The correct option is C the greatest value of sum of coefficients is 32
In the expansion (xsinp+x−1cosp)10,
the general term isTr+1= 10Cr(xsinp)10−r(x−1cosp)r
For this term to be independent of x, 10−2r=0
⇒r=5.
∴ Independent term in the given expansion is 10C5sin5pcos5p=10C5sin52p32
which is greatest when sin2p=1 i.e., greatest value is 10!25(5!)2
and least when sin2p=−1
i.e., p=(4n−1)π4,n∈Z
For sum of coefficients in the expansion of (sinp+cosp)10, by substituting x=1, we get (1+sin2p)5 which is least when sin2p=−1
Hence, least sum of coefficients is zero.
Greatest sum of coefficient occurs when sin2p=1.
Hence, greatest sum is 25=32.