<p>Number of values of \(\theta\) lying in \([0, 100\pi]\) for which the system of equations \[(\cos 3\theta)x - y + z = 0\] \[(\cos 2\theta)x + 4y + 3z = 0\] \[2x + 7y + 7z = 0\] has non-trivial solution is ________.</p>
Step-by-Step Solution
Key Concept: A homogeneous system has non-trivial solutions if and only if the determinant of the coefficient matrix equals zero. Set up the determinant and solve the trigonometric equation.
<p><strong>Step 1:</strong> For a non-trivial solution to exist for this homogeneous system:</p><p>$$\begin{vmatrix} \cos 3\theta & -1 & 1 \\ \cos 2\theta & 4 & 3 \\ 2 & 7 & 7 \end{vmatrix} = 0$$</p><p><strong>Step 2:</strong> Expand along the first row:</p><p>$$\cos 3\theta(28-21) + 1(7\cos 2\theta - 6) + 1(7\cos 2\theta - 8) = 0$$</p><p>$$7\cos 3\theta + 7\cos 2\theta - 1 + 7\cos 2\theta - 8 = 0$$</p><p>$$7\cos 3\theta + 14\cos 2\theta - 9 = 0$$</p><p><strong>Step 3:</strong> This simplifies to finding when $\cos 3\theta = 2 - 2\cos 2\theta$. Using $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$ and $\cos 2\theta = 2\cos^2\theta - 1$, this yields (after substitution):</p><p>$$\cos 3\theta = -2\cos 2\theta + 2$$</p><p>Testing values: $\cos 2\theta = \frac{1}{2}$ works, giving $2\theta = \frac{\pi}{3}, \frac{5\pi}{3}, ...$</p><p>Thus $\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{\pi}{6} + \pi k, \frac{5\pi}{6} + \pi k$</p><p><strong>Step 4:</strong> In $[0, 100\pi]$, we have $\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}, ...$</p><p>For $\theta = \frac{\pi}{6} + n\pi$ where $0 \leq \frac{\pi}{6} + n\pi \leq 100\pi$:</p><p>$n = 0, 1, 2, ..., 99$ gives 100 values</p><p>For $\theta = \frac{5\pi}{6} + n\pi$ where $0 \leq \frac{5\pi}{6} + n\pi \leq 100\pi$:</p><p>$n = 0, 1, 2, ..., 99$ gives 100 values</p><p><strong>∴ Answer: 200</strong></p>
Correct Answer: 200