plot with length x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@  and breadth y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  and then sold a triangular part of it whose base is y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  and height is z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36D7@ . Find the area of the remaining part of the plot.

Solution

Given,

Arjun bought a rectangular plot with length x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  and breadth y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D7@

MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3716@  area of rectangular plot

=l×b =x×y =xy MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaWGSbGaey41aqRaamOyaaqaaiabg2da9iaadIhacqGHxdaTcaWG 5baabaGaeyypa0JaamiEaiaadMhaaaaa@42ED@

Also, given triangular part of it whose base is y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D7@  and height is z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36D8@  so, area of triangular part

= 1 2 ×y×z = 1 2 ×yz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiaabgdaaeaacaaIYaaaaiabgEna0kaadMhacqGHxdaT caWG6baabaGaeyypa0ZaaSaaaeaacaqGXaaabaGaaGOmaaaacqGHxd aTcaWG5bGaamOEaaaaaa@452A@

Area of remaining part of the plot

= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DF@  Area of rectangular plot MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4eGaaa@368F@  Area of triangular plot

=xy 1 2 yz =y( x 1 2 z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaWG4bGaamyEaiabgkHiTmaalaaabaGaaeymaaqaaiaaikdaaaGa amyEaiaadQhaaeaacqGH9aqpcaWG5bWaaeWaaeaacaWG4bGaeyOeI0 YaaSaaaeaacaqGXaaabaGaaGOmaaaacaWG6baacaGLOaGaayzkaaaa aaa@453F@

Question: 75

Amisha has a square plot of side m and another triangular plot with base and height each equal to m. What is the total area of both plots?

Solution

Given,
side of square plot =m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam yBaaaa@37D1@  and
height & base of triangular plot =m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam yBaaaa@37D1@

Area of square plot, (side) 2 = m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaabo hacaqGPbGaaeizaiaabwgacaGGPaWaaWbaaSqabeaacaaIYaaaaOGa eyypa0JaamyBamaaCaaaleqabaGaaGOmaaaaaaa@3EB7@

Area of triangular plot,

( 1 2 ×h×b ) = 1 2 ×m×m = m 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaadaqada qaamaalaaabaGaaGymaaqaaiaaikdaaaGaey41aqRaamiAaiabgEna 0kaadkgaaiaawIcacaGLPaaaaeaacqGH9aqpdaWcaaqaaiaaigdaae aacaaIYaaaaiabgEna0kaad2gacqGHxdaTcaWGTbaabaGaeyypa0Za aSaaaeaacaWGTbWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaaaaaa aa@4B48@

Total area of both plots

= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DF@  Area of square plot + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaScaaa@36BB@  Area of triangular plot

= m 2 + m 2 2 = 2 m 2 + m 2 2 = 3 m 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaWGTbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSYaaSaaaeaacaWG TbWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaaaaaeaacqGH9aqpda WcaaqaaiaaikdacaWGTbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIa amyBamaaCaaaleqabaGaaGOmaaaaaOqaaiaaikdaaaaabaGaeyypa0 ZaaSaaaeaacaaIZaGaamyBamaaCaaaleqabaGaaGOmaaaaaOqaaiaa ikdaaaaaaaa@480C@    

[taking LCM of 1&2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaacA cacaaIYaaaaa@37FA@  is 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaaaa@3695@  ]

Question: 76

A taxi service charges Rs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacdaGaa8hiai aa=jfacaWFZbaaaa@384C@  8 per km and levies a fixed charge of Rs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacdaGaa8hiai aa=jfacaWFZbaaaa@384C@  50.Write an algebraic expression for the above situation, if the taxi is hired for x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@  km.

Solution

As per the given information, taxi service charged Rs 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOuaiaabo hacaqGGaGaaGioaaaa@3909@  per km and fixed charged of 50. If taxi is hired for x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D6@  km. Then, algebraic expression for the situation =8×x+50=8x+50 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ioaiabgEna0kaadIhacqGHRaWkcaaI1aGaaGimaiabg2da9iaaiIda caWG4bGaey4kaSIaaGynaiaaicdaaaa@4230@

Hence, the required expression is 8x+50. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaadI hacqGHRaWkcaaI1aGaaGimaiaac6caaaa@3AA5@  

Question: 77

Shiv works in a mall and gets paid Rs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOuaiaabo haaaa@37A4@  50 per hour. Last week he worked for 7 hours and this week he will work for x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@  hours. Write an algebraic expression for the money paid to him for both the weeks.

