Question

The solutions of the equation : 

• A. π/6, 5π/6
• B. 5π/12, 7π/12
• C. 7π/12, 11π/12
• D. π/12, π/6

Hint:

Row operations to create a common factor across a row or column are usually the first step to solving determinant equations.

Answer 1

The Correct Option is C

Step 1: Apply \( R_1 \rightarrow R_1 + R_2 \): The first row becomes \( [2, 2, 2\sin^2 x + \cos^2 x] \). Not helpful.
Step 2: Apply \( R_1 \rightarrow R_1 + R_2 + R_3 \): First row sum: \( (1+\sin^2 x) + \cos^2 x + 4\sin 2x = 2 + 4\sin 2x \). All elements in the first row become \( 2 + 4\sin 2x \).
Step 3: Factoring out: \( (2 + 4\sin 2x)\begin{vmatrix} 1 & 1 & 1 \\ \ldots & \ldots & \ldots \end{vmatrix} = 0 \).
Step 4: \( 2 + 4\sin 2x = 0 \implies \sin 2x = -\frac{1}{2} \).
Step 5: Since \( 0 < x < \pi \), we have \( 0 < 2x < 2\pi \). Thus, \( 2x = \frac{7\pi}{6} \) or \( 2x = \frac{11\pi}{6} \).
Step 6: \( x = \frac{7\pi}{12}, \frac{11\pi}{12} \).