Question

If the $(r+1)$ th term in the expansion of $\left(\sqrt[3]{\frac{a}{\sqrt{b}}}+\sqrt{\frac{b}{\sqrt[3]{a}}}\right)^{21}$ has the same power of $a$ and $b$, then the value of $r$ is

• A. 9
• B. 10
• C. 8
• D. 6

Answer 1

Answer: (A)

We have $T_{r+1}={ }^{21} C_{r}\left(\sqrt[3]{\frac{a}{\sqrt{b}}}\right)^{21-r}\left(\sqrt{\frac{b}{\sqrt[3]{a}}}\right)^{r}$
$={ }^{21} C_{r} a^{7-(r / 2)} b^{(2 / 3) r-(7 / 2)}$
As the powers of $a$ and $d$ are the same, $7-\frac{r}{2}=\frac{2}{3} r-\frac{7}{2}$
$\Rightarrow r=9$