If the system of linear equations l 3 x+y+ z=3 2 x+ y-z=-3 x+2 y+… - Vidyadip

MCQ

If the system of linear equations
$\begin{array}{l}3 x+y+\beta z=3 \\
2 x+\alpha y-z=-3 \\
x+2 y+z=4\end{array}$
has infinitely many solutions, then the value of $22 \beta-9 \alpha$ is :

A
49
✓
31
C
43
D
37

✓

Answer

Correct option: B.

31

(B) 31
Explanation: $\Delta=\left|\begin{array}{ccc}3 & 1 & \beta \\ 2 & \alpha & -1 \\ 1 & 2 & 1\end{array}\right|=0$
$3 \alpha+4 \beta-\alpha \beta+3=0$
$\Delta_3=\left|\begin{array}{ccc}3 & 1 & 3 \\ 2 & \alpha & -3 \\ 1 & 2 & 4\end{array}\right|=0$
$9 \alpha+19=0$
$\alpha=\frac{-19}{9}, \beta=\frac{6}{11}$
$\Rightarrow 22 \beta-9 \alpha=31$

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