Three digit numbers $x17, 3y6$ and $12z$ where $x, y, z$ are integers from $0$ to $9$, are divisible by a fixed constant $k$. Then the determinant $\left| {\,\begin{array}{*{20}{c}}x&3&1\\7&6&z\\1&y&2\end{array}\,} \right|$ + $48$ must be divisible by

Consider system of equations  $ x + y -az = 1$  ;  $2x + ay + z = 1$   ; $ax + y -z = 2$

Find values of $x$ for which $\left|\begin{array}{ll}3 & x \\ x & 1\end{array}\right|=\left|\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right|$

Let $\mathrm{A}$ be a square matrix of order $3 \times 3$ , then $|\mathrm{k A}|$ is equal to

If $a > 0$and discriminant of $a{x^2} + 2bx + c$is negative, then $\left| {\,\begin{array}{*{20}{c}}a&b&{ax + b}\\b&c&{bx + c}\\{ax + b}&{bx + c}&0\end{array}\,} \right|$ is

If $a, b, c$ are sides of a scalene triangle, then the value of $\left| \begin{array}{*{20}{c}}
a&b&c\\
b&c&a\\
c&a&b
\end{array} \right|$ is