KVPY 2020-21 SA Exam Previous Papers

Question : 2 of 80

Marks: +1, -0

Let

p_1(x) = x^3 -2020x^2 +b_1x + c_1

and

p_2(x) = x^3 -2021x^2 + b_2x + c_2

be polynomials having two common roots α and β. Suppose there exist polynomials

q_1(x) \and q_2(x)

such that

p_1(x) q_1(x) + p_2(x)q_2(x) = x^2 -3x + 2

. Thenthe correct identity is

Solution:

Let

p_{1}(x)=x^{3}-2020 x^{2}+b_{1} x+c_{1}

=(x-\α)(x-\β)(x-\γ)

and

p_{2}(x)=x^{3}-2021 x^{2}+b_ 2 x+c _2

=(x-\α)(x-\β)(x-\δ)

∵

p_{1}(x) \.q_{1}(x)+p_{2}(x) q_{2}(x)=x^{2}-3 x+2

Comparing the coefficient of

x^3

we get

q_{1}(x)=-q_{2}(x)=q(x)(

say

)

(x-\α)(x-\β)[q(x)(\δ-\γ)]=(x-1)(x-2)

Hence

\α=1, \β=2, \γ=2017

and

\δ=2018

p_{1}(x)=(x-1)(x-2)(x-2017)

\⇒p_{1}(3)=-4028

p_{2}(x)=(x-1)(x-2)(x-2018)

p_{1}(3)+p_{2}(1)+4028=0

Go to Question:

Prev Question

Next Question