The ordered pair $(a, b)$, for which the system of linear equations  $3 x-2 y+z=b$  ;  $5 x-8 y+9 z=3$  ;  $2 x+y+a z=-1$ has no solution, is

If the system of equations $2x + 3y - z = 0$, $x + ky - 2z = 0$ and  $2x - y + z = 0$ has a non -trivial solution $(x, y, z)$, then $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k$ is equal to

If $a,b,c$ be positive and not all equal, then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right|$ is

The system of linear equations $\lambda x+2 y+2 z=5$ ; $2 \lambda x+3 y+5 z=8$ ; $4 x+\lambda y+6 z=10$ has

Let $\mathrm{A}(-1,1)$ and $\mathrm{B}(2,3)$ be two points and $\mathrm{P}$ be a variable point above the line $A B$ such that the area of $\triangle \mathrm{PAB}$ is $10$ . If the locus of $\mathrm{P}$ is $\mathrm{ax}+\mathrm{by}=15$, then $5 a+2 b$ is :

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

$a x+y=0$

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