For the following Assertion and Reason, the correct option is:
Assertion: The pH of water increases with increase in temperature.
Reason: The dissociation of water into H$$^+$$ and OH$$^-$$ is an exothermic reaction.

We begin with the auto-ionisation (self-dissociation) of water, which can be written as

$$\mathrm{H_2O(l)\; \rightleftharpoons \; H^+(aq) \;+\; OH^-(aq)}.$$

For this equilibrium, the equilibrium constant is called the ionic product of water and is denoted by $$K_w$$. Mathematically,

$$K_w \;=\;[H^+][OH^-].$$

In pure water the concentrations of $$H^+$$ and $$OH^-$$ are equal, so we may write

$$[H^+] \;=\;[OH^-] \;=\;\sqrt{K_w}.$$

The definition of pH is

$$\displaystyle\text{pH} \;=\; -\log_{10}[H^+].$$

Combining these facts, for pure water we get

$$\text{pH} \;=\; -\log_{10}\!\left(\sqrt{K_w}\right) \;=\; -\dfrac{1}{2}\log_{10}K_w.$$

Now we recall Le-Chatelier’s principle. If a reaction is endothermic (absorbs heat, $$\Delta H > 0$$), then raising the temperature shifts the equilibrium to the right, increasing the value of the equilibrium constant. Conversely, if a reaction is exothermic (releases heat, $$\Delta H < 0$$), raising the temperature would decrease the equilibrium constant.

Experimental thermodynamic data show that the dissociation of water is actually endothermic, i.e.

$$\Delta H^\circ > 0.$$

Therefore, as temperature rises, $$K_w$$ increases. Substituting this increase into the earlier expression

$$\text{pH} \;=\; -\dfrac{1}{2}\log_{10}K_w,$$

we see that a larger $$K_w$$ makes $$\log_{10}K_w$$ larger, its negative half becomes more negative, so the numerical value of pH decreases. Hence water becomes slightly acidic at higher temperatures; its pH goes below 7.

Thus the assertion “The pH of water increases with increase in temperature” is false.

Next, the reason states that the dissociation of water is an exothermic reaction. As just noted, measured enthalpy change $$\Delta H^\circ>0$$, so the process is endothermic, not exothermic. Therefore the reason is also false.

Both the assertion and the reason are false, and hence the reason cannot possibly explain the assertion.

Hence, the correct answer is Option B.