In respect of the equation $k = A e^{-E_a/RT}$ in chemical kinetics,which one of the following statements is correct?

If $E_f$ and $E_r$ are the activation energies of forward and reverse reactions and the reaction is known to be exothermic,then

Write the logarithmic form of the Arrhenius equation $k = A e^{-\frac{E_a}{RT}}$.

For the reaction $A_2 + B_2 \rightleftharpoons 2 AB$,the activation energies $(E_a)$ for the forward and backward reactions are $180 \ kJ \ mol^{-1}$ and $200 \ kJ \ mol^{-1}$ respectively. If a catalyst lowers the $E_a$ for both reactions by $100 \ kJ \ mol^{-1}$,which of the following statements is correct?

$A \rightarrow B$. The molecule $A$ changes into its isomeric form $B$ following first-order kinetics at a temperature of $1000 \ K$. If the energy barrier with respect to reactant energy for such isomeric transformation is $191.48 \ kJ \ mol^{-1}$ and the frequency factor is $10^{20} \ s^{-1}$,the time required for $50 \%$ of molecules of $A$ to become $B$ is $..............$ picoseconds (nearest integer). $[R = 8.314 \ J \ K^{-1} \ mol^{-1}]$

The rate of reaction is tripled for a $10^\circ C$ rise in temperature. The increase in the reaction rate if the temperature is increased by $60^\circ C$ is $............$ times.