JEE Main 2025 — Determinants Question with Solution
JEE Main 2025 (23 Jan Shift 1)
Question
If the system of equations
has infinitely many solutions, then is equal to
Choose an option
Show full solutionCorrect option: D
Correct answer
D
Step-by-step explanation
$\begin{aligned}
& (\lambda-1) x+(\lambda-4) y+\lambda z=5 \\
& \lambda x+(\lambda-1) y+(\lambda-4) z=7 \\
& (\lambda+1) x+(\lambda+2) y-(\lambda+2) z=9
\end{aligned}$
For infinitely many solutions
$\begin{aligned}
& \mathrm{D}=\left|\begin{array}{ccc}
\lambda-1 & \lambda-4 & \lambda \\
\lambda & \lambda-1 & \lambda-4 \\
\lambda+1 & \lambda+2 & -(\lambda+2)
\end{array}\right|=0 \\
& (\lambda-3)(2 \lambda+1)=0 \\
& \mathrm{D}_{\mathrm{x}}=\left|\begin{array}{ccc}
5 & \lambda-4 & \lambda \\
7 & \lambda-1 & \lambda-4 \\
9 & \lambda+2 & -(\lambda+2)
\end{array}\right|=0 \\
& 2(3-\lambda)(23-2 \lambda)=0 \\
& \lambda=3 \\
& \therefore \lambda^2+\lambda=9+3=12
\end{aligned}$
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