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KVPY2020MathematicsQuadratic Equation

Let a ,   b ,   c be non-zero real roots of the equation x 3 + a x 2 + b x + c = 0 . Then

Options

  1. Athere are infinitely many such triples a ,   b ,   c
  2. Bthere is exactly one such triple a ,   b ,   c
  3. Cthere are exactly two such triples a ,   b ,   c
  4. Dthere are exactly three such triples a ,   b ,   c

Correct answer

C. there are exactly two such triples a ,   b ,   c

Step-by-step solution

x 3 + a x 2 + b x + c = 0 = ( x - a ) ( x - b ) ( x - c ) a + b + c = - a ⇒ 2 a + b + c = 0   … i a b + b c + c a = b   … ii a b c = - c ⇒ a b = - 1 [ ∵ c ≠ 0 ]   … iii Also a is a root of equation ⇒ 2 a 3 + a b + c = 0 ⇒ 2 a 3 - 1 + c = 0 ⇒ c = 1 - 2 a 3 from i 2 a 2 + a b + a c = 0 2 a 2 - 1 + a 1 - 2 a 3 = 0 2 a 2 - 2 a 4 + a - 1 = 0 2 a 2 ( 1 - a ) ( 1 + a ) + ( a - 1 ) = 0 ⇒ ( 1 - a ) 2 a 2 ( a + 1 ) - 1 = 0 ⇒ a = 1 or 2 a 3

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