1. Formula for capillary rise

For a liquid rising in a capillary tube,

$h = \frac{2T\cos\theta}{\rho g r}$

where:

• $T$ = surface tension of the liquid
• $\theta$ = angle of contact
• $\rho$ = density of the liquid
• $r$ = radius of capillary tube
• $g$ = acceleration due to gravity

For water in a clean glass capillary, $\theta \approx 0^\circ$, so

$\cos\theta \approx 1$

Hence,

$h \propto \frac{T}{\rho}$

2. Effect of temperature

When water is heated:

• Surface tension $T$ decreases significantly.
• Density $\rho$ also decreases slightly.

The decrease in surface tension is the dominant effect, so the ratio $\dfrac{T}{\rho}$ decreases. Therefore, the capillary rise $h$ decreases with increase in temperature.

So,

$h_{\text{hot}} < h_{\text{cold}}$

3. Check Statement I

Statement I says: the height of capillary rise will be smaller in hot water.

This is true.

4. Check Statement II

Statement II says: the height of capillary rise will be smaller in cold water.

This is the opposite of the correct result, so it is false.

5. Evaluate options

• A: Both false — incorrect
• B: Both true — incorrect
• C: Statement I true, Statement II false — correct
• D: Statement I false, Statement II true — incorrect

Therefore, the correct option is

$\boxed{\text{C}}$