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-

If $f(\theta ) =\left| {\begin{array}{*{20}{c}}
1&{\cos {\mkern 1mu} \theta }&1\\
{ - \sin {\mkern 1mu} \theta }&1&{ - \cos {\mkern 1mu} \theta }\\
{ - 1}&{\sin {\mkern 1mu} \theta }&1
\end{array}} \right|$ and $A$ and $B$ are respectively the maximum and the minimum values of $f(\theta )$, then $(A , B)$ is equal to

Prove that the determinant $\left|\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right|$ is independent of $\theta$

The value of $\lambda $ for which the system of equations $2x - y - z = 12,$ $x - 2y + z = - 4,$ $x + y + \lambda z = 4$ has no solution is

The following system of linear equations  $7 x+6 y-2 z=0$ ; $3 x+4 y+2 z=0$ ; ${x}-2{y}-6{z}=0,$ has

Consider the system of linear equations

$x+y+z=5, x+2 y+\lambda^2 z=9$

$x+3 y+\lambda z=\mu$, where $\lambda, \mu \in R$. Then, which of the following statement is NOT correct?