Question

If reciprocal of every root of the equation $2 x ^3-7 x ^2+ px - q =0$, (where $p , q$ are positive integers) is also a root then $(2 p-q)$ is divisible by

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (A), (B), (C)

$ 2 x ^3-7 x ^2+ px + q =0$
Let the root are $\alpha, \frac{1}{\alpha}, 1$
$\therefore \alpha+\frac{1}{\alpha}+1=\frac{7}{2} $
$\Rightarrow 2\left(\alpha^2+\alpha+1\right)=7 \alpha$
$\Rightarrow 2 \alpha^2-5 \alpha+2 \Rightarrow \alpha=2 \text { or } \frac{1}{2} $
$\therefore \text { roots are } 2, \frac{1}{2}, 1$
$ \frac{q}{2}=1 \Rightarrow q =2$
$\text { and } p-q=5 \Rightarrow p=7$
$\therefore 2 p-q=14-2=12 $