If the following system of linear equations

$2 x+y+z=5$

$x-y+z=3$

$x+y+a z=b$

has no solution, then :

Which of the following is correct?

Let $\omega = - \frac{1}{2} + i\frac{{\sqrt 3 }}{2}$. Then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{ - 1 - {\omega ^2}}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^4}}\end{array}\,} \right|$ is

Find equation of line joining $(1,2)$ and $(3,6)$ using determinates

Let $\lambda \in R .$ The system of linear equations

$2 x_{1}-4 x_{2}+\lambda x_{3}=1$

$x_{1}-6 x_{2}+x_{3}=2$

$\lambda x_{1}-10 x_{2}+4 x_{3}=3$  is inconsistent for 

$\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{m{a_1}}&{{b_1}}\\{{a_2}}&{m{a_2}}&{{b_2}}\\{{a_3}}&{m{a_3}}&{{b_3}}\end{array}\,} \right| = $