The number of θ ∈ (0, 4π) for which the system of linear equations <br> 3(sin 3θ)x - y + z = 2 <br> 3(cos 2θ)x + 4y + 3z = 3 <br> 6x + 7y + 7z = 9 <br> has no solution is :
Step-by-Step Solution
Key Concept: A system of linear equations has no solution if the determinant of the coefficient matrix is zero and the augmented matrix is inconsistent.
The system has no solution if the determinant of the coefficient matrix \Delta = 0 and at least one of \Delta x, \Delta y, or \Delta z is non-zero. The coefficient matrix is [[3sin(3\theta), -1, 1], [3cos(2\theta), 4, 3], [6, 7, 7]]. Calculating the determinant: \Delta = 3sin(3\theta)(28-21) + 1(21cos(2\theta)-18) + 1(21cos(2\theta)-24) = 21sin(3\theta) + 42cos(2\theta) - 42 = 0. This simplifies to sin(3\theta) + 2cos(2\theta) = 2. Solving this trigonometric equation for \theta \in (0, 4\pi) yields the number of values.
Correct Answer: B