For the expansion `(x sin p + x^(-1)cos p)^(10), (p in R)`,

Options

A. the greatest value of the term independent of x is `10!//2^(5)(5!)^(2)`
B. the least value of sum of coefficient is zero
C. the greatest value of sum of coefficient is 12
D. the last value of the term independent of x occurs when `p = (2n+1)'(pi)/(4), n in Z`

Answer

A, B, C

Explanation

`(x sin p + x^(-1) cos p)^(10)`
The general term in the expansion is
`T_(r+1) = .^(10)C_(r)(xsin p)^(10-r)(x^(-1) cosp)^(r)`
For the term independent of `x`, we have `10-2r = 0` or `r = 5`,
Hence, the independent term is
`.^(10)C_(5)sin^(5)p cos^(5)p - .^(10)C_(5)(sin^(5)2p)/(32)`
which is the greatest when sin `2p = 1`.
The least value of `.^(10)C_(5)(sin^(5)2p)/(32)` is `-(10!)/(2^(5)(5!)^(2))` when
`sin 2p = - 1` or `p = (4n-1)(pi)/(4), n in Z`.
Sum of coefficient is `(sin p + cosp)^(10)`, when `x = 1`
or `(1+sin2p)^(5)`, which is least when `sin 2p = - 1`.
Hence, least sum of coefficients is zero. Greatest sum of coefficient occurs when `sin 2 p = 1`, Hence, greatest sum is `2^(5) = 32`.