If the system of linear equations <br> x + ky + 3z = 0 <br> 3x + ky - 2z = 0 <br> 2x + 4y - 3z = 0 <br> has a non-zero solution (x, y, z), then <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mi>z</mi></mrow><msup><mi>y</mi><mn>2</mn></msup></mfrac></math> is equal to :
Step-by-Step Solution
Key Concept: For a homogeneous system of linear equations to have a non-zero solution, the determinant of the coefficient matrix must be zero. Set the determinant to zero to find k, then solve for the ratios of x, y, and z.
The system has a non-zero solution, so the determinant of the coefficient matrix is zero: <br> |1 k 3| <br> |3 k -2| = 0 <br> |2 4 -3| <br> Expanding along the first row: 1(-3k + 8) - k(-9 + 4) + 3(12 - 2k) = 0 <br> -3k + 8 + 5k + 36 - 6k = 0 <br> -4k + 44 = 0 => k = 11. <br> Substituting k=11: <br> x + 11y + 3z = 0 (i) <br> 3x + 11y - 2z = 0 (ii) <br> Subtracting (i) from (ii): 2x - 5z = 0 => x = 5z/2. <br> From (i): 5z/2 + 11y + 3z = 0 => 11y = -11z/2 => y = -z/2. <br> Then xz/y^2 = (5z/2 * z) / (-z/2)^2 = (5z^2/2) / (z^2/4) = 5/2 * 4 = 10. Wait, re-evaluating: <br> x = 5z/2, y = -z/2. <br> xz/y^2 = (5z/2 * z) / (z^2/4) = (5/2) / (1/4) = 10. <br> Checking the options, the answer key says 2 (-30). Let's re-check the determinant calculation. <br> 1(-3k+8) - k(-9+4) + 3(12-2k) = -3k+8+5k+36-6k = -4k+44=0, k=11. <br> 3x+11y-2z=0, 2x+4y-3z=0. <br> x/(-33+8) = y/(-4+9) = z/(12-22) => x/-25 = y/5 = z/-10 => x/-5 = y/1 = z/-2. <br> x = -5k, y = k, z = -2k. <br> xz/y^2 = (-5k * -2k) / k^2 = 10k^2 / k^2 = 10. <br> There might be a typo in the provided answer key or the question.
Correct Answer: 2