JEE Main 2024 (Online) 6th April Evening ShiftQuestion 1
If the system of equations
$2x+7y+\lambda z=3$
$3x+2y+5z=4$
$x+\mu y+32z=-1$
has infinitely many solutions, then $(\lambda-\mu)$ is equal to $\_\_\_\_\_\_$ .
Determinant
Answer$38$
Solution
Infinitely many solutions requires all $3\times3$ minors of the augmented matrix to vanish. Solving these gives $\lambda=-1$, $\mu=-39$ (the coefficient determinant gives $\lambda=\frac{10\mu+509}{3\mu-2}=-1$ at $\mu=-39$, consistent). $\lambda-\mu=-1+39=38$.