Solution

Given, money paid to shiv =Rs50per h. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG jbVlaabkfacaqGZbGaaGynaiaaicdacaaMe8UaaeiCaiaabwgacaqG YbGaaeiiaiaabIgacaqGUaaaaa@424C@

MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3717@  Money paid last week =Rs50×7 =Rs350 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaMe8UaaeOuaiaabohacaaMc8UaaGynaiaaicdacqGHxdaTcaaI 3aaabaGaeyypa0JaaGjbVlaabkfacaqGZbGaaGPaVlaaiodacaaI1a GaaGimaaaaaa@4838@

So, money paid this week =Rs50×x =Rs50x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaMe8UaaeOuaiaabohacaaMc8UaaGynaiaaicdacqGHxdaTcaWG 4baabaGaeyypa0JaaGjbVlaabkfacaqGZbGaaGPaVlaaiwdacaaIWa GaamiEaaaaaa@48B4@

Total money paid to shiv =Rs(350+50x) =Rs50(x+7) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaMe8UaaeOuaiaabohacaaMc8UaaiikaiaaiodacaaI1aGaaGim aiabgUcaRiaaiwdacaaIWaGaamiEaiaacMcaaeaacqGH9aqpcaaMe8 UaaeOuaiaabohacaaMc8UaaGynaiaaicdacaGGOaGaamiEaiabgUca RiaaiEdacaGGPaaaaaa@4E09@

Question: 78

Sonu and Raj have to collect different kinds of leaves for science project. They go to a park where Sonu collects 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik daaaa@374F@  leaves and Raj collects x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@  leaves.

After some time Sonu loses 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maaaa@3695@  leaves and Raj collects 2x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaadI haaaa@3791@  leaves. Write an algebraic expression to find the total number of leaves collected by both of them.

Solution

According to the question,

Sonu collected leaves =123 =9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaIXaGaaGOmaiabgkHiTiaaiodaaeaacqGH9aqpcaaI5aaaaaa@3BCF@

Raj collected leaves =x+2x =3x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaWG4bGaey4kaSIaaGOmaiaadIhaaeaacqGH9aqpcaaIZaGaamiE aaaaaa@3D3D@

MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3717@  Total leaves collected =9+3x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG yoaiabgUcaRiaaiodacaWG4baaaa@3A3E@

Hence, the required expression is 9+3x. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiabgU caRiaaiodacaWG4bGaaiOlaaaa@39EA@

Question: 79

A school has a rectangular play ground with length x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@  and breadth y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  and a square lawn with side x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@  as shown in the figure given below.

CH10 EX Q79

What is the total perimeter of both of them combined together?

Solution

Given, Length of rectangular playground, AB=x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiaabk eacqGH9aqpcaWG4baaaa@3965@

& breadth of rectangular playground, BC=y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOqaiaabo eacqGH9aqpcaWG5baaaa@3968@

FCDE is a square, i.e., FC=CD=EF=DE=x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOraiaabo eacqGH9aqpcaqGdbGaaeiraiabg2da9iaabweacaqGgbGaeyypa0Ja aeiraiaabweacqGH9aqpcaWG4baaaa@412A@

ABCF is a rectangle, i.e., AB=FC=xandBC=AF=y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiaabk eacqGH9aqpcaqGgbGaae4qaiabg2da9iaadIhacaaMe8Uaaeyyaiaa b6gacaqGKbGaaGjbVlaabkeacaqGdbGaeyypa0JaaeyqaiaabAeacq GH9aqpcaWG5baaaa@47F2@

Now, perimeter of combined (playground + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaScaaa@36BB@  lawn)
= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DF@  Sum of all sides

=AB+BC+CD+DE+EF+FA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaae yqaiaabkeacqGHRaWkcaqGcbGaae4qaiabgUcaRiaaboeacaqGebGa ey4kaSIaaeiraiaabweacqGHRaWkcaqGfbGaaeOraiabgUcaRiaabA eacaqGbbaaaa@4497@

=x+y+x+x+x+y=4x+2y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaam iEaiabgUcaRiaadMhacqGHRaWkcaWG4bGaey4kaSIaamiEaiabgUca RiaadIhacqGHRaWkcaWG5bGaeyypa0JaaGinaiaadIhacqGHRaWkca aIYaGaamyEaaaa@4696@

Question: 80

The rate of planting the grass is Rs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOuaiaabo haaaa@37A4@ x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36D5@  per square meter. Find the cost of planting the grass on a triangular lawn whose base is y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D6@  meters and height is z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36D7@  meters.

