Binomial Theorem
Finding Values Using Middle Term
Grade 11
Question:
<p>If the middle term of \(\left(\frac{1}{x} + x \sin x\right)^{10}\) is equal to \(7\frac{7}{8}\), then value of \(x\) is</p>
<p>(a) \(\frac{\pi}{6}\)</p>
<p>(b) \(\frac{\pi}{6}\)</p>
<p>(c) 1</p>
<p>(d) -1</p>
Step-by-Step Solution
Key Concept: Identify the middle term in a binomial expansion and use the given condition to find the value of the variable.
<p><strong>Solution:</strong> For the expansion of $\left(\frac{1}{x} + x\sin x\right)^{10}$, the middle term is the 6th term ($r=5$).</p><p>$T_6 = \binom{10}{5}\left(\frac{1}{x}\right)^5(x\sin x)^5 = \binom{10}{5}(\sin x)^5$</p><p>$\binom{10}{5} = 252$</p><p>Given: $252(\sin x)^5 = 7\frac{7}{8} = \frac{63}{8}$</p><p>$(\sin x)^5 = \frac{63}{8 \times 252} = \frac{1}{32}$</p><p>$\sin x = \frac{1}{2}$</p><p>∴ $x = \frac{\pi}{6}$</p>
Correct Answer: A