For the reaction $N_2 + 3H_2 \rightleftharpoons 2NH_3$,the equilibrium constant is $k$. What is the equilibrium constant for the reaction $2N_2 + 6H_2 \rightleftharpoons 4NH_3$?

$15$ moles of $H_2$ and $5.2$ moles of $I_2$ are mixed and allowed to attain equilibrium at $500 \, ^oC$. At equilibrium,the concentration of $HI$ is found to be $10$ moles. The equilibrium constant for the formation of $HI$ is

$4.5 \ mol$ each of hydrogen and iodine are heated in a $10 \ L$ closed vessel. At equilibrium,$3 \ mol$ of $HI$ is formed. The equilibrium constant for the reaction $H_2(g) + I_2(g) \rightleftharpoons 2HI(g)$ will be:

For the reaction equilibrium $N_2O_4(g) \rightleftharpoons 2NO_2(g)$,the concentrations of $N_2O_4$ and $NO_2$ at equilibrium are $4.8 \times 10^{-2} \ mol \ L^{-1}$ and $1.2 \times 10^{-2} \ mol \ L^{-1}$ respectively. The value of $K_c$ for the reaction is:

$K_1, K_2$ and $K_3$ are the equilibrium constants of the following reactions $(I), (II)$ and $(III)$ respectively.
$(I) \, N_2 + 2O_2 \rightleftharpoons 2NO_2$
$(II) \, 2NO_2 \rightleftharpoons N_2 + 2O_2$
$(III) \, NO_2 \rightleftharpoons \frac{1}{2} N_2 + O_2$
The correct relation from the following is

If the equilibrium constant for the reaction $2AB \rightleftharpoons A_2 + B_2$ is $49$,what is the equilibrium constant for $AB \rightleftharpoons \frac{1}{2}A_2 + \frac{1}{2}B_2$?