If for the system of linear equations having infinite solutions \( \begin{aligned} & (\lambda-4) x+(\lambda-2) y+\la…
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If for the system of linear equations having infinite solutions
\(
\begin{aligned}
& (\lambda-4) x+(\lambda-2) y+\lambda z=0 \\
& 2 x-3 y+5 z=0 \\
& x+2 y+6 z=0
\end{aligned}
\)
then \(\lambda^2+\lambda\) is
✓ Correct answer: b)
\(\frac{9}{2}\)
Explanation
\(\begin{aligned}
&\text { Given system has infinite solution }\\
&\begin{aligned}
& \Rightarrow\left|\begin{array}{ccc}
\lambda-4 & \lambda-2 & \lambda \\
2 & -3 & 5 \\
1 & 2 & 6
\end{array}\right|=0 \\
& \Rightarrow(\lambda-4)(-28)-(\lambda-2) 7+\lambda \times 7=0 \\
& \Rightarrow-4(\lambda-4)-(\lambda-2)+d=0 \\
& \Rightarrow-4 \lambda+16-\lambda+2+\lambda=0 \\
& =-4 \lambda=-18 \quad\lambda=\frac{9}{2}
\end{aligned}
\end{aligned}\)
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