Activation energy of any chemical reaction can be calculated if one knows the value of

Which of the following plots is in accordance with the Arrhenius equation?

The rate constant of a reaction is increased $4$ times after the addition of a catalyst to the reaction mixture at the same temperature of $27^{\circ} C$. The change in the activation energy of this reaction is (Take $\ln(1/4) = -1.386, R = 8.314 \ J \ K^{-1} \ mol^{-1}$)

Consider an endothermic reaction $X \to Y$ with the activation energies $E_b$ and $E_f$ for the backward and forward reactions,respectively. In general:

Fill in the blank :
$1.$ In collision theory the molecules are hard .........
$2.$ In a reality the molecules are hard ..........
$3.$ Collision take place with .......... species.

Which statement is true with respect to a catalyst?