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 system is inconsistent (i.e., the augmented matrix has a higher rank than the coefficient matrix).
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 for which the system is inconsistent.
Correct Answer: 2