Question

Let $\alpha, \beta$ be the roots of $x^2-x+p=0$ and $\gamma, \delta$ be the roots of $x^2-4 x+q=0$. If $\alpha, \beta, \gamma, \delta$ are in G.P., then the integral values of $p$ and $q$ respectively, are

• A. $-2,-32$
• B. $-2,3$
• C. $-6,3$
• D. $-6,-32$

Answer 1

Answer: (A)

$\text { Let } \beta=\alpha r , \gamma=\alpha r ^2, \delta=\alpha r ^3$
$\alpha+\beta=1 \Rightarrow \alpha(1+ r )=1 $
$\gamma+\delta=4 \Rightarrow \alpha(1+ r ) r ^2=4 \Rightarrow r ^2=4 \Rightarrow r = \pm 2 $
$r =2 \Rightarrow \alpha=\frac{1}{3}, b =\frac{2}{3} \text { and } p \text { is not an integer. } $
$\therefore r =-2, \alpha=-1, \beta=2, \gamma=-4, \delta=8 $
$p =\alpha \cdot \beta=-2, q =\gamma \cdot \delta=-32 $