Consider the system of linear equations x+y+z=5, x+2 y+ ^2 z=9, x… - Vidyadip

MCQ

Consider the system of linear equations $x+y+z=5, x+2 y+\lambda^2 z=9$, $x+3 y+\lambda z=\mu$ where $\lambda, \mu \in R$. Then, which of the following statement is $\text{NOT}$ correct?

A
System has infinite number of solution if $\lambda=1$ and $\mu=13$
B
System is inconsistent if $\lambda=1$ and $\mu \neq 13$
C
System is consistent if $\lambda \neq 1$ and μ = 13
D
System has unique solution if $\lambda \neq 1$ and $\mu \neq 13$

✓

Answer

$\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & \lambda^2 \\ 1 & 3 & \lambda\end{array}\right|=0$
$\Rightarrow 2 \lambda^2-\lambda-1=0 \\ \lambda=1,-\frac{1}{2}$
$\left|\begin{array}{ccc}1 & 1 & 5 \\ 2 & \lambda^2 & 9 \\ 3 & \lambda & \mu\end{array}\right|=0$
$\Rightarrow \mu=13$
Infinite solution $\lambda=1$ & $\mu=13$
For unique $\operatorname{sol}^{ n } \lambda \neq 1$
If $\lambda \neq 1$ and $\mu \neq 13$
Considering the case when $\lambda=-\frac{1}{2}$ and $\mu \neq 13$ this will generate no soluton case

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