Consider the system of linear equation x + y + z = 4μ, x + 2y + 2 =...

Given: 

System of linear equations,

$x + y + z = 4\mu,$

$x + 2y + 2\lambdaz = 10\mu,$

$x + 3y + 4\lambda z 2 = {\mu}^{2} + 15$.

⇒ \Delta = \left|\begin{cases}1 & 1 & 1 \\ 1 & 2 & 2\lambda \\ 1 & 3 & 4{\lambda}^{2}\end{cases}\right|

$⇒ \Delta = \left(8{\lambda}^{2} - 6\lambda\right) - \left(4{\lambda}^{2} - 2\lambda\right) + \left(3 - 2\right)$

$⇒ \Delta = 8{\lambda}^{2} - 6\lambda - 4{\lambda}^{2} + 2\lambda + 1$

$⇒ \Delta = 4{\lambda}^{2} - 4\lambda + 1$

$⇒ \Delta = ( 2\lambda - 1{)}^{2}$

For unique solution, $\Delta \ne 0$.

$⇒ 2\lambda - 1 \ne 0$

$⇒ \lambda \ne \dfrac{1}{2}$

Let, $\Delta = 0, \lambda = \dfrac{1}{2}$

We know that, for infinite solutions, ${\Delta}_{x} = {\Delta}_{y} = {\Delta}_{z} = 0$

⇒ {\Delta}_{x} = \left|\begin{cases}4\mu & 1 & 1 \\ 10\mu & 2 & 1 \\ {\mu}^{2} + 15 & 3 & 1\end{cases}\right|

$⇒ {\Delta}_{x} = 4\mu\left(2 - 3\right) - \left(10\mu - {\mu}^{2} - 15\right) + \left(30\mu - 2{\mu}^{2} - 30\right)$

$⇒ {\Delta}_{x} = - 4\mu - 10\mu + {\mu}^{2} + 15 + 30\mu - 2{\mu}^{2} - 30$

$⇒ {\Delta}_{x} = - {\mu}^{2} + 16\mu - 15$

$⇒ ( \mu - 15 ) ( \mu - 1 ) = 0$

So, for infinite solution $\lambda = \dfrac{1}{2}, \mu = 1$ or $15$.