<p>If \(Z\) is a non-real complex number, then find the minimum value of \(\dfrac{\text{Im}\, Z^5}{(\text{Im}\, Z)^5}\).</p>
Step-by-Step Solution
Key Concept: Express Z = x + iy and expand Z⁵ using the binomial theorem, then isolate the imaginary part. The ratio simplifies to a function of t = x/y, which can be minimized using calculus or AM-GM inequality.
<p><strong>Step 1:</strong> Let Z = x + iy where x, y ∈ ℝ and y ≠ 0 (non-real).</p><p><strong>Step 2:</strong> Expand Z⁵ = (x + iy)⁵ using binomial theorem:</p><p>Z⁵ = x⁵ + 5x⁴(iy) + 10x³(iy)² + 10x²(iy)³ + 5x(iy)⁴ + (iy)⁵</p><p>Z⁵ = x⁵ + 5ix⁴y - 10x³y² - 10ix²y³ + 5xy⁴ + iy⁵</p><p><strong>Step 3:</strong> Extract imaginary part:</p><p>Im(Z⁵) = 5x⁴y - 10x²y³ + y⁵ = y(5x⁴ - 10x²y² + y⁴)</p><p><strong>Step 4:</strong> Form the ratio:</p><p>$$\frac{\text{Im}\,Z^5}{(\text{Im}\,Z)^5} = \frac{y(5x⁴ - 10x²y² + y⁴)}{y⁵} = \frac{5x⁴ - 10x²y² + y⁴}{y⁴}$$</p><p><strong>Step 5:</strong> Let t = x²/y². Then:</p><p>$$f(t) = 5t² - 10t + 1$$</p><p><strong>Step 6:</strong> Find minimum using calculus:</p><p>$$f'(t) = 10t - 10 = 0 \implies t = 1$$</p><p><strong>Step 7:</strong> Check second derivative: f''(t) = 10 > 0 (minimum confirmed).</p><p>At t = 1: f(1) = 5(1) - 10(1) + 1 = -4</p><p>∴ <strong>Minimum value = -4</strong></p>
Correct Answer: -4