<p>Maximum sum of coefficient in the expansion of \((1 - x\sin\theta + x^2)^n\) is</p>
Step-by-Step Solution
Key Concept: The sum of coefficients in any polynomial expansion is found by substituting x=1. Here, you must maximize f(1) = (1 - sin θ + 1)ⁿ = (2 - sin θ)ⁿ by minimizing sin θ.
<p><strong>Step 1:</strong> To find the sum of all coefficients in the expansion of (1 - x sin θ + x²)ⁿ, substitute x = 1.</p><p><strong>Step 2:</strong> Sum of coefficients = (1 - sin θ + 1)ⁿ = (2 - sin θ)ⁿ</p><p><strong>Step 3:</strong> To maximize (2 - sin θ)ⁿ, we must maximize the base (2 - sin θ).</p><p><strong>Step 4:</strong> Since sin θ ∈ [-1, 1], the expression (2 - sin θ) is maximized when sin θ is minimum, i.e., sin θ = -1.</p><p><strong>Step 5:</strong> Maximum value = (2 - (-1))ⁿ = (2 + 1)ⁿ = 3ⁿ</p><p>∴ Answer: <strong>3ⁿ</strong> (Option C)</p>
Correct Answer: C