The greatest value of the term independent of x, as a varies over R, in the expansion of xcos⁡α$$+$$$$sin$$⁡α$$x20is$$

Question

The greatest value of the term independent of x, as a varies over R, in the expansion of xcos⁡α$$+$$$$sin$$⁡α$$x20is$$

Infinity Learn · Accepted Answer

$$Tr+1$$, the $$(r+1)thterm$$ in the expansion of xcos⁡α$$+$$$$sin$$⁡α$$x20is$$  $$20Cr(xcos$$⁡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$$)10$$            $$=20C10$$[$$cos$$⁡α$$sin$$⁡α]$$10Thus$$, the greatest possible value of β is  $$20C101210$$.

The greatest value of the term independent of $x$ , as a varies over $R$ , in the expansion of ${(x \cos α + \frac{\sin α}{x})}^{20}$ is

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

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The greatest value of the term independent of $x$ , as a varies over $R$ , in the expansion of ${(x \cos α + \frac{\sin α}{x})}^{20}$ is