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3 and 4 .Determinants and Matrices
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If the system of equations $2 x-y+z=4$, $5 x+\lambda y+3 z=12$,$100 x-47 y+\mu z=212$ has infinitely many solutions, then $\mu-2 \lambda$ is equal to
A$56$
B$57$
C$55$
D$59$
(JEE MAIN-2025)
Solution
$\Delta=0 \Rightarrow\left|\begin{array}{ccc}2 & -1 & 1 \\ 5 & \lambda & 3 \\ 100 & -47 & \mu\end{array}\right|=0$
$2(\lambda \mu+141)+(5 \mu-300)-235-100 \lambda=0 \ldots$
$\Delta_3=0 \Rightarrow\left|\begin{array}{ccc}2 & -1 & 4 \\ 5 & \lambda & 12 \\ 100 & -47 & 212\end{array}\right|=0$
$6 \lambda=-12 \Rightarrow \lambda=-2$
Put $\lambda=2$ in$……. (1)$
$2(-2 \mu+141)+5 \mu-300-235+200=0$
$\mu=53$
$\therefore 57$
$2(\lambda \mu+141)+(5 \mu-300)-235-100 \lambda=0 \ldots$
$\Delta_3=0 \Rightarrow\left|\begin{array}{ccc}2 & -1 & 4 \\ 5 & \lambda & 12 \\ 100 & -47 & 212\end{array}\right|=0$
$6 \lambda=-12 \Rightarrow \lambda=-2$
Put $\lambda=2$ in$……. (1)$
$2(-2 \mu+141)+5 \mu-300-235+200=0$
$\mu=53$
$\therefore 57$
Standard 12
Mathematics
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