Let the system of linear equations $x +2 y + z =2$, $\alpha x +3 y - z =\alpha,-\alpha x + y +2 z =-\alpha$ be inconsistent. Then $\alpha$ is equal to

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

Evaluate $\left|\begin{array}{cc}x & x+1 \\ x-1 & x\end{array}\right|$

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 number of solutions of the equations $x + 4y - z = 0,$ $3x - 4y - z = 0,\,x - 3y + z = 0$ is

The number of solution of the following equations ${x_2} - {x_3} = 1,\,\, - {x_1} + 2{x_3} = - 2,$ ${x_1} - 2{x_2} = 3$ is