The greatest value of the term independent of x, as a varies over R, in the expansion of xcos⁡α+sin⁡αx20is

Tr+1, the (r+1)thterm in the expansion of xcos⁡α+sin⁡αx20is  20Cr(xcos⁡α)20−rsin⁡αxr                  =20Crx20−2r(cos⁡α)20−r(sin⁡α)rFor this term to be independent of x, we set 20–2r=0 ⇒    r=10.Let   β=Term independent of x, then        β=20C10(cos⁡α)10(sin⁡α)10            =20C10[cos⁡αsin⁡α]10Thus, the greatest possible value of β is  20C101210.