Determinants
Determinant condition for system
nta_pyq_2025_apr
Grade 12
Question:
If the system of equation 2x +$\lambday$+$3z = 5$$3x + 2y - z = 7$$4x + 5y$+ $\mu$$z = 9$has infinitely many solutions, then ($\lambda$+ $\mu$ ) is equal to : 2 2
Step-by-Step Solution
Key Concept: Apply the matrix property for determinant condition for system and reduce it to determinant or parameter equations.
|2$\lambda$3 | | | (3) $\Delta$ = 0 $\Rightarrow$ | 3$2 -1$| = 0 |4 5 $\mu$ | $\Rightarrow$ 2(2$\mu$ + 5) +$\lambda$(-$4 - 3$$\mu$$) + 3(7) = 0$$\Rightarrow$ 4$\mu$ - 3$\lambda$$\mu$ - 4$\lambda$+$31 = 0$...(1) |2$\lambda$5| | | $\Delta$$3 = 0$$\Rightarrow$ 3 2$7 = 0$| | |4 5 9| $\Rightarrow$$2(-17)$+$\lambda$($1) + 5(7) = 0$$\Rightarrow$$\lambda$= -1 from equation (1) 4$\mu$ + 3$\mu$ +$4 + 31 = 0$$\Rightarrow$ $\mu$ = -5 2 2 ∴$\lambda$+ $\mu$ = 26
Correct Answer: 3