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-

The system of equations $kx + y + z =1$ $x + ky + z = k$ and $x + y + zk = k ^{2}$ has no solution if $k$ is equal to

If $a, b, c$ are three complex numbers such that $a^2 + b^2 + c^2 = 0$ and  $\left| {\begin{array}{*{20}{c}}
{\left( {{b^2} + {c^2}} \right)}&{ab}&{ac}\\
{ab}&{\left( {{c^2} + {a^2}} \right)}&{bc}\\
{ac}&{bc}&{\left( {{a^2} + {b^2}} \right)}
\end{array}} \right| = K{a^2}{b^2}{c^2}$ then value of $K$ is

The values of the determinant $\left| {\,\begin{array}{*{20}{c}}1&{\cos (\alpha - \beta )}&{\cos \alpha }\\{\cos (\alpha - \beta )}&1&{\cos \beta }\\{\cos \alpha }&{\cos \beta }&1\end{array}\,} \right|$ is

The system of equations $kx + 2y\,-z = 1$  ;  $(k\,-\,1)y\,-2z = 2$  ;  $(k + 2)z = 3$ has unique solution, if $k$ is equal to

If $A, B, C$ are the angles of triangle then the value of determinant $\left| {\begin{array}{*{20}{c}}
  {\sin \,2A}&{\sin \,C}&{\sin \,B} \\ 
  {\sin \,C}&{\sin \,2B}&{\sin A} \\ 
  {\sin \,B}&{\sin \,A}&{\sin \,2C} 
\end{array}} \right|$ is