If the system of equations, $x + 2y – 3z = 1$, $(k + 3)z = 3,$ $(2k + 1)x + z = 0$is inconsistent, then the value of $ k$  is

Evaluate $\Delta=\left|\begin{array}{lll}3 & 2 & 3 \\ 2 & 2 & 3 \\ 3 & 2 & 3\end{array}\right|$

If the system of equations

$ 2 x+7 y+\lambda z=3 $

$ 3 x+2 y+5 z=4 $

$ x+\mu y+32 z=-1$

has infinitely many solutions, then $(\lambda-\mu)$ is equal to $\qquad$

If a system of the equation ${(\alpha + 1)^3}x + {(\alpha + 2)^3}y – {(\alpha + 3)^3} = 0$ and $(\alpha + 1)x + (\alpha + 2)y – (\alpha + 3) = 0,x + y – 1 = 0$ is constant. what is the value of $\alpha $

If $C = 2\cos \theta $, then the value of the determinant $\Delta = \left| {\,\begin{array}{*{20}{c}}C&1&0\\1&C&1\\6&1&C\end{array}\,} \right|$ is