Chemistry - How does the rate constant change with the change of the temperature?

Answer B is technically correct although C could also be accepted. Assuming $E_A$ to be constant, at low temperatures because of the inverse $T$ in the exponential $\exp(-E_A/(R\cdot \text{small number}) \equiv \exp(-\text{big})$ which is a small number. At high temperatures $\exp(-E_A/(R\cdot \text{big number})\equiv \exp(-\text{small})$ is a big number, so the rate constant increases with temperature.

At temperature such that the exponential $\to 1$ the rate constant is $A$ for all practical purposes. Usually this limit is not reached unless $E_A$ is very small and then as $A$ is also a function of temperature this ($A$) term becomes now important. Generally, however, for the vast majority of reactions the exponential term is the most important.