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} $
Now, $ f(a) < 0 $
$\Rightarrow 81+9 p-9 < 0 $
$9 p < -72 \Rightarrow p < -8$