<p>If |k| = 5 and 0° < θ < 360°, then the number of different solutions of 3 cos θ + 4 sin θ = k is</p>
Step-by-Step Solution
Key Concept: Convert a cos θ + b sin θ to the form R sin(θ + φ) and find the range.
<p><strong>Step 1:</strong> We can write 3 cos θ + 4 sin θ = 5 sin(θ + φ), where tan φ = 3/4</p><p><strong>Step 2:</strong> The maximum value of 3 cos θ + 4 sin θ is √(3² + 4²) = √25 = 5</p><p><strong>Step 3:</strong> The minimum value is -5</p><p><strong>Step 4:</strong> Since |k| = 5, we have k = ±5</p><p><strong>Step 5:</strong> The equation 3 cos θ + 4 sin θ = 5 has solutions (when sin(θ + φ) = 1), and 3 cos θ + 4 sin θ = -5 has solutions (when sin(θ + φ) = -1)</p><p>∴ Answer is (b).</p>
Correct Answer: a