First slide

Binomial theorem for positive integral Index

Question

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

Moderate

A

$^{20} C_{10}$
B

$^{20} C_{15}$
C

$^{20} C_{19}$
D

$^{20} C_{10} {(\frac{1}{2})}^{10}$

Solution

$T_{r + 1,}$ the $(r + 1) t h$ term in the expansion of ${(x \cos α + \frac{\sin α}{x})}^{20}$ is $^{20} C_{r} (x \cos α)^{20 - r} {(\frac{\sin α}{x})}^{r}$
   $=^{20} C_{r} x^{20 - 2 r} (\cos α)^{20 - r} (\sin α)^{r}$
For this term to be independent of $x$ , we set $20 - 2 r = 0$
$\Rightarrow r = 10 .$
Let    $β =$ Term independent of $x$ , then
   $β =^{20} C_{10} (\cos α)^{10} (\sin α)^{10}$
   $=^{20} C_{10} [\cos α \sin α]^{10}$
Thus, the greatest possible value of $β$ is $^{20} C_{10} {(\frac{1}{2})}^{10} .$

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