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 $A$, $B$ and $C$ are square matrices of order $3$ such that $A = \left[ {\begin{array}{*{20}{c}}   x&0&1 \\    0&y&0 \\    0&0&z  \end{array}} \right]$ and $\left| B \right| = 36$, $\left| C \right| = 4$,  $\left( {x,y,z \in N} \right)$ and $\left| {ABC} \right| = 1152$ then the minimum value of $x + y + z$ is

If the system of equations $x + 2y + 3z = 4 , x + py + 2z = 3 , x + 4y + \mu z = 3$ has an infinite number of solutions , then :

Find the equation of the line joining $\mathrm{A}(1,3)$ and $\mathrm{B}(0,0)$ using determinants and find $\mathrm{k}$ if $\mathrm{D}(\mathrm{k}, 0)$ is a point such that area of triangle $\mathrm{ABD}$ is $3 \,\mathrm{sq}$ $\mathrm{units}$.

Evaluate the determinants

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

If a system of the equation ${(\alpha + 1)^3}x + {(\alpha + 2)^3}y - {(\alpha + 3)^3} = 0$ and $(\alpha + 1)x + (\alpha + 2)y - (\alpha + 3) = 0,x + y - 1 = 0$ is constant. what is the value of $\alpha $