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KVPY2016MathematicsQuadratic Equation

The polynomial equation x³-3 a x²+ (27 a²+9 ) x+2016=0 has -

Options

  1. Aexactly one real root for any real a
  2. Bthree real roots for any real a
  3. Cthree real roots for any a 0 , and exactly one real root for any a < 0
  4. Dthree real roots for any a 0 , and exactly one real root for any a >0

Correct answer

A. exactly one real root for any real a

Step-by-step solution

aligned f ^ ( x ) &=3 x ²-6 ax +27 a ²+9 &=3 [ x ²-2 ax +9 a ²+3 ]=3 (( x - a )²+8 a ²+3 ) aligned f^ (x) is + ve for x R so f(x) is monotonic for x R .

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