Evaluate the determinant $\Delta = \begin{vmatrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0 \end{vmatrix}$.

$\left|\begin{array}{cc}\sin \frac{2 \pi}{9} & \cos \frac{2 \pi}{9} \\ \sin \frac{5 \pi}{18} & \cos \frac{5 \pi}{18}\end{array}\right|=$ . . . . . . .

The area of a triangle with vertices $(-3, 0)$,$(3, 0)$,and $(0, k)$ is $9$ sq units. Find the value of $k$.

If $a \neq b \neq c$,the value of $x$ which satisfies the equation $\left| \begin{array}{ccc} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{array} \right| = 0$ is

Let $P$ be a matrix of order $3 \times 3$ such that all the entries in $P$ are from the set $\{-1, 0, 1\}$. Then,the maximum possible value of the determinant of $P$ is:

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: