If the system of equations l ( -1) x+( -4) y+ z=5 x+( -1) y+( -4)… - Vidyadip

MCQ

If the system of equations
$
\begin{array}{l}
(\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{array}
$
has infinitely many solutions, then $\lambda^2+\lambda$ is equal to

A
10
✓
12
C
6
D
20

✓

Answer

Correct option: B.

12

(B)
$
\begin{array}{l}
(\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{array}
$
For infinitely many solutions
$
\begin{array}{l}
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 \\
D_{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{array}
$

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