If the system of equations $x + y+  z = 5$ ; $x + 2y + 3z = 9$ ; $x + 3y + \alpha z = \beta $ has infinitely many solutions, then $\beta  - \alpha $ equals

If the system of equations

$ 11 x+y+\lambda z=-5 $

$ 2 x+3 y+5 z=3 $

$ 8 x-19 y-39 z=\mu$

has infinitely many solutions, then $\lambda^4-\mu$ is equal to :

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 $k=$ .......

The least value of the product $xyz$ for which the determinant $\left| {\begin{array}{*{20}{c}}
  x&1&1 \\ 
  1&y&1 \\ 
  1&1&z 
\end{array}} \right|$ is non-negative, is 

If $A=\left[\begin{array}{lll}1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9\end{array}\right],$ find $|A|$.

If the points $(2k, k), (k, 2k)$ and $(k, k)$ with $k > 0$ enclose a triangle of area $18$ square unit then centroid of triangle is equal to