If $A = \begin{bmatrix} 0 & k & k \\ k & -4 & -6 \\ k & -3 & -5 \end{bmatrix}$ is a singular matrix,then the value of $k$ is:

If $\alpha, \beta, \gamma$ are the roots of the equation $\left|\begin{array}{ccc}x & 2 & 2 \\ 2 & x & 2 \\ 2 & 2 & x\end{array}\right|=0$ and $\min (\alpha, \beta, \gamma)=\alpha$, then $2 \alpha+3 \beta+4 \gamma=$

If $a, b, c$ are in $A.P.$,find the value of $\left|\begin{array}{ccc} 2y+4 & 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c \end{array}\right|$.

Evaluate the determinant: $\left|\begin{array}{ll}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right|$

If $A$,$B$ and $C$ are the angles of a triangle,then the value of the determinant $\left| \begin{array}{ccc} -1 + \cos B & \cos C + \cos B & \cos B \\ \cos C + \cos A & -1 + \cos A & \cos A \\ -1 + \cos B & -1 + \cos A & -1 \end{array} \right|$ is:

The area of a triangle is $4$ sq. units,and its vertices are $(-2, 0)$,$(0, 4)$,and $(0, k)$. Find the value of $k$.