Determinants Question 2
Question 2 - 2024 (01 Feb Shift 2)
Let the system of equations $x+2 y+3 z=5,2 x+3 y+z=9,4 x+3 y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2 \mu$ is equal to :
(1) 28
(2) 17
(3) 22
(4) 15
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Answer (2)
Solution
$x+2 y+3 z=5$
$2 x+3 y+z=9$
$4 x+3 y+\lambda z=\mu$
for infinite following $\Delta=\Delta_{1}=\Delta_{2}=\Delta_{3}=0$
$\Delta=\left|\begin{array}{lll}1 & 2 & 3 \ 2 & 3 & 1 \ 4 & 3 & \lambda\end{array}\right|=0 \Rightarrow \lambda=-13$
$\Delta_{1}=\left|\begin{array}{ccc}5 & 2 & 3 \ 9 & 3 & 1 \ \mu & 3 & -13\end{array}\right|=0 \Rightarrow \mu=15$
$\Delta_{2}=\left|\begin{array}{ccc}1 & 5 & 3 \ 2 & 9 & 1 \ 4 & 15 & -13\end{array}\right|=0$
$\Delta_{3}=\left|\begin{array}{ccc}1 & 2 & 5 \ 2 & 3 & 9 \ 4 & 3 & 15\end{array}\right|=0$
for $\lambda=-13, \mu=15$ system of equation has infinite solution hence $\lambda+2 \mu=17$
