Question

The number of q∈ (0, 4π) for which the system of linear equations
3(sin 3θ) x – y + z = 2
3(cos 2θ) x + 4y + 3z = 3
6x + 7y + 7z = 9
has no solution, is

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

The Correct Option is B

 To determine the number of values for \(\theta \in (0, 4\pi)\) for which the given system of linear equations has no solution, we check the condition for inconsistency in the system.

The given equations are:

1. \(3\sin 3\theta \cdot x - y + z = 2\)
2. \(3\cos 2\theta \cdot x + 4y + 3z = 3\)
3. \(6x + 7y + 7z = 9\)

The condition for a system of linear equations to have no solution is that the determinant of the coefficient matrix must be zero, and the augmented matrix must provide additional inconsistencies.

Let's write the coefficient matrix \(A\) and the augmented matrix:

	x	y	z
1	\(3\sin 3\theta\)	-1	1
2	\(3\cos 2\theta\)	4	3
3	6	7	7

Calculate the determinant of the coefficient matrix:

\(\text{Det}(A) = \begin{vmatrix} 3\sin 3\theta & -1 & 1 \\ 3\cos 2\theta & 4 & 3 \\ 6 & 7 & 7 \end{vmatrix}\)

By expanding this determinant, we find:

\(\text{Det}(A) = 3\sin 3\theta(28 - 21) - 1(21 - 18) + 1(21 - 24)\)

Simplifying this gives:

\(\text{Det}(A) = 21\sin 3\theta - 3 - 3\)

\(\text{Det}(A) = 21\sin 3\theta - 6\)

For the determinant to be zero:

\(21\sin 3\theta = 6 \Rightarrow \sin 3\theta = \frac{2}{7}\)

Values of \(\theta\) that satisfy this within \((0, 4\pi)\) can be found by solving \(3\theta = \sin^{-1}\left(\frac{2}{7}\right)\) plus the general solution for sine:

\(3\theta = n\pi + (-1)^n\sin^{-1}\left(\frac{2}{7}\right)\)

For \(\theta \in (0, 4\pi)\), \(3\theta\) must range from \((0, 12\pi)\):

• Calculate \(\sin^{-1}\left(\frac{2}{7}\right)\) = \(\approx 0.289\)

Solving these within \((0, 12\pi)\) gives multiple solutions. Calculate how many \(3\theta\) fit within this range:

Values from the periodicity of sine \((n = 0 \text{ to } 11)\), correspondingly giving \(3\theta \approx 0.289, \pi - 0.289, \pi + 0.289, 2\pi - 0.289, \ldots, 11\pi + 0.289\), so ensure all possibilities for \(\theta\) that resolve \(3\theta = n\pi + (-1)^n\sin^{-1}\left(\frac{2}{7}\right)\) are counted.

There are 7 valid \(\theta\) within this range, leading to the system having no solution for \(7\) such values.

Thus, the number of \(\theta \in (0, 4\pi)\) for which the given system has no solution is 7.