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Q.
The term independent of a in the expansion of $\left(1+\sqrt{ a }+\frac{1}{\sqrt{ a }-1}\right)^{-30}$ is
Binomial Theorem
Solution:
$\left(1+\sqrt{a}+\frac{1}{\sqrt{a}-1}\right)^{-30}$
$=\left(\frac{ a }{\sqrt{ a }-1}\right)^{-30}$
$=\left(\frac{\sqrt{a}-1}{a}\right)^{30}$
$=\frac{1}{a^{30}}(1-\sqrt{a})^{30}$
$=\frac{1}{a^{30}}\left\{{ }^{30} C_{0}-{ }^{30} C_{1} \sqrt{a}+\ldots+{ }^{30} C_{30}(\sqrt{a})^{30}\right\}$
There is no term independent of $a$.