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KVPY2019MathematicsQuadratic Equation

Let f x = x 6 - 2 x 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

Options

  1. A- 5
  2. B0
  3. C4
  4. D5

Correct answer

B. 0

Step-by-step solution

Given, f x = x 6 - 2 x 5 + x 3 + x 2 - x - 1 = x 2 x 4 - x 3 - x 2 - 1 - x x 4 - x 3 - x 2 - 1 + 2 x 2 - 2 x - 1 ⇒ f x = x 2 - x g x + 2 x 2 - 2 x - 1 ∴   f a = 2 a 2 - 2 a - 1 ∴   Σ f a = f a + f b + c + f d = 2 Σ a 2 - 2 Σ a - Σ 1 = 2 Σ a 2 - 2 Σ a b - 2 Σ a - 4 ∵   a , b , c , d are roots of equation g x = x 4 - x 3 - x 2 - 1 = 0 then Σ a = 1 ,   Σ a b = - 1 ∴   Σ f a = 2 1 2 - 2 - 1 - 2 1 - 4 = 6 - 2 - 4 = 0

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