Consider system of equations in $x$ , $y$ and $z$

$12x + by + cz = 0$ ;   $ax + 24y + cz = 0$  ;   $ax + by + 36z = 0$ .

(where $a$ , $b$ , $c$ are real numbers, $a \ne 12$ , $b \ne 24$ , $c \ne 36$ ).

If system of equation has solution and $z \ne 0$, then value of  $\frac{1}{{a - 12}} + \frac{2}{{b - 24}} + \frac{3}{{c - 36}}$ is

Suppose $D = \left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|$ and $D' = \left| {\,\begin{array}{*{20}{c}}{{a_1} + p{b_1}}&{{b_1} + q{c_1}}&{{c_1} + r{a_1}}\\{{a_2} + p{b_2}}&{{b_2} + q{c_2}}&{{c_2} + r{a_2}}\\{{a_3} + p{b_3}}&{{b_3} + q{c_3}}&{{c_3} + r{a_3}}\end{array}\,} \right|$, then

Let $p$ and $p+2$ be prime numbers and let $\Delta=\left|\begin{array}{ccc}p ! & (p+1) ! & (p+2) ! \\ (p+1) ! & (p+2) ! & (p+3) ! \\ (p+2) ! & (p+3) ! & (p+4) !\end{array}\right|$ Then the sum of the maximum values of $\alpha$ and $\beta$, such that $p ^{\alpha}$ and $( p +2)^{\beta}$ divide $\Delta$, is $........$

For what value of $k$ to the following system of equations possess a non-trivial solution ?

$x + ky + 3z = 0$   ;    $3x + ky + 2z = 0$  ; $2x + 3y + 4z = 0$

Consider the system of linear equations

$-x+y+2 z=0$

$3 x-a y+5 z=1$

$2 x-2 y-a z=7$

Let $S_{1}$ be the set of all $\mathrm{a} \in {R}$ for which the system is inconsistent and $S_{2}$ be the set of all $a \in {R}$ for which the system has infinitely many solutions. If $n\left(S_{1}\right)$ and $n\left(S_{2}\right)$ denote the number of elements in $S_{1}$ and $\mathrm{S}_{2}$ respectively, then

If ${x^a}{y^b} = {e^m},{x^c}{y^d} = {e^n},{\Delta _1} = \left| {\,\begin{array}{*{20}{c}}m&b\\n&d\end{array}\,} \right|\,\,{\Delta _2} = \left| {\,\begin{array}{*{20}{c}}a&m\\c&n\end{array}\,} \right|$ and ${\Delta _3} = \left| {\,\begin{array}{*{20}{c}}a&b\\c&d\end{array}\,} \right|$, then the values of $x$  and $y$  are respectively