JEE MAIN - Mathematics (2025 - 23rd January Evening Shift - No. 10)
The system of equations
$$\begin{aligned} & x+y+z=6, \\ & x+2 y+5 z=9, \\ & x+5 y+\lambda z=\mu, \end{aligned}$$
has no solution if
$\lambda=17, \mu=18$
$\lambda=17, \mu \neq 18$
$\lambda=15, \mu \neq 17$
$\lambda \neq 17, \mu \neq 18$
Explanation
$$\begin{aligned}
& \mathrm{D}=\left|\begin{array}{lll}
1 & 1 & 1 \\
1 & 2 & 5 \\
1 & 5 & \lambda
\end{array}\right|=0 \\
& \lambda=17 \\
& D_z=\left|\begin{array}{lll}
1 & 1 & 6 \\
1 & 2 & 9 \\
1 & 5 & \mu
\end{array}\right| \neq 0 \\
& \mu \neq 18
\end{aligned}$$
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