The determinant $\left| {\begin{array}{*{20}{c}}{\cos \,\,(\theta \, + \,\phi )}&{ - \,\sin \,\,(\theta \, + \,\phi )}&{\cos \,2\phi }\\{\sin \,\theta }&{\cos \,\theta }&{\sin \,\phi }\\{ - \,\cos \,\theta }&{\sin \,\theta }&{\cos \,\phi }\end{array}} \right|$ is :

If $A$, $B$ and $C$ are square matrices of order $3$ such that $A = \left[ {\begin{array}{*{20}{c}}   x&0&1 \\    0&y&0 \\    0&0&z  \end{array}} \right]$ and $\left| B \right| = 36$, $\left| C \right| = 4$,  $\left( {x,y,z \in N} \right)$ and $\left| {ABC} \right| = 1152$ then the minimum value of $x + y + z$ is

If $A\, = \,\left[ \begin{gathered}
  1\ \ \ \,1\ \ \ \,2\ \ \  \hfill \\
  0\ \ \ \,2\ \ \ \,1\ \ \  \hfill \\
  1\ \ \ \,0\ \ \ \,2\ \ \  \hfill \\ 
\end{gathered}  \right]$ and $A^3 = (aA-I) (bA-I)$,where $a, b$ are integers and $I$ is a $3 × 3$ unit matrix then value of $(a + b)$ is equal to

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 $\qquad$

The area of a triangle is $5$ and two of its vertices are $A(2, 1), B(3, -2)$. The third  vertex which lies on line $y = x + 3$ is-

Consider the following system of questions $\alpha x+2 y+z=1$  ;  $2 \alpha x+3 y+z=1$  ;  $3 x+\alpha y+2 z=\beta$ . For some $\alpha, \beta \in R$. Then which of the following is NOT correct.