Solution

Given,
base of triangular lawn is y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36D7@  meters and height z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36D8@  meters.

Area of triangular lawn
= 1 2 ×y×z = 1 2 yz m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpdaWcaaqaaiaaigdaaeaacaaIYaaaaiabgEna0kaadMhacqGHxdaT caWG6baabaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaacaWG5b GaamOEaiaaysW7caqGTbWaaWbaaSqabeaacaaIYaaaaaaaaa@4688@

Cost of planting the grass on lawn
= 1 2 yz×x =Rs 1 2 xyz MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaaigdaaeaacaaIYaaaaiaadMhacaWG6bGaey41aqRa amiEaaqaaiabg2da9iaaysW7caqGsbGaae4CamaalaaabaGaaGymaa qaaiaaikdaaaGaamiEaiaadMhacaWG6baaaaa@465C@

CH10 EX Q80_sol_self

Question: 81

Find the perimeter of the figure given below:

 

Solution

We know that, perimeter is the sum of all sides. Perimeter of the given figure
=AB+BC+CD+DA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaae yqaiaabkeacqGHRaWkcaqGcbGaae4qaiabgUcaRiaaboeacaqGebGa ey4kaSIaaeiraiaabgeaaaa@3FB1@

=2(x+y)+(5xy)+2(x+y)+(5xy) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaacIcacaWG4bGaey4kaSIaamyEaiaacMcacqGHRaWkcaGGOaGa aGynaiaadIhacqGHsislcaWG5bGaaiykaiabgUcaRiaaikdacaGGOa GaamiEaiabgUcaRiaadMhacaGGPaGaey4kaSIaaiikaiaaiwdacaWG 4bGaeyOeI0IaamyEaiaacMcaaaa@4D69@

=2x+2y+5xy+2x+2y+5xy MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaadIhacqGHRaWkcaaIYaGaamyEaiabgUcaRiaaiwdacaWG4bGa eyOeI0IaamyEaiabgUcaRiaaikdacaWG4bGaey4kaSIaaGOmaiaadM hacqGHRaWkcaaI1aGaamiEaiabgkHiTiaadMhaaaa@497D@

On combining the like terms,

=5x+2x+5x+2xy+2yy+2y =14x+2y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGH9a qpcaaI1aGaamiEaiabgUcaRiaaikdacaWG4bGaey4kaSIaaGynaiaa dIhacqGHRaWkcaaIYaGaamiEaiabgkHiTiaadMhacqGHRaWkcaaIYa GaamyEaiabgkHiTiaadMhacqGHRaWkcaaIYaGaamyEaaqaaiabg2da 9iaaigdacaaI0aGaamiEaiabgUcaRiaaikdacaWG5baaaaa@4F9C@

Question: 82

In a rectangular plot, 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaaaa@3698@  square flower beds of side ( x+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bGaey4kaSIaaGOmaaGaayjkaiaawMcaaaaa@39FC@  meters each have been laid (see figure given below). Find the total cost of fencing the flower beds at the cost of Rs50 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacdaGaa8hiai aabkfacaqGZbGaaGPaVlaaiwdacaaIWaaaaa@3B54@  per 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaaIWaaaaa@3808@  meters.

CH10 EX Q82

Solution

Given,
side of one square flower bed
=(x+2) m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaadIhacqGHRaWkcaaIYaGaaiykaiaabccacaqGTbaaaa@3C66@

Perimeter of one square flower bed =4(side)=4(x+2) m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG inaiaacIcacaqGZbGaaeyAaiaabsgacaqGLbGaaiykaiabg2da9iaa isdacaGGOaGaamiEaiabgUcaRiaaikdacaGGPaGaaeiiaiaab2gaaa a@43F2@

Now, total perimeter of such square flower beds
=5x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ynaiaadIhaaaa@389B@  perimeter of 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaaaa@3694@  square

=5×4(x+2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG ynaiabgEna0kaaisdacaGGOaGaamiEaiabgUcaRiaaikdacaGGPaaa aa@3E67@

=20(x+2) m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OmaiaaicdacaGGOaGaamiEaiabgUcaRiaaikdacaGGPaGaaeiiaiaa b2gaaaa@3DDC@

