JEE Main 2024 (Online) 30th January Evening ShiftQuestion 1
Consider the system of linear equations $x+y+z=5$, $x+2y+\lambda^{2}z=9$, $x+3y+\lambda z=\mu$, where $\lambda,\mu\in\mathbb{R}$. Then, which of the following statement is NOT correct?
Determinant
Solution
$\Delta=\begin{vmatrix}1&1&1\\1&2&\lambda^2\\1&3&\lambda\end{vmatrix}=-(\lambda-1)(2\lambda+1)$, zero at $\lambda=1$ and $\lambda=-\frac12$. $\lambda=1$: consistent iff $\mu=13$ (infinite), inconsistent otherwise: (B),(D) hold. $\lambda=-\frac12,\mu=13$ is consistent (rank 2 = 2), and $\lambda\ne1,\mu=13$ is consistent otherwise by $\Delta\ne0$: (A) holds. (C) fails: for $\lambda=-\frac12$ ($\ne1$) and $\mu\ne13$ the system has no unique solution since $\Delta=0$. So (C) is NOT correct. Option C.