Determinants Question 8
Question 8 - 2024 (30 Jan Shift 2)
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?
(1) System has infinite number of solution if $\lambda=1$ and $\mu=13$
(2) System is inconsistent if $\lambda=1$ and $\mu \neq 13$
(3) System is consistent if $\lambda \neq 1$ and $\mu=13$
(4) System has unique solution if $\lambda \neq 1$ and $\mu \neq 13$
Show Answer
Answer (4)
Solution
$\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & \lambda^{2} \\ 1 & 3 & \lambda\end{array}\right|=0$
$\Rightarrow 2 \lambda^{2}-\lambda-1=0$
$\lambda=1,-\frac{1}{2}$
$\left|\begin{array}{ccc}1 & 1 & 5 \\ 2 & \lambda^{2} & 9 \\ 3 & \lambda & \mu\end{array}\right|=0 \Rightarrow \mu=13$
Infinite solution $\lambda=1 \& \mu=13$
For unique $\operatorname{sol}^{n} \lambda \neq 1$
For no $\operatorname{sol}^{n} \lambda=1 \& \mu \neq 13$
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
