Question

The coefficient of $x^{-30}$ in ${\left(3x^2+\frac{2}{3x^2}\right)}^{15}$ is :

• A. ${}^{15}{C}_2{.3}^8{.2}^7$
• B. $(\frac{2}{3}{)}^{15}$
• C. $(\frac{3}{2}{)}^{15}$
• D. $3^{1}4{.2}^2$

Answer 1

The correct option is B $(\frac{2}{3}{)}^{15}$

Given $(3x^2+\frac{2}{3x^2}{)}^{15}$

let the $(n+1{)}^{th}$ term have $x^{-30}$ as variable

$T_{n-a}{=}^{15}{C}_n(3x^2{)}^{15-n}{\left(\frac{2}{3x^2}\right)}^n$

$T_{n-a}{=}^{15}{C}_n{3}^{15-2n}{2}^n{x}^{30-4n}$

we have $x^{-30}$

On comparing powers

we get

$30-4n=-30$

$4n=60$

$n=15$

we have ${15}^{th}$ term of expansion i.e.

coefficient of $x^{-30}{=}^{15}{C}_{15}(3x^2{)}^{15-45}{\left(\frac{2}{3x^2}\right)}^{15}$

$Answer=1\times 1.{\left(\frac{2}{3}\right)}^{15}$