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Consider the system of linear equation $x+y+z=$ $4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15$, where $\lambda, \mu \in R$. Which one of the following statements is $NOT$ correct?
The system has unique solution if $\lambda \neq \frac{1}{2}$ and $\mu \neq 1,15$
The system is inconsistent if $\lambda=\frac{1}{2}$ and $\mu \neq 1$
The system has infinite number of solutions if $\lambda=\frac{1}{2}$ and $\mu=15$
The system is consistent if $\lambda \neq \frac{1}{2}$
Solution
$ x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda $ $ { }^2 z=\mu^2+15$
$\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 2 & 2 \lambda \\ 1 & 3 & 4 \lambda^2\end{array}\right|=(2 \lambda-1)^2$
For unique solution $\Delta \neq 0,2 \lambda-1 \neq 0,\left(\lambda \neq \frac{1}{2}\right)$
Let $\Delta=0, \lambda=\frac{1}{2}$
$\Delta_{\mathrm{y}}=0, \Delta_{\mathrm{x}}=\Delta_{\mathrm{z}}=\left|\begin{array}{ccc}4 \mu & 1 & 1 \\ 10 \mu & 2 & 1 \\ \mu^2+15 & 3 & 1\end{array}\right|$
$=(\mu-15)(\mu-1)$
For infinite solution $\lambda=\frac{1}{2}, \mu=1$ or 15
