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3 and 4 .Determinants and Matrices
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If the system of equations $x+2 y-3 z=2$, $2 x+\lambda y+5 z=5$, $14 x+3 y+\mu z=33$ has infinitely many solutions, then $\lambda+\mu$ is equal to:
A$13$
B$10$
C$11$
D$12$
(JEE MAIN-2025)
Solution
$D =\left|\begin{array}{ccc}1 & 2 & -3 \\ 2 & \lambda & 5 \\ 14 & 3 & \mu\end{array}\right|=0, \lambda \mu+42 \lambda-4 \mu+107=0$
$D_1=2 \lambda \mu+99 \lambda-10 \mu+255$
$D_2=13-\mu$
$D_3=5 \lambda+5$
$D_2=0 \Rightarrow \mu=13 \ D_3=0 \Rightarrow \lambda=-1$
check \ verify for these values $D \& D_2=0$
$D_1=2 \lambda \mu+99 \lambda-10 \mu+255$
$D_2=13-\mu$
$D_3=5 \lambda+5$
$D_2=0 \Rightarrow \mu=13 \ D_3=0 \Rightarrow \lambda=-1$
check \ verify for these values $D \& D_2=0$
Standard 12
Mathematics
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