Question

If the system of equations
2x + y - z = 5
2x -5y + λz = μ
x + 2y - 5z = 7
has infinitely many solutions, then (λ + μ)² +  (λ - μ)² is equal to

• A. 904
• B. 912
• C. 916
• D. 920

Answer 1

The Correct Option is C

Step 1: Find the determinant \( \Delta \) of the coefficient matrix. The coefficient matrix is: \[ \Delta = \begin{vmatrix} 2 & 1 & -1 2 & -5 & \lambda 1 & 2 & -5 \end{vmatrix} \] Expanding the determinant: \[ \Delta = 2 \begin{vmatrix} -5 & \lambda 2 & -5 \end{vmatrix} - 1 \begin{vmatrix} 2 & \lambda 1 & -5 \end{vmatrix} + (-1) \begin{vmatrix} 2 & -5 1 & 2 \end{vmatrix} \] \[ = 2\left((-5)(-5) - (\lambda)(2)\right) - 1\left(2(-5) - \lambda(1)\right) - \left(2(2) - (-5)(1)\right) \] \[ = 2(25 - 2\lambda) - 1(-10 - \lambda) - (4 + 5) \] \[ = 2(25 - 2\lambda) + (10 + \lambda) - 9 \] \[ = 50 - 4\lambda + 10 + \lambda - 9 = 51 - 3\lambda = 0 \] Thus, \[ \lambda = 17 \] 
Step 2: Solve for \( \mu \) using \( \Delta_x = 0 \). For \( \Delta_x \), we substitute the first column of the matrix with the constants. The matrix is: \[ \Delta_x = \begin{vmatrix} 5 & 1 & -1 \mu & -5 & 17 7 & 2 & -5 \end{vmatrix} \] Expanding the determinant: \[ \Delta_x = 5 \begin{vmatrix} -5 & 17 2 & -5 \end{vmatrix} - 1 \begin{vmatrix} \mu & 17 7 & -5 \end{vmatrix} + (-1) \begin{vmatrix} \mu & -5 7 & 2 \end{vmatrix} \] \[ = 5\left((-5)(-5) - 17(2)\right) - 1\left(\mu(-5) - 17(7)\right) - \left(\mu(2) - (-5)(7)\right) \] \[ = 5(25 - 34) - 1(-5\mu - 119) - (\mu(2) + 35) \] \[ = 5(-9) + 5\mu + 119 - 2\mu - 35 = -45 + 5\mu + 119 - 2\mu - 35 = 39 + 3\mu = 0 \] \[ \Rightarrow \mu = -13 \] 
Step 3: Calculate \( (\lambda + \mu)^2 + (\lambda - \mu)^2 \). Substitute \( \lambda = 17 \) and \( \mu = -13 \): \[ (\lambda + \mu)^2 + (\lambda - \mu)^2 = (17 - 13)^2 + (17 + 13)^2 \] \[ = 4^2 + 30^2 = 16 + 900 = 916 \]