If the system of equation <br>x + (\sqrt{2}\sin\alpha)y + (\sqrt{2}\cos\alpha)z = 0<br>x + (\cos\alpha)y + (\sin\alpha)z = 0<br>x + (\sin\alpha)y - (\cos\alpha)z = 0<br>has a non-trivial solution, then \alpha =
Step-by-Step Solution
Key Concept: For a system of homogeneous linear equations to have a non-trivial solution, the determinant of the coefficient matrix must be zero.
The system has a non-trivial solution if the determinant of the coefficient matrix is zero:<br>| 1 & \sqrt{2}\sin\alpha & \sqrt{2}\cos\alpha |<br>| 1 & \cos\alpha & \sin\alpha |<br>| 1 & \sin\alpha & -\cos\alpha | = 0<br>Expanding the determinant:<br>1(-\cos^2\alpha - \sin^2\alpha) - \sqrt{2}\sin\alpha(-\cos\alpha - \sin\alpha) + \sqrt{2}\cos\alpha(\sin\alpha - \cos\alpha) = 0<br>-1 + \sqrt{2}\sin\alpha\cos\alpha + \sqrt{2}\sin^2\alpha + \sqrt{2}\sin\alpha\cos\alpha - \sqrt{2}\cos^2\alpha = 0<br>-1 + 2\sqrt{2}\sin\alpha\cos\alpha - \sqrt{2}(\cos^2\alpha - \sin^2\alpha) = 0<br>-1 + \sqrt{2}\sin(2\alpha) - \sqrt{2}\cos(2\alpha) = 0<br>\sqrt{2}(\sin(2\alpha) - \cos(2\alpha)) = 1<br>\sin(2\alpha) - \cos(2\alpha) = 1/\sqrt{2}<br>\sin(2\alpha - \pi/4) = 1/2<br>2\alpha - \pi/4 = \pi/6 or 5\pi/6<br>2\alpha = 5\pi/12 or 13\pi/12<br>\alpha = 5\pi/24 or 13\pi/24. Checking the options, 5\pi/24 is present.
Correct Answer: B