Cost of fencing of 100 m= Rs50 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dacaaIWaGaaeiiaiaab2gacqGH9aqpimaacaWFGaGaaeOuaiaaboha caaMc8UaaGynaiaaicdaaaa@401C@

Cost of 1 m= Rs 50 100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaabc cacaqGTbGaeyypa0dcdaGaa8hiaiaabkfacaqGZbGaaGPaVpaalaaa baGaaGynaiaaicdaaeaacaaIXaGaaGimaiaaicdaaaaaaa@40E7@

Cost of 20(x+2) m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaic dacaGGOaGaamiEaiabgUcaRiaaikdacaGGPaGaaeiiaiaab2gaaaa@3CD6@

= 50 100 ×20(x+2) =10(x+2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpdaWcaaqaaiaaiwdacaaIWaaabaGaaGymaiaaicdacaaIWaaaaiab gEna0kaaikdacaaIWaGaaiikaiaadIhacqGHRaWkcaaIYaGaaiykaa qaaiabg2da9iaaigdacaaIWaGaaiikaiaadIhacqGHRaWkcaaIYaGa aiykaaaaaa@488D@

= Rs(10x+20) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0dcda Gaa8hiaiaa=jfacaWFZbGaaGPaVlaacIcacaaIXaGaaGimaiaadIha cqGHRaWkcaaIYaGaaGimaiaacMcaaaa@4100@

Question: 83

A wire is (7x3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaiE dacaWG4bGaeyOeI0IaaG4maiaacMcaaaa@3A99@  meters long. A length of (3x4) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaaio dacaWG4bGaeyOeI0IaaGinaiaacMcaaaa@3A96@  meters is cut for use. Now, answer the following questions:

a.    How much wire is left?

b.   If this left out wire is used for making an equilateral triangle. What is the length of each side of the triangle so formed?

Solution

Given,
length of wire =(7x3) m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaiEdacaWG4bGaeyOeI0IaaG4maiaacMcacaqGGaGaaeyBaaaa @3D33@  and wire cut for use has length =(3x4) m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaiodacaWG4bGaeyOeI0IaaGinaiaacMcacaqGGaGaaeyBaaaa @3D30@

a. Left wire =(7x3)(3x4) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaiEdacaWG4bGaeyOeI0IaaG4maiaacMcacqGHsislcaGGOaGa aG4maiaadIhacqGHsislcaaI0aGaaiykaaaa@414B@

=7x33x+4 =7x3x3+4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqaaeaacqGH9a qpcaaI3aGaamiEaiabgkHiTiaaiodacqGHsislcaaIZaGaamiEaiab gUcaRiaaisdaaeaacqGH9aqpcaaI3aGaamiEaiabgkHiTiaaiodaca WG4bGaeyOeI0IaaG4maiabgUcaRiaaisdaaaaa@4749@

=(4x+1) m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaisdacaWG4bGaey4kaSIaaGymaiaacMcacaqGGaGaaeyBaaaa @3D23@

b. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3717@  Left wire =(4x+1) m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaisdacaWG4bGaey4kaSIaaGymaiaacMcacaqGGaGaaeyBaaaa @3D23@

MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyinIWfaaa@3717@  Perimeter of equilateral triangle = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0daaa@36DF@  Length of wire left

3×(side)=4x+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4TaaG 4maiabgEna0kaacIcacaqGZbGaaeyAaiaabsgacaqGLbGaaiykaiab g2da9iaaisdacaWG4bGaey4kaSIaaGymaaaa@4471@

side= 4x+1 3 = 1 3 (4x+1) m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taae 4CaiaabMgacaqGKbGaaeyzaiabg2da9maalaaabaGaaGinaiaadIha cqGHRaWkcaaIXaaabaGaaG4maaaacqGH9aqpdaWcaaqaaiaaigdaae aacaaIZaaaaiaacIcacaaI0aGaamiEaiabgUcaRiaaigdacaGGPaGa aeiiaiaab2gaaaa@49E4@

Question: 84

Rohan's mother gave him Rs3x y 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH84rpq0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatCvAUf KttLearyWrHrwDNLgaiuaacqWFGaaicaqGsbGaae4CaiaaykW7caaM c8UaaG4maiaaysW7caWG4bGaamyEamaaCaaaleqabaGaaGOmaaaaaa a@4742@  and his father gave him Rs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipE0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOuaiaabo haaaa@37A4@ 5(