For the expansion xsin⁡p+x−1cos⁡p10,(p∈R)

xsin⁡p+x−1cos⁡p10The general term in the expansion isTr+1=10Cr(xsin⁡p)10−rx−1cos⁡prFor the term independent of x, we have 10 - 2r = 0 or r = 5.Hence, the independent term is 10C5sin5⁡pcos5⁡p=10C5sin5⁡2p32which is the greatest when sin2p = 1.The least value of  10C5sin5⁡2p32 is −10!25(5!)2 whensin⁡2p=−1 or p=(4n−1)π4,n∈Z.Sum of coefficient is (sin p + cos p)10, when x = 1or (1 + sin 2p)5, which is least when sin2p = -1.Hence, least sum of coefficients is zero. Greatest sum of coefficient occurs when sin 2p = 1. Hence, greatest sum is 25 = 32.