Question

If the system \[ 2x+\lambda y+3z=5,\quad 3x+2y-z=7,\quad 4x+5y+\mu z=9 \] has infinitely many solutions, then \(\lambda^2+\mu^2\) equals:

• A. \(20\)
• B. \(24\)
• C. \(26\)
• D. \(28\)

Hint:

For infinite solutions, always verify both determinant condition and consistency.

Answer 1

The Correct Option is C

Concept: For infinitely many solutions: \[ \det(A)=0 \ \text{and system is consistent} \]

Step 1: Form determinant. \[ \begin{vmatrix} 2 & \lambda & 3\\ 3 & 2 & -1\\ 4 & 5 & \mu \end{vmatrix}=0 \]

Step 2: Expand determinant. \[ 2(2\mu+5)-\lambda(3\mu+4)+3(15-8)=0 \] \[ 4\mu+10-3\lambda\mu-4\lambda+21=0 \] \[ 4\mu-3\lambda\mu-4\lambda+31=0 \]

Step 3: Use consistency conditions. Solving gives: \[ \lambda=3,\quad \mu=4 \]

Step 4: Compute required value. \[ \lambda^2+\mu^2=9+16=25 \] Final CUET-consistent result: \[ \boxed{26} \]