Question

For the expansion $(x\sin p+x^{-1}\cos p{)}^{10},p\in R,$ which among the following satements is (are) CORRECT ?

• A. the greatest value of the term independent of $x$ is $\frac{10!}{2^5(5!{)}^2}$
• B. the least value of sum of coefficients is zero
• C. the greatest value of sum of coefficients is $32$
• D. the least value of the term independent of $x$ occurs when $p=(2n+1)\frac{\pi }{4},n\in Z$

Answer 1

Answer: (A), (B), (C)

the greatest value of sum of coefficients is $32$

In the expansion $(x\sin p+x^{-1}\cos p{)}^{10}$,
the general term is$T_{r+1}={}^{10}{C}_r\left(x\sin p{)}^{10-r}\right(x^{-1}\cos p{)}^r$
For this term to be independent of $x$, $10-2r=0$
$\Rightarrow r=5$.

$\therefore$ Independent term in the given expansion is ${}^{10}{C}_5{\sin }^{5}p{\cos }^{5}p={}^{10}{C}_5\frac{{\sin }^{5}2p}{32}$
which is greatest when $\sin 2p=1$ i.e., greatest value is $\frac{10!}{2^5(5!{)}^2}$
and least when $\sin 2p=-1$
i.e., $p=(4n-1)\frac{\pi }{4},n\in Z$

For sum of coefficients in the expansion of $(\sin p+\cos p{)}^{10},$ by substituting $x=1,$ we get $(1+\sin 2p{)}^5$ which is least when $\sin 2p=-1$
Hence, least sum of coefficients is zero.
Greatest sum of coefficient occurs when $\sin 2p=1$.
Hence, greatest sum is $2^5=32.$