If ${\Delta _1} = \left| {\begin{array}{*{20}{c}}
  x&{\sin \,\theta }&{\cos \,\theta } \\ 
  {\sin \,\theta }&{ – x}&1 \\ 
  {\cos \,\theta }&1&x 
\end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{*{20}{c}}
  x&{\sin \,2\theta }&{\cos \,\,2\theta } \\ 
  {\sin \,2\theta }&{ – x}&1 \\ 
  {\cos \,\,2\theta }&1&x 
\end{array}} \right|$, $x \ne 0$ ; then for all $\theta  \in \left( {0,\frac{\pi }{2}} \right)$

Let the system of linear equations  $x+y+k z=2$ ;  $2 x+3 y-z=1$ ; $3 x+4 y+2 z=k$ , have infinitely many solutions. Then the system $( k +1) x +(2 k -1) y =7$ ; $(2 k +1) x +( k +5) y =10 \text { has : }$

For all values of $A,B,C$ and $P,Q,R$, the value of $\left| {\,\begin{array}{*{20}{c}}{\cos (A – P)}&{\cos (A – Q)}&{\cos (A – R)}\\{\cos (B – P)}&{\cos (B – Q)}&{\cos (B – R)}\\{\cos (C – P)}&{\cos (C – Q)}&{\cos (C – R)}\end{array}\,} \right|$ is

The value of $x,$ if $\left| {\,\begin{array}{*{20}{c}}{ – x}&1&0\\1&{ – x}&1\\0&1&{ – x}\end{array}\,} \right| = 0 $ is equal to

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