Question

$\alpha ,\beta$ be the roots of $x^2-3x+a=0$ and $\gamma ,\beta$ the roots of $x^2-12x+b=0$ and numbers $\alpha ,\beta ,\gamma ,\delta$ (in order) form an increasing G.P., then

• A. $a=3,b=12$
• B. $a=12,b=3$
• C. $a=2,b=32$
• D. $a=4,b=16$

Answer 1

The correct option is C $a=2,b=32$

$\alpha ,\beta ,\gamma ,\delta$ being G.P. they may be taken as $k,kr,k{r}^2,k{r}^3$.
$S=k(1+r)=3,k{r}^2(1+r)=12$
$\therefore 3r^2=12$ or $r=2\therefore k=1$
$P=k^{2}r=a,k^2{r}^5=b$.
Putting for $r$ and $k,a=2,b=32$.