The number of values of α for which the system of equations x + y +...

Given linear equations are

$x + y + z = \alpha$

$\alphax + 2\alphay + 3z = - 1$

$x + 3\alphay + 5z = 4$

For this system of linear equations to have inconsistent solution

D = \left|\begin{cases}1 & 1 & 1 \\ \alpha & 2\alpha & 3 \\ 1 & 3\alpha & 5\end{cases}\right| = 0

$⇒ {\left(\alpha - 1\right)}^{2} = 0$

$⇒ \alpha = 1$

Now for $\alpha = 1$

{D}_{1} = \left|\begin{cases}1 & 1 & 1 \\ - 1 & 2 & 3 \\ 4 & 3 & 5\end{cases}\right|

$= \left(10 - 9\right) - \left(- 5 - 12\right) + \left(- 3 - 8\right)$

$= 1 + 17 - 11 \ne 0$

Hence, for $\alpha = 1$ the system of equation has inconsistent solution.