<p>Let complex number <i>z</i> satisfy the inequality <i>2 ≤ |z| ≤ 4</i>. A point P is selected in this region at random. The probability that argument of P lies in the interval <i>[-π/4, π/4]</i> is <i>1/K</i>, then <i>K</i> = ?</p>
Step-by-Step Solution
Key Concept: The probability is the ratio of the area in the annular region where the argument lies in [-π/4, π/4] to the total area of the annulus. The argument condition defines a sector, so we need to find what fraction of the full annulus this sector occupies.
<p><strong>Step 1:</strong> Identify the region. We have an annulus (ring) with inner radius r₁ = 2 and outer radius r₂ = 4.</p><p><strong>Step 2:</strong> Calculate the total area of the annulus.</p><p>Total Area = π(4²) - π(2²) = 16π - 4π = 12π</p><p><strong>Step 3:</strong> Identify the sector defined by the argument condition. The argument of z lies in [-π/4, π/4], which represents a sector of angle θ = π/4 - (-π/4) = π/2 radians.</p><p><strong>Step 4:</strong> Calculate the area of the annular sector with angle π/2 (which is 1/4 of the full circle).</p><p>Favorable Area = (π/2)/(2π) × 12π = (1/4) × 12π = 3π</p><p><strong>Step 5:</strong> Calculate the probability.</p><p>Probability = Favorable Area / Total Area = 3π / 12π = 1/4</p><p><strong>Step 6:</strong> Given that probability = 1/K, we have:</p><p>1/K = 1/4</p><p>Therefore, K = 4</p><p><strong>∴ Answer:</strong> 4</p>
Correct Answer: 4