KVPY SA Exam 2018 Question Paper

Question : 5 of 80

Marks: +1, -0

The number of polynomials p(x) with integer coefficients such that the curve y = p(x) passes through (2,2) and (4, 5) is

Solution:

y = P(x) = a_0 + a_1 × a_2 x^2 + … +a_nx_n

a_0,a_1,a_2,a_3,...... a_n ∈ I

2 = P(2) ….(1)
5 = P(4) ….(2)
By (1) & (2)
⇒

3 = a_1 (4 -2) + a_2 (4^2 -2^2)

+ a_3(4^3 -2^3) +..... + a_n(4^n -2^n)

Clearly RHS is even and LHS is odd no polynomial exists.

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