First slide

Binomial theorem for positive integral Index

Question

If the middle term of ${(\frac{1}{x} + xsin x)}^{10}$ is equal to $7 \frac{7}{8},$ then value of x is

Moderate

A

$2 nπ + \frac{π}{6}$
B

$nπ + \frac{π}{c}$
C

$nπ + (- 1)^{n} \frac{π}{6}$
D

$nπ + (- 1)^{n} \frac{π}{3}$

Solution

${(\frac{1}{x} + xsin x)}^{10}$
Here, n=10 (even)
$\begin{aligned} \Rightarrow Middle term = (\frac{10}{2} + 1) th = 6 th \\ T_{6} =^{10} C_{5} {(\frac{1}{x})}^{10 - 5} (xsin x)^{5} \\ \Rightarrow 252 (\sin x)^{5} = \frac{63}{8} \Rightarrow (\sin x)^{5} = \frac{1}{32} \\ \Rightarrow \sin x = \frac{1}{2} \Rightarrow \sin x = \sin π / 6 \\ ∴ x = nπ + (- 1)^{n} \frac{π}{6} \end{aligned}$

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