Question

If the system of equation $$ 2x + \lambda y + 3z = 5 \\3x + 2y - z = 7 \\4x + 5y + \mu z = 9 $$ has infinitely many solutions, then $ \lambda^2 + \mu^2 $ is equal to:

• A. 22
• B. 18
• C. 26
• D. 30

Hint:

To determine when a system has infinitely many solutions, compute the determinant of the coefficient matrix. If the determinant is zero, the system has infinitely many solutions. For such systems, use the conditions derived from the matrix to find the values of the parameters \( \lambda \) and \( \mu \).

Answer 1

The Correct Option is C

The given system of equations is: \[ 2x + \lambda y + 3z = 5 3x + 2y - z = 7 \\4x + 5y + \mu z = 9 \] To check for infinitely many solutions, we use the determinant of the coefficient matrix. The coefficient matrix is: \[ \Delta = \begin{vmatrix} 2 & \lambda & 3 3 & 2 & -1 \\ 4 & 5 & \mu \end{vmatrix} \] \[ \Delta = 0 \quad \text{(for infinitely many solutions)} \] Expanding the determinant: \[ \Delta = 2 \begin{vmatrix} 2 & -1 5 & \mu \end{vmatrix} - \lambda \begin{vmatrix} 3 & -1 \\ 4 & \mu \end{vmatrix} + 3 \begin{vmatrix} 3 & 2 \\ 4 & 5 \end{vmatrix} \] \[ = 2(\lambda \mu - (-5)) - \lambda (3 \mu - (-4)) + 3(15 - 8) \] \[ = 2(\lambda \mu + 5) - \lambda (3 \mu + 4) + 3(7) \] \[ = 2\lambda \mu + 10 - \lambda (3 \mu + 4) + 21 \] \[ = 2\lambda \mu + 10 - \lambda 3 \mu - 4 \lambda + 21 \] After solving the system, we find: \[ \Delta_3 = 0 \quad \text{and} \quad 2(7) + \lambda(1) + 5(7) = 0 \] Solving for \( \lambda \) and \( \mu \), we find \( \lambda = -1 \) and \( \mu = -5 \). Hence, \[ \lambda^2 + \mu^2 = (-1)^2 + (-5)^2 = 1 + 25 = 26 \]