Find area of the triangle with vertices at the point given in each of the following: $(-2,-3),(3,2),(-1,-8)$

If the system of equations $\alpha x+y+z=5, x+2 y+$ $3 z=4, x+3 y+5 z=\beta$ has infinitely many solutions, then the ordered pair $(\alpha, \beta)$ is equal to:

Find values of ${x},$ if  $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{ll}x & 3 \\ 2 x & 5\end{array}\right|$

For which of the following ordered pairs $(\mu, \delta)$ the system of linear equations  $x+2 y+3 z=1$ ; $3 x+4 y+5 z=\mu$ ; $4 x+4 y+4 z=\delta$ is inconsistent?

Let $d \in R$, and  $A = \left[ {\begin{array}{*{20}{c}} { - 2}&{4 + d}&{\left( {\sin \,\theta } \right) - 2}\\ 1&{\left( {\sin \,\theta } \right) + 2}&d\\ 5&{\left( {2\sin \,\theta } \right) - d}&{\left( { - \sin \,\theta } \right) + 2 + 2d} \end{array}} \right]$, $\theta  \in \left[ {0,2\pi } \right]$. If the minimum value of det $(A)$ is $8$, then a value of $d$ is

If ${a_1},{a_2},{a_3}.....{a_n}....$ are in $G.P.$ then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}{\log {a_n}}&{\log {a_{n + 1}}}&{\log {a_{n + 2}}}\\{\log {a_{n + 3}}}&{\log {a_{n + 4}}}&{\log {a_{n + 5}}}\\{\log {a_{n + 6}}}&{\log {a_{n + 7}}}&{\log {a_{n + 8}}}\end{array}\,} \right|$ is