<p>If the middle term of <span>\(\left(\dfrac{1}{x} + x\sin x\right)^{10}\)</span> is equal to <span>\(7\dfrac{7}{8}\)</span>, then find the value of <span>\(x\)</span>.</p>
Step-by-Step Solution
Key Concept: The middle term of (a+b)^10 is the 6th term. Use the binomial expansion formula T_{r+1} = C(n,r)a^(n-r)b^r with r=5, then solve the resulting trigonometric equation by recognizing the numerical value leads to sin x = 1/2.
<p><strong>Step 1:</strong> For (a+b)^10, the middle term is T₆ (the 6th term, since r+1=6 gives r=5).</p><p><strong>Step 2:</strong> Apply binomial formula: T₆ = C(10,5)·(1/x)^5·(x sin x)^5 = C(10,5)·x^(-5)·x^5·sin^5 x = 252 sin^5 x</p><p><strong>Step 3:</strong> Convert 7⁷⁄₈ to improper fraction: 7⁷⁄₈ = 63/8</p><p><strong>Step 4:</strong> Set up equation: 252 sin^5 x = 63/8</p><p><strong>Step 5:</strong> Simplify: sin^5 x = (63/8)/252 = 63/(8×252) = 1/32 = (1/2)^5</p><p><strong>Step 6:</strong> Therefore: sin x = 1/2</p><p><strong>Step 7:</strong> General solution for sin x = 1/2 is: x = nπ + (−1)ⁿ(π/6), where n ∈ Z</p><p>∴ Answer: <strong>x = nπ + (−1)ⁿ π/6, n ∈ Z</strong></p>
Correct Answer: x = nπ + (−1)ⁿ π/6, n ∈ Z