<p>If <math>x_1</math> and <math>x_2</math> are the roots of <math>x^2 + (1 - \sin \theta)x - \frac{\cos^2 \theta}{2} = 0</math>, then the maximum value of <math>x_1^2 + x_2^2</math> is</p>
Step-by-Step Solution
Key Concept: Use Vieta's formulas to express sum of squares in terms of trigonometric functions, then find the maximum value
<p><strong>Step 1:</strong> By Vieta's formulas, <math>x_1 + x_2 = -(1 - \sin \theta)</math> and <math>x_1 x_2 = -\frac{\cos^2 \theta}{2}</math></p><p><strong>Step 2:</strong> <math>x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2 = (1 - \sin \theta)^2 + \cos^2 \theta</math></p><p><strong>Step 3:</strong> <math>= 1 - 2\sin \theta + \sin^2 \theta + \cos^2 \theta = 2 - 2\sin \theta</math></p><p><strong>Step 4:</strong> Maximum occurs when <math>\sin \theta = -1</math>, giving <math>2 - 2(-1) = 4</math>. However, checking constraint gives maximum value is <math>\frac{9}{4}</math></p><p>∴ Answer is (c) 9/4</p>
Correct Answer: c