<p>The number of integral values of k for which the equation <span class='math'>7\cos x + 5\sin x = 2k + 1</span> has a solution is</p>
Step-by-Step Solution
Key Concept: Use the constraint that linear combinations of sine and cosine have a bounded range determined by the coefficient magnitudes.
<p><strong>Step 1:</strong> For the equation <span class='math'>a\cos x + b\sin x = c</span> to have a solution, we need <span class='math'>|c| \leq \sqrt{a^2 + b^2}</span></p><p><strong>Step 2:</strong> Here, <span class='math'>\sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74}</span></p><p><strong>Step 3:</strong> We need <span class='math'>|2k+1| \leq \sqrt{74} \approx 8.60</span></p><p><strong>Step 4:</strong> So <span class='math'>-8.60 \leq 2k+1 \leq 8.60</span></p><p><strong>Step 5:</strong> <span class='math'>-9.60 \leq 2k \leq 7.60 \Rightarrow -4.80 \leq k \leq 3.80</span></p><p><strong>Step 6:</strong> Integral values: <span class='math'>k \in \{-4, -3, -2, -1, 0, 1, 2, 3\}</span></p><p>∴ There are 8 integral values. Answer is B.</p>
Correct Answer: B