Question

The greatest value of the term independent of $x$, as $\alpha$ varies over $R$ in the expansion of
$\left(x\cos \alpha +\frac{\sin \alpha }{x}\right)$ is :

• A. ${}^{20}{C}_{10}$
• B. ${}^{20}{C}_{19}$
• C. ${}^{20}{C}_6$
• D. ${}^{20}{C}_6{\left(\frac{1}{2}\right)}^{10}$

Answer 1

The correct option is A ${}^{20}{C}_{10}$

$(xcosa+\frac{sina}{x}{)}^{20}$

$\therefore$ The term independent of x = ?

$(xcosa+\frac{sina}{x}{)}^{20}{=}^{20}{C}_a(xcosa{)}^{20}+..{.}^{20}{C}_{10}\left(xcosa{)}^{10}\right(\frac{sina}{x}{)}^{10}+...$

So ${10}^{th}$ term ${\Rightarrow }^{20}{C}_{10}\left(x^{10}\right)\left(cosa{)}^{10}\right(sina{)}^{10}(\frac{1}{x}{)}^{10}$

So ${10}^{th}$ term is independent of x

and coefficient is $\Rightarrow {(}^{20}{C}_{10})$ (Option A is correct)