KVPY SX SB Exam 2016 Question Paper

Let $\mathit{f}\left(\mathit{x}\right)={\mathit{x}}^3-27\mathit{x}$
$\mathit{f}\prime \left(\mathit{x}\right)=3{\mathit{x}}^2-27=3\left({\mathit{x}}^2-9\right)$As sum of the roots is zero, so if two roots are integer then 3rd root has to be integer
Now put x = 6t
$216{\mathit{t}}^3-27\times 6\mathit{t}+\mathit{k}=0$
$54\left(4{\mathit{t}}^3-3\mathit{t}\right)+\mathit{k}=0$
Put t = cos θ
54 cos 3θ = – k
Now for 3θ = 0, 2π we get integral solution
So two values of 'k'