The number of solutions of the equations $x + 4y - z = 0,$ $3x - 4y - z = 0,\,x - 3y + z = 0$ is

For how many diff erent values of $a$ does the following system have at least two distinct solutions?

$a x+y=0$

$x+(a+10) y=0$

Evaluate the determinants

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

Find values of $\mathrm{k}$ if area of triangle is $4$ square units and vertices are $(-2,0),(0,4),(0, \mathrm{k})$

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations

$x+y+z=5$    ;    $x+2 y+3 z=\mu$   ;     $x+3 y+\lambda z=1$

is constructed. If $\mathrm{p}$ is the probability that the system has a unique solution and $\mathrm{q}$ is the probability that the system has no solution, then :

$2x + 3y + 4z = 9$,$4x + 9y + 3z = 10,$$5x + 10y + 5z = 11$ then the value of $ x$ is