Question

Let coefficient of $x^2$ and $x^3$ in the expansion of $(3+a x)^9$ are equal. If a lies between the roots of the equation $49 x^2+7 p x-9=0$, then the range of $p$ is $(-\infty,-\lambda)$. Find the value of $\lambda$

Answer 1

Coefficient of $x^2=$ coefficient of $x^3$
${ }^9 C _2 3^7 a ^2={ }^9 C _3 3^6 a ^3$
$\therefore a =\frac{9}{7}$
$\text { Now, } f ( a )<0 $
$\Rightarrow 81+9 p -9<0$
$9 p <-72 \Rightarrow p <-8$