Determinants
Linear equations with parameter
nta_pyq_2025_apr
Grade 12
Question:
If the system of linear equations$3x + y$+$\betaz$= 3 2x +$\alphay$- z = -3$x + 2y + z = 4$has infinitely many solutions, then the value of 22$\beta$- 9$\alpha$is :
Step-by-Step Solution
Key Concept: Use the standard setup for linear equations with parameter and simplify the resulting algebraic condition.
|3 1$\beta$| | | (2) $\Delta$ = | 2$\alpha$-1 | = 0 |1 2 1 | 3$\alpha$+ 4$\beta$-$\alpha$$\beta$+$3 = 0$|3 1 3 | | | $\Delta$$3 = 2$$\alpha$-$3 = 0$| | |1 2 4 | 9$\alpha$+$19 = 0 -19$6$\alpha$= ,$\beta$= 9 11 $\Rightarrow$ 22$\beta$- 9$\alpha$= 31
Correct Answer: 2