The greatest value of the term independent of X, as α varies over R, in the expansion of xcos⁡α+sin⁡αx20 is

The general term in the expansion of xcos⁡α+sin⁡αx20 is 20Cr(xcos⁡α)20−rsin⁡αxr=20Crx20−2r(cos⁡α)20−r(sin⁡α)rFor this term to be independent of X ,we get20-2r=0 + r=10 Let β = Term independent of x=20C10(cos α)10 (sin α)10=20C10(cos α sin α )10=20C10 sin⁡2α210Thus, the greatest possible value of  β is 20C101210.