The value of $a$ for which the matrix $A = \begin{bmatrix} a & 2 \\ 2 & 4 \end{bmatrix}$ is singular is:

Evaluate the determinant: $\left|\begin{array}{rrr}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0\end{array}\right|$

If $a, b, c$ are distinct and rational numbers,then the value of the determinant $\left| \begin{array}{ccc} (a^2 + b^2 + c^2) & (ab + bc + ca) & (ab + bc + ca) \\ (ab + bc + ca) & (a^2 + b^2 + c^2) & (ab + bc + ca) \\ (ab + bc + ca) & (ab + bc + ca) & (a^2 + b^2 + c^2) \end{array} \right|$ is always:

For any $a, b, c \in R$,the determinant $\left|\begin{array}{lll}bc & b+c & 1 \\ ca & c+a & 1 \\ ab & a+b & 1\end{array}\right|$ is equal to

The number of real values of $t$ such that the system of homogeneous equations
$\begin{aligned}
t x+(t+1) y+(t-1) z &=0 \\
(t+1) x+t y+(t+2) z &=0 \\
(t-1) x+(t+2) y+t z &=0
\end{aligned}$
has non-trivial solutions is

Find the area of the triangle with vertices $(a \cos \theta, b \sin \theta)$,$(-a \sin \theta, b \cos \theta)$,and $(-a \cos \theta, -b \sin \theta)$.