What is the slope of the graph between $\ln K$ and $\frac{1}{T}$ according to the Arrhenius equation?

When the temperature of a reaction is increased from $300 \ K$ to $310 \ K$,the rate of the reaction increases by $2.5$ times. If the rate constant of the reaction at $300 \ K$ is $K$,what will be the rate constant at $310 \ K$?

For a first-order reaction,the time taken to complete $10\%$ at $298\, K$ is equal to the time taken to complete $25\%$ at $308\, K$. The activation energy of the reaction is ................ $kJ/mol$.

When the temperature changes from $20\,^oC$ to $50\,^oC$,the rate of reaction becomes three times. The activation energy $(E_a)$ for the reaction is $.... \, kJ \, mol^{-1}$ $(R = 8.314 \, J \, K^{-1} \, mol^{-1})$.

The rate constant for the reaction,$COCl_{2(g)} \longrightarrow CO_{(g)} + Cl_{2(g)}$ is given by $\ln[k / (min^{-1})] = -11067 / T(K) + 31.33$. The temperature at which the rate of this reaction will be doubled from that at $25^{\circ} C$ is $..... \, ^{\circ} C$.

Consider the following graph of the kinetic energy distribution among molecules at temperature $T.$ If the temperature is increased,how would the resulting graph differ from the one above $:-$