If the system of equations $x+y+z=6 \,; \,2 x+5 y+\alpha z=\beta  \,; \, x+2 y+3 z=14$ has infinitely many solutions, then $\alpha+\beta$ is equal to.

Let $\theta \in\left(0, \frac{\pi}{2}\right)$. If the system of linear equations

$\left(1+\cos ^{2} \theta\right) x+\sin ^{2} \theta y+4 \sin 3 \theta z=0$

$\cos ^{2} \theta x+\left(1+\sin ^{2} \theta\right) y+4 \sin 3 \theta z=0$

$\cos ^{2} \theta x+\sin ^{2} \theta y+(1+4 \sin 3 \theta) z=0$

has a non-trivial solution, then the value of $\theta$ is :

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

Find equation of line joining $(1,2)$ and $(3,6)$ using determinates

If $A_1B_1C_1,\, A_2B_2C_2,\, A_3B_3C_3$ are three digit number each of which is divisible by $k$ and $\Delta  = \left| {\begin{array}{*{20}{c}}
  {{A_1}{\kern 1pt} }&{{B_1}}&{{C_1}} \\ 
  {{A_2}}&{{B_2}}&{{C_2}} \\ 
  {{A_3}}&{{B_3}}&{{C_3}} 
\end{array}} \right|$ ; then $\Delta $ is divisible by

For positive numbers $x,y$ and $z$  the numerical value of the determinant $\left| {\,\begin{array}{*{20}{c}}1&{{{\log }_x}y}&{{{\log }_x}z}\\{{{\log }_y}x}&1&{{{\log }_y}z}\\{{{\log }_z}x}&{{{\log }_z}y}&1\end{array}\,} \right|$is