Step-by-step analysis:

1. Analyze Statement - 1: "The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down."

   • Case 1: Hose held vertically up. The water is ejected upwards with an initial velocity. Due to gravity acting downwards, the water decelerates as it moves up. According to the equation of motion for an object under constant acceleration, $v^2 = u^2 + 2as$, where $u$ is the initial velocity, $v$ is the final velocity, $a$ is the acceleration, and $s$ is the displacement. For upward motion, $a = -g$ and $s = h$ (height). $v^2 = u^2 - 2gh$ This shows that the velocity $v$ of the water decreases as the height $h$ increases. Now, we use the principle of continuity for a steady, incompressible flow, which states that the volume flow rate ($Q = Av$) is constant, where $A$ is the cross-sectional area of the stream and $v$ is the fluid velocity. $A \times v = \text{constant}$ This implies that $A \propto \frac{1}{v}$. Since the velocity $v$ decreases as the water goes up, the cross-sectional area $A$ must increase. An increase in area means the stream spreads out. This part of the statement is correct.

   • Case 2: Hose held vertically down. The water is ejected downwards with an initial velocity. Due to gravity acting downwards, the water accelerates as it falls. Using the same equation of motion, for downward motion, $a = g$ and $s = h$ (depth). $v^2 = u^2 + 2gh$ This shows that the velocity $v$ of the water increases as it falls. Again, from the equation of continuity, $A \times v = \text{constant}$, which means $A \propto \frac{1}{v}$. Since the velocity $v$ increases as the water falls, the cross-sectional area $A$ must decrease. A decrease in area means the stream narrows down. This part of the statement is also correct.

   • Conclusion for Statement - 1: The statement accurately describes the physical behavior of a water stream under gravity. Therefore, Statement - 1 is True.

2. Analyze Statement - 2: "In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant."

   • This statement is the definition of the principle of continuity for an incompressible fluid.
   • For a steady flow, the mass of fluid entering a section of a pipe or stream per unit time must equal the mass of fluid leaving it per unit time. This is the principle of conservation of mass.
   • The mass flow rate is given by $\dot{m} = \rho A v$, where $\rho$ is the density of the fluid.
   • So, $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$.
   • For an incompressible fluid, the density $\rho$ is constant (i.e., $\rho_1 = \rho_2$).
   • Therefore, the equation simplifies to $A_1 v_1 = A_2 v_2$, which means the