Question

If the coefficients of $x^2$ and $x^3$ in the expansion of $(3+a x)^9$ are equal, then the value of $a$ is

• A. $\frac{7}{9}$
• B. $\frac{7}{8}$
• C. $\frac{9}{7}$
• D. $\frac{9}{8}$

Answer 1

Answer: (C)

The general term in the expansion of $(3+a x)^9$ is
$T_{r+1}={ }^9 C_r 3^{9-r} a^r x^r$
For coefficient of $x^2$, put $r=2$.
$\therefore T_{2+1}={ }^9 C_2 3^{9-2} a^2 x^2$
$\Rightarrow$ Coefficient of $x^2={ }^9 C_2 3^7 a^2 ....$(i)
For coefficient of $x^3$, put $r=3$.
$\therefore T_{3+1} ={ }^9 C_3 3^{9-3} a^3 x^3 $
$={ }^9 C_3 3^6 a^3 x^3$
$\Rightarrow $ Coefficient of $ x^3 ={ }^9 C _3 3^6 a^3....$(ii)
Given, coefficient of $x^2=$ Coefficient of $x^3$
$ { }^9 C _2 3^7 a^2={ }^9 C _3 3^6 a^3 \text { [from Eqs. (i) and (ii)] } $
$ \Rightarrow \frac{9 \times 8}{2} \times 3 \times 1=\frac{9 \times 8 \times 7}{6} \times 1 \times a $
$ \Rightarrow \frac{3}{2}=\frac{7}{6} \times a \Rightarrow a=\frac{9}{7} $