If ${2^{{a_1}}},{2^{{a_2}}},{2^{{a_3}}},{......2^{{a_n}}}$ are in $G.P.$ then $\left| {\begin{array}{*{20}{c}}
  {{a_1}}&{{a_2}}&{{a_3}} \\ 
  {{a_{n + 1}}}&{{a_{n + 2}}}&{{a_{n + 3}}} \\ 
  {{a_{2n + 1}}}&{{a_{2n + 2}}}&{{a_{2n + 3}}} 
\end{array}} \right|$ is equal to

Let $\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in R$. If $x(\alpha, 1,2)+y(1, \beta, 2)$ $+z(2,3, \gamma)=(0,0,0)$ for some $x, y, z \in R, x y z \neq$ 0 , then $6 \alpha+4 \beta+\gamma$ is equal to..............

$x + ky - z = 0,3x - ky - z = 0$ and $x - 3y + z = 0$ has non-zero solution for $k =$

If $\left| {\begin{array}{*{20}{c}}
  {^9{C_4}}&{^9{C_5}}&{^{10}{C_r}} \\ 
  {^{10}{C_6}}&{^{10}{C_7}}&{^{11}{C_{r + 2}}} \\ 
  {^{11}{C_8}}&{^{11}{C_9}}&{^{12}{C_{r + 4}}} 
\end{array}} \right| = 0$ then $r$ is equal to 

$S$ denote the set of all real values of $\lambda$ such that the system of equations  $\lambda x + y + z =1$ ; $x +\lambda y + z =1$ ; $x + y +\lambda z =1$ is inconsistent, then $\sum_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)$ is equal to

Let $\alpha, \beta$ and $\gamma$ be real numbers. consider the following system of linear equations

$x+2 y+z=7$

$x+\alpha z=11$

$2 x-3 y+\beta z=\gamma$

Match each entry in List - $I$ to the correct entries in List-$II$

Then the system has