JEE MAIN - Mathematics (2024 - 30th January Evening Shift - No. 16)
Consider the system of linear equations $$x+y+z=5, x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu$$, where $$\lambda, \mu \in \mathbb{R}$$. Then, which of the following statement is NOT correct?
System is consistent if $$\lambda \neq 1$$ and $$\mu=13$$
System is inconsistent if $$\lambda=1$$ and $$\mu \neq 13$$
System has unique solution if $$\lambda \neq 1$$ and $$\mu \neq 13$$
System has infinite number of solutions if $$\lambda=1$$ and $$\mu=13$$
Explanation
$$\begin{aligned} & \left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & \lambda^2 \\ 1 & 3 & \lambda \end{array}\right|=0 \\ & \Rightarrow 2 \lambda^2-\lambda-1=0 \\ & \lambda=1,-\frac{1}{2} \\ & \left|\begin{array}{ccc} 1 & 1 & 5 \\ 2 & \lambda^2 & 9 \\ 3 & \lambda & \mu \end{array}\right|=0 \Rightarrow \mu=13 \end{aligned}$$
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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