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KVPY2016MathematicsQuadratic Equation

Number of integers k for which the equation x ³-27 x + k =0 has at least two distinct integer roots is -

Options

  1. A1
  2. B2
  3. C3
  4. D4

Correct answer

B. 2

Step-by-step solution

Let f(x)=x³-27 xf^ (x)=3 x²-27=3 (x²-9 ) As sum of the roots is zero, so if two roots are integer then 3^ rd root has to be integer Now put x =6 t 216 t³-27 6 t+k=054 (4 t³-3 t )+k=0 Put t = 54 3 =- k Now for 3 =0,2 we get integral solution So two values of 'k'

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