Question

Find the term of the expansion of ${(\sqrt[3]{3x^{-2}}+x)}^7$ containing $x$ in the second power.

• A. $T_4$
• B. $T_5$
• C. $T_6$
• D. $T_7$

Answer 1

Answer: (B)

$T_5$

Note:- The general term or $(r+1{)}^{th}$ term of $(a+b{)}^n$ is $T_{r+1}{=}^n{C}_r{a}^{n-r}{b}^r$

$T_{r+1}{=}^7{C}_r{\left(\sqrt[3]{3x^{-2}}\right)}^{7-r}{\left(x\right)}^r$

$\Rightarrow T_{r+1}{=}^7{C}_r{\left(x^{-2}\right)}^{7-r/3}{x}^r$

$\Rightarrow T_{r+1}{=}^7{C}_r{x}^{-2/3(7-r)+r}$

Now, if power of $x$ must be $2$, then $r$ is:-

$\therefore x^{-2/3(7-r)+r}=x^2$

$\Rightarrow \frac{-2}{3}(7-r)+r=2$

$\Rightarrow \frac{-14}{3}+\frac{2}{3}r+r=2$

$\Rightarrow \frac{5r}{3}=\frac{20}{3}$

$\Rightarrow r=4$

Therefore $T_{4+1}=T_5$ is the term which contains $x^2$