Question

$\alpha ,\beta$ be the roots of the equation $x^2-3x+a=0$ and $\gamma ,\delta$ the roots of $x^2-12x+b=0$ and numbers $\alpha ,\beta ,\gamma ,\delta$ (in this order) form an increasing $GP$., 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 $a=2,b=32$

$x^2-3x+a=0$
For this to be true, roots have to be real or $b^2-4ac\ge 0⟹9-4a\ge 0ora\ge 9/4⟹a\ge 2$
So, $a=2$. The roots $x^2-3x+a=0$ of are $1,2.$
$x^2-12x+b=0b^2-4ac\ge 0⟹144-4b\ge 0orb\ge 36$
But, given the roots of both equations are in increasing GP. Hence, $b$ must be $32$.
The roots of $x^2-12x+b=0$ are $4,8$.
Now, we can say all the 4 roots are $1,2,4,8$ which are in increasing GP.
Therefore, ${}^{\prime }{a}^{\prime }=2$ and ${}^{\prime }{b}^{\prime }=32$.