Match the following

Column - I		Column - II
I.	If the system of equations  $2x+3ky+\left(3k+4\right)z=0$ $x+\left(k+4\right)y+\left(4k+2\right)z=0$ $x+2\left(k+4\right)y+\left(3k+4\right)z=0$ has non-trivial solution then k = ___	P.	1
II.	Rank of the matrix $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$	Q.	$\frac{1}{2}\left(or\right)-8$
III.	If $\left[\begin{array}{ccc}1& 2& x\\ 4& -1& 7\\ 2& 4& -6\end{array}\right]$ is a singular matrix then $\text{x}=\_\_\_$	R.	-3

(1) Det A = O $\left|\begin{array}{ccc}2& 3k& 3k+4\\ 1& k+4& 4k+2\\ 1& 2k+8& 3k+4\end{array}\right|=0$

$2\left[\left(k+4\right)\left(3k+4\right)-\left(4k+2\right)\left(2k+8\right)\right]-3k\left[3k+4-4k-2\right]+\left(3k+4\right)\left[2k+8-k-4\right]=0$

$2\left[3k^2+16k+16-8k^2-36k-16\right]-3k\left(2-k\right)+\left(3k+4\right)\left(k+4\right)=0$

$-10k^2-40k+6k^2-6k+16k+16=0$

$\Rightarrow -2\left[2k^2+15k-8\right]=0\left(2k-1\right)\left(k+8\right)=0$

$\therefore k=\frac{1}{2}\left(or\right)k=-8$

(2) $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ Det of $3\times 3$ and $2\times 2$ matrix = 0

$\therefore Rank=1$

(3) Condition: $\text{Det A }=0 \text{1}\left(-\text{22}\right)-\text{2}\left(-\text{38}\right)+\text{ x }\left(\text{18}\right)=0$ 

$\text{18x }+\text{ 54 }=0$

$\text{x }=-\text{ 3}$