A jeweler has bars of $18$-carat gold and $12$- carat gold. How much of each must be melted together to obtain a bar of $16$-carat gold, weighing $120 \mathrm{gm}$? (Given: Pure gold is $24$-carat).

Find the value of $k$ for which the following pair of linear equations have infinitely many solutions.
$2x+3y=7$,
$\left(k-1\right)x+\left(k+2\right)y=3k$

For what value of $\mathrm{k}$ does the following pair of linear equations have infinitely many solutions?

$10\mathrm{x}+5\mathrm{y}-(\mathrm{k}-5)=0, 20\mathrm{x}+10\mathrm{y}-\mathrm{k}=0$

For what value of $\mathrm{k}$ will the following pair of linear equations have no solution?

$2\mathrm{x}+3\mathrm{y}=9, 6\mathrm{x}+(\mathrm{k}-2) \mathrm{y}=(3\mathrm{k}-2)$