Question

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

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

Answer 1

Answer: (A)

$-2,-32$

$\alpha$ & $\beta$ are the roots of the equation $x^2-x+p$

$\gamma$ & $\delta$ are the roots of the equation $x^2-4x+q$

Since, $\alpha ,\beta ,\gamma ,\delta$ are in G.P

Let $\alpha =a$, $\beta =ar$, $\gamma =a{r}^2$, $\delta =a{r}^3$

$\alpha +\beta =a+ar=1$ $[\because$ Sum of the roots =1] ...(1)

$\gamma +\delta =a{r}^2+a{r}^3=r^2(a+ar)=4$ $[\because$ Sum of the roots =4] ...(2)

From (1) & (2), we get $r^2=4\Rightarrow r=\pm 2$

Substituting $r$ in (1), we get $a=-1,\frac{1}{3}$

Now, $\alpha \beta =p=a^{2}r=-2,\frac{2}{9}$

and $\gamma \delta =q=a^2{r}^5=-32,\frac{32}{9}$
Ans: A