The addition of a catalyst during a chemical reaction alters which of the following quantities?

The temperature at which the rate constants of the two gaseous reactions given below become equal is . . . . . . $K$. (Nearest integer).
$X \longrightarrow Y \quad k_1 = 10^6 e^{\frac{-30000}{T}}$
$P \longrightarrow Q \quad k_2 = 10^4 e^{\frac{-24000}{T}}$
Given: $\ln 10 = 2.303$

On introducing a catalyst at $500 \, K,$ the rate constant of a first order reaction increases $2.718$ times. If the activation energy in the presence of a catalyst is $4.15 \, kJ \, mol^{-1},$ then what will be $E_a$ in the absence of a catalyst? (Value of $e = 2.718$ and $R = 8.314 \times 10^{-3} \, kJ \, K^{-1} \, mol^{-1}$)

In most cases,for a rise of $10 \ K$ temperature,the rate constant is doubled or tripled. This is due to the reason that

The activation energy of a first-order reaction at $25\,^{\circ}C$ is $30\,kJ/mol$. In the presence of a catalyst,the activation energy of the same reaction at $25\,^{\circ}C$ becomes $24\,kJ/mol$. The rate of the reaction in the presence of the catalyst will be how many times the original rate?

The activation energy of a certain reaction is $87 \ kJ \ mol^{-1}$. When the temperature is decreased from $37^{\circ} C$ to $15^{\circ} C$,what is the ratio of the rate constant at $37^{\circ} C$ to that at $15^{\circ} C$ for this reaction (in $/ 1$)?