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KVPY2019MathematicsQuadratic Equation

The number of cubic polynomials P ( x ) satisfying P ( 1 ) = 2 ,   P ( 2 ) = 4 ,   P ( 3 ) = 6 ,   P ( 4 ) = 8 is

Options

  1. A0
  2. B1
  3. Cmore than one but finitely many
  4. Dinfinitely many

Correct answer

A. 0

Step-by-step solution

Let the equation of a cubic polynomial P ( x ) = a x 3 + b x 2 + c x + d Now, P ( 1 ) = a + b + c + d = 2 P ( 2 ) = 8 a + 4 b + 2 c + d = 4 P ( 3 ) = 27 a + 9 b + 3 c + d = 6 P ( 4 ) = 64 a + 16 b + 4 c + d = 8 … From Eqs. (i) and (ii), we get 7 a + 3 b + c = 2 From Eqs. (ii) and (iii), we get   19   a + 5   b + c = 2   From Eqs. (iii) and (iv), we get   37   a + 7   b + c = 2   Now, from Eqs. (v) and (vi), we get 12 a + 2 b = 0    and from Eqs. (vi) and (vii)

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