**Question:**

For which of the following ordered pairs $(\mu, \delta)$, the system of linear equations

$x+2 y+3 z=1$

$3 x+4 y+5 z=\mu$

$4 x+4 y+4 z=\delta$

is inconsistent?

Correct Option: 1

**Solution:**

From the given linear equation, we get

$D=\left|\begin{array}{lll}1 & 2 & 3 \\ 3 & 4 & 5 \\ 4 & 4 & 4\end{array}\right|\left(R_{3} \rightarrow R_{3}-2 R_{2}+3 R_{3}\right)$

$=\left|\begin{array}{lll}1 & 2 & 3 \\ 3 & 4 & 5 \\ 0 & 0 & 0\end{array}\right|=0$

Now, let $P_{3}=4 x+4 y+4 z-\delta=0$. If the system has

solutions it will have infinite solution.

Hence, $3 \alpha+\beta=4$ and $4 \alpha+2 \beta=4$

$\Rightarrow \alpha=2$ and $\beta=-2$

So, for infinite solution $2 \mu-2=\delta$

$\Rightarrow$ For $2 \mu \neq \delta+2$ system is inconsistent

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