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KVPY2018MathematicsQuadratic Equation

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

Options

  1. A0
  2. B1
  3. Cmore than 1 but finite
  4. Dinfinite

Correct answer

A. 0

Step-by-step solution

Let P ( x ) = a n x n + a n - 1 x n - 1 + a n - 2 x n - 2 + …    + a 1 x + a 0 a 0 , a 1 , a 2 … ∈ I Given, P ( 2 ) = 2 and P ( 4 ) = 5 2 = a n 2 n + a n - 1 2 n - 1 + a n - 2 2 n - 2 + …    + a 1 2 + a 0 … (i) 5 = a n 4 n + a n - 1 4 n - 1 + a n - 2 4 n - 2 + … + 4 a 1 + a 0 … (ii) On subtracting Eq. (i) from Eq. (ii), we get 3 = a n 4 n - 2 n + a n - 1 4 n - 1 - 2 n - 1 + … + 2 a 1 Clearly, L H S is odd number and R H S is even number. ∴ No

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