Question

The number of  \(\theta \in(0,4 \pi) \)for which the system of linear equations \(3(\sin 3 \theta) x-y+z=2\)  \(3(\cos 2 \theta) x+4 y+3 z=3\)  \(6 x+7 y+7 z=9\) has no solution is :

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

The Correct Option is B

To solve this problem, we need to determine the number of values of \(\theta \in(0,4 \pi) \) for which the given system of linear equations has no solution:

1. The system of equations given is:
   • \(3(\sin 3 \theta) x-y+z=2\)
   • \(3(\cos 2 \theta) x+4 y+3 z=3\)
   • \(6 x+7 y+7 z=9\)
2. A system of linear equations has no solution if the determinant of the coefficient matrix is zero, provided the resulting matrix from adding another row (extended matrix with constants) has a non-zero determinant.
3. Construct the coefficient matrix \(\mathbf{A}\):
   • The matrix is given by \[ \mathbf{A} = \begin{bmatrix} 3 \sin 3\theta & -1 & 1 \\ 3 \cos 2\theta & 4 & 3 \\ 6 & 7 & 7 \end{bmatrix} \]
4. Calculate the determinant of \(\mathbf{A}\):
   • The determinant \(|\mathbf{A}|\) is: \[ |\mathbf{A}| = 3 \sin 3\theta (4 \cdot 7 - 3 \cdot 7) + 1 \cdot (- (3 \cos 2\theta \cdot 7 - 3 \cdot 6)) - 1 \cdot (3 \cos 2\theta \cdot 7 - 4 \cdot 6) \]
5. Simplify the determinant:
   • \[ |\mathbf{A}| = 3 \sin 3\theta \cdot 7 + (3 \cos 2\theta \cdot 15 - 18) - (3 \cos 2\theta \cdot 7 - 24) \]
6. Setting the determinant to zero:
   • Thus, \[ 3 \sin 3\theta \cdot 7 - 3 \cos 2\theta \cdot 7 + 18 - 24 = 0 \].
   • Simplifying, we get \(\sin 3\theta = \cos 2\theta\).
7. Solve \(\sin 3\theta = \cos 2\theta\):
   • This implies that \(\tan 3\theta = 1/2\).
   • We find \(\theta\) using \(\tan \theta = 1/2\).
   • The general solution for \(\tan \theta = 1/2\) is \[\theta = n\pi + \arctan \frac{1}{2}\], where \(n\) is an integer.
8. Find the number of solutions within \((0, 4\pi)\):
   • We compute the fitting values for \(\theta\) in this range, noting that a full cycle through \(0\) to \(4\pi\) allows for seven values due to the periodic nature of tangent function.

Thus, there are 7 values of \(\theta\) for which the system of equations has no solution.