What is the relationship between the temperature $T$ and the activation energy $E_a$ in the Arrhenius equation?

For an endothermic reaction $X \rightarrow Y$,the activation energies for the forward and backward reactions are $E_f$ and $E_b$ respectively. Then,in general:

In a first order reaction at $27\,^oC$ and $47\,^oC$,$50\%$ of the reaction is complete in $30\,\min$ and $10\,\min$ respectively. Calculate the energy of activation $(E_a)$.

The relation between the fraction of molecules $\left( \frac{N_E}{N_T} \right)$ and temperature is explained by the Boltzmann and Maxwell distribution graph.

The rate of a chemical reaction doubles for every $10\,^{\circ}C$ rise of temperature. If the temperature is raised by $50\,^{\circ}C,$ the rate of the reaction increases by about ......... times.

The energies of activation for forward and reverse reactions for $A_2 + B_2 \rightleftharpoons 2AB$ are $180 \, kJ \, mol^{-1}$ and $200 \, kJ \, mol^{-1}$ respectively. The presence of a catalyst lowers the activation energy of both (forward and reverse) reactions by $100 \, kJ \, mol^{-1}.$ The enthalpy change of the reaction $(A_2 + B_2 \rightarrow 2AB)$ in the presence of a catalyst will be (in $kJ \, mol^{-1}$)