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 homogeneous system of 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) | = 0 <br> | 1, sin(alpha), -cos(alpha) | <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 + 2sqrt(2)sin(alpha)cos(alpha) + sqrt(2)(sin^2(alpha) - cos^2(alpha)) = 0 <br> -1 + sqrt(2)sin(2alpha) - sqrt(2)cos(2alpha) = 0 <br> sqrt(2)(sin(2alpha) - cos(2alpha)) = 1 <br> sin(2alpha - pi/4) = 1/sqrt(2) = sin(pi/4) <br> 2alpha - pi/4 = pi/4 => 2alpha = pi/2 => alpha = pi/4 (not in options) <br> Or 2alpha - pi/4 = 3pi/4 => 2alpha = pi => alpha = pi/2 (not in options) <br> Checking the options, for alpha = 5pi/24: <br> 2alpha = 5pi/12 <br> sin(5pi/12 - 3pi/12) = sin(2pi/12) = sin(pi/6) = 1/2 != 1/sqrt(2). <br> Re-evaluating the determinant expansion: <br> 1(-cos^2 - sin^2) - sqrt(2)sin(-cos - sin) + sqrt(2)cos(sin - cos) = -1 + sqrt(2)sin*cos + sqrt(2)sin^2 + sqrt(2)sin*cos - sqrt(2)cos^2 = -1 + 2sqrt(2)sin*cos - sqrt(2)(cos^2 - sin^2) = -1 + sqrt(2)sin(2alpha) - sqrt(2)cos(2alpha) = 0. <br> sqrt(2)(sin(2alpha) - cos(2alpha)) = 1 => sin(2alpha - pi/4) = 1/sqrt(2). <br> 2alpha - pi/4 = pi/4 => 2alpha = pi/2 => alpha = pi/4. <br> 2alpha - pi/4 = 3pi/4 => 2alpha = pi => alpha = pi/2. <br> 2alpha - pi/4 = 9pi/4 => 2alpha = 10pi/4 = 5pi/2 => alpha = 5pi/4. <br> 2alpha - pi/4 = -5pi/4 => 2alpha = -pi => alpha = -pi/2. <br> Let's re-check the determinant calculation. <br> | 1 sqrt(2)sin sqrt(2)cos | <br> | 1 cos sin | <br> | 1 sin -cos | <br> R2 -> R2-R1, R3 -> R3-R1: <br> | 1 sqrt(2)sin sqrt(2)cos | <br> | 0 cos-sqrt(2)sin sin-sqrt(2)cos | <br> | 0 sin-sqrt(2)sin -cos-sqrt(2)cos | <br> (cos-sqrt(2)sin)(-cos-sqrt(2)cos) - (sin-sqrt(2)cos)(sin-sqrt(2)sin) = 0 <br> -cos^2 - sqrt(2)cos^2 + sqrt(2)sin*cos + 2sin*cos - (sin^2 - sqrt(2)sin^2 - sqrt(2)sin*cos + 2sin*cos) = 0 <br> -cos^2 - sqrt(2)cos^2 + sqrt(2)sin*cos - sin^2 + sqrt(2)sin^2 + sqrt(2)sin*cos = 0 <br> -1 - sqrt(2)(cos^2 - sin^2) + 2sqrt(2)sin*cos = 0 <br> -1 - sqrt(2)cos(2alpha) + sqrt(2)sin(2alpha) = 0 <br> sqrt(2)(sin(2alpha) - cos(2alpha)) = 1. <br> sin(2alpha - pi/4) = 1/sqrt(2). <br> 2alpha - pi/4 = pi/4 + 2npi or 3pi/4 + 2npi. <br> 2alpha = pi/2 + 2npi => alpha = pi/4 + npi. <br> 2alpha = pi + 2npi => alpha = pi/2 + npi. <br> None of the options match. Let's re-read the question. Maybe it's sin(alpha) and cos(alpha) without sqrt(2)? No, it's there. Let's check option B: alpha = 5pi/24. 2alpha = 5pi/12 = 75 degrees. sin(75) - cos(75) = sin(75) - sin(15) = 2cos(45)sin(30) = 2 * (1/sqrt(2)) * (1/2) = 1/sqrt(2). <br> sqrt(2) * (1/sqrt(2)) = 1. Correct.
Correct Answer: 2