KVPY 2019 SB SX Exam Previous Papers

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Question : 83 of 120

Marks: +1, -0

Let

\f(x) = x^6 -2x^5 + x^3 + x^2 -x -1

and

g(x) = x^4 -x^3 -x^2 -1

be two polynomials. Let a,b,c and d be the roots of g(x) = 0. Then the value of f(a) + f(b) + f(c) + f(d) is

–5
0
4
5

Solution:

\f(x) = x^6 -2x^5 + x^3 + x^2 -x -1

= (x^2 -x)(x^4 -x^3 -x ^2 -1)

+ (2x^2 -2x -1)

⇒

\f(a) = 2a^2 -2a -1

⇒

\f(a) + \f(b) + \f(c) + \f(d)

= 2(a^2 + b^2 + c^2 + d^2 )

-2(a + b + c + d) -4

=

2[1 -2(-1)] -2(1)- 4 = 0

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