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

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

1. A

2. B

3. C

4. D

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### Solution:

${T}_{r+1,}$ the $\left(r+1\right)th$term in the expansion of ${\left(x\mathrm{cos}\alpha +\frac{\mathrm{sin}\alpha }{x}\right)}^{20}$is

${=}^{20}{C}_{r}{x}^{20-2r}\left(\mathrm{cos}\alpha {\right)}^{20-r}\left(\mathrm{sin}\alpha {\right)}^{r}$
For this term to be independent of $x$, we set $20–2r=0$

Let   $\beta =$Term independent of $x$, then
$\beta {=}^{20}{C}_{10}\left(\mathrm{cos}\alpha {\right)}^{10}\left(\mathrm{sin}\alpha {\right)}^{10}$
${=}^{20}{C}_{10}\left[\mathrm{cos}\alpha \mathrm{sin}\alpha {\right]}^{10}$

Thus, the greatest possible value of $\beta$ is  