<p>The term independent of \(x\) in the expansion of \(\left(1 - \dfrac{1}{x} + 3x^5\right)\left(2x^2 - \dfrac{1}{x}\right)^8\) is:</p>
Step-by-Step Solution
Key Concept: The independent term comes from multiplying terms from (1 - 1/x + 3x⁵) with terms from (2x² - 1/x)⁸ such that the total power of x equals zero. Use the binomial expansion to find the general term of (2x² - 1/x)⁸, then systematically combine with each term in the first bracket.
<p><strong>Step 1:</strong> Find the general term in (2x² - 1/x)⁸</p><p>Using binomial theorem: T_{r+1} = C(8,r)(2x²)^{8-r}(-1/x)^r = C(8,r)·2^{8-r}·(-1)^r·x^{16-2r-r} = C(8,r)·2^{8-r}·(-1)^r·x^{16-3r}</p><p><strong>Step 2:</strong> For independent term from 1 × (2x² - 1/x)⁸, need 16 - 3r = 0</p><p>This gives r = 16/3 (not an integer, so no contribution)</p><p><strong>Step 3:</strong> For independent term from (-1/x) × (2x² - 1/x)⁸, need x^{16-3r} · x^{-1} = x⁰</p><p>So 16 - 3r - 1 = 0 → r = 5</p><p>Contribution: -1 · C(8,5)·2³·(-1)⁵ = -1 · 56 · 8 · (-1) = 448</p><p><strong>Step 4:</strong> For independent term from 3x⁵ × (2x² - 1/x)⁸, need x^{16-3r} · x⁵ = x⁰</p><p>So 16 - 3r + 5 = 0 → r = 7</p><p>Contribution: 3 · C(8,7)·2¹·(-1)⁷ = 3 · 8 · 2 · (-1) = -48</p><p><strong>Step 5:</strong> Total constant term = 0 + 448 - 48 = 400</p><p>∴ Answer: B</p>
Correct Answer: B