Choose the correct answer. If the middle term of $\Big(\frac{1}{\text{x}}+\text{x}\sin\text{x}\Big)^{10}$ is equal to $7\frac{7}{8},$ then value of x is: Hint: \(\text{T}_6=\ ^{10}\text{C}_5\frac{1}{\text{x}^5}.\text{x}^5\…

Question

Choose the correct answer.

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

Hint: $\text{T}_6=\ ^{10}\text{C}_5\frac{1}{\text{x}^5}.\text{x}^5\ \sin^5\text{x}=\frac{63}{8}\Rightarrow\sin^5\text{x}=\frac{1}{2^5}\sin\frac{1}{2}$

$\Rightarrow\text{x}=\text{n}\pi+(-1)^\text{n}\frac{\pi}{6}$

Aicharya · Accepted Answer

Solution: Given expression is $\Big(\frac{1}{\text{x}}+\text{x}\sin\text{x}\Big)^{10}$ Since, n = 10 (even), so there is only one middle term which is, 6 th term. $\therefore\text{T}_6=\text{T}_{5+1}=\ ^{10}\text{C}_5\Big(\frac{1}{\text{x}}\Big)^{10-5}(\text{x}\sin\text{x})^5$ $\Rightarrow\frac{63}{8}=\ ^{10}\text{C}_5\sin^5\text{x}$ (given) $\Rightarrow\frac{63}{8 }=252\times\sin^5\text{x}\Rightarrow\sin^5\text{x}=\frac{1}{32}$ $\Rightarrow\sin\text{x}=\frac{1}{2}\Rightarrow\sin\text{x}=\sin\frac{\pi}{6}$ $\Rightarrow\text{x}=\text{n}\pi+(-1)^\text{n}\frac{\pi}{6}$

Choose the correct answer. If the middle term of \(\Big(\frac{1}{\text{x}}+\text{x}\sin\text{x}\Big)^{10}\)…

Solution