Consider the system of linear equations x+y+z=5, x+2 y+ ^2 z=9 x+3 y+ z= , where , R. Then, which of the following statement is NOT correct?

D. System has unique solution if 1 and 13

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

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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 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 $\mu=13$
✓
System has unique solution if $\lambda \neq 1$ and $\mu \neq 13$

✓

Answer

Correct option: D.

System has unique solution if $\lambda \neq 1$ and $\mu \neq 13$

d
$\begin{aligned} & \left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & \lambda^2 \\ 1 & 3 & \lambda\end{array}\right|=0 \\ & \Rightarrow 2 \lambda^2-\lambda-1=0 \\ & \lambda=1,-\frac{1}{2} \\ & \left|\begin{array}{ccc}1 & 1 & 5 \\ 2 & \lambda^2 & 9 \\ 3 & \lambda & \mu\end{array}\right|=0 \Rightarrow \mu=13\end{aligned}$

Infinite solution $\lambda=1 \& \mu=13$

For unique $\operatorname{sol}^{\mathrm{n}} \lambda \neq 1$

For no $\operatorname{sol}^{\mathrm{n}} \lambda=1 \& \mu \neq 13$

If $\lambda \neq 1$ and $\mu \neq 13$

Considering the case when $\lambda=-\frac{1}{2}$ and $\mu \neq 13$ this will generate no solution case

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