If the system of equations  $x +y + z = 6$ ; $x + 2y + 3z= 10$ ; $x + 2y + \lambda z = 0$ has a unique solution, then $\lambda $ is not equal to

If the system of equations

$x-2 y+3 z=9$

$2 x+y+z=b$

$x-7 y+a z=24$

has infinitely many solutions, then $a – b$ is equal to

If $A = \left| {\,\begin{array}{*{20}{c}}{\sin (\theta + \alpha )}&{\cos (\theta + \alpha )}&1\\{\sin (\theta + \beta )}&{\cos (\theta + \beta )}&1\\{\sin (\theta + \gamma )}&{\cos (\theta + \gamma )}&1\end{array}\,} \right|$ ,then

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$, then det. $[( I+ B)^{50} -50B]$ is equal to

If $x, y, z$ are in arithmetic progression with common difference $d , x \neq 3 d ,$ and the
determinant of the matrix $\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]$ is zero, then the value of $k ^{2}$ is ….. .