Question

If the coefficients of $x^2$ and $x^3$ in the expansion of ${(3+ax)}^9$ be same, then the value of ${}^{\prime }{a}^{\prime }$ is

• A. $3/7$
• B. $7/3$
• C. $7/9$
• D. $9/7$

Answer 1

The correct option is D $9/7$

Since,
${(3+ax)}^9{=}^9{C}_0{3}^9{+}^9{C}_1{3}^8\left(ax\right){+}^9{C}_2{3}^7{\left(ax\right)}^2{+}^9{C}_3{3}^6{\left(ax\right)}^3+\dots$
Since, coefficient of $x^2=$ coefficient of $x^3$
$\Rightarrow {}^9{C}_2{3}^7{a}^2{=}^9{C}_3{3}^6{a}^3$
$\Rightarrow \frac{{}^9{C}_2}{{}^9{C}_3}\cdot 3=a$
$\Rightarrow \frac{\frac{9\times 8}{2\times 1}}{\frac{9\times 8\times 7}{3\times 2}}\times 3=a$
$\Rightarrow a=\frac{9}{7}$