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

The number of $\theta \in(0,4 \pi)$ for which the system of linear equations

$3(\sin 3 \theta) x-y+z=2$, $3(\cos 2 \theta) x+4 y+3 z=3$, $6 x+7 y+7 z=9$ has no solution is.

Let $S$ be the set of all values of $\theta \in[-\pi, \pi]$ for which the system of linear equations

$x+y+\sqrt{3} z=0$

$-x+(\tan \theta) y+\sqrt{7} z=0$

$x+y+(\tan \theta) z=0$

has non-trivial solution. Then $\frac{120}{\pi} \sum_{\theta \in s} \theta$ is equal to

If $\alpha+\beta+\gamma=2 \pi$, then the system of equations

$x+(\cos \gamma) y+(\cos \beta) z=0$

$(\cos \gamma) x+y+(\cos \alpha) z=0$

$(\cos \beta) x+(\cos \alpha) y+z=0$

has :

The following system of linear equations  $2 x+3 y+2 z=9$ ; $3 x+2 y+2 z=9$  ;$x-y+4 z=8$

Let $x, y, z > 0$ are respectively $2^{nd}, 3^{rd}, 4^{th}$ term of $G.P.$ and $\Delta  = \left| {\begin{array}{*{20}{c}}
{{X^k}}&{{X^{k + 1}}}&{{X^{k + 2}}}\\
{{Y^k}}&{{Y^{k + 1}}}&{{Y^{k + 2}}}\\
{{Z^k}}&{{Z^{k + 1}}}&{{Z^{k + 2}}}
\end{array}} \right| = {\left( {r - 1} \right)^2}\left( {1 - \frac{1}{{{r^2}}}} \right)$ , (where $r$ is common ratio), then $k=$ .......