Question

The greatest value of the term independent for $x$ in the expansion of ${(x\sin p+x^{-1}\cos p)}^{10}$, $p\in R$, is

• A. $2^5$
• B. $\frac{10!}{(5!{)}^2}$
• C. $\frac{1}{2^5}$.$\frac{10!}{(5!{)}^2}$
• D. None of the above

Answer 1

The correct option is C $\frac{1}{2^5}$.$\frac{10!}{(5!{)}^2}$

The term independent of $x$ will be

${}^{10}{C}_5\sin \left(\alpha {)}^5\cos \right(\alpha {)}^5$

$=\frac{10!}{32(5!)(5!)}\left[32\sin \right(\alpha {)}^5\cos \left(\alpha {)}^5\right]$

$=\frac{10!}{32(5!)(5!)}\left[{\sin }^5\right(2\alpha \left)\right]$

Hence maximum value is for $\alpha ={45}^0$

Maximum value being $\frac{10!}{32(5!)(5!)}$.