Question

The coefficient of the term independent of $a$ in the expansion of ${(1+\sqrt{a}+\frac{1}{\sqrt{a}-1})}^{-30}$ is

• A. ${}^{30}{C}_{20}$
• B. $0$
• C. ${}^{30}{C}_{10}$
• D. None of these

Answer 1

The correct option is B $0$

${(1+\sqrt{a}+\frac{1}{\sqrt{a}-1})}^{-30}$
$={\left(\frac{a}{\sqrt{a}-1}\right)}^{-30}$
$={\left(\frac{\sqrt{a}-1}{a}\right)}^{30}$
$=\frac{1}{a^{30}}(1-\sqrt{a}{)}^{30}$
$=\frac{1}{a^{30}}[{}^{30}{C}_0-{}^{30}{C}_1\sqrt{a}+.....+{}^{30}{C}_{30}(\sqrt{a}{)}^{30}]$
Hence no term is independent of $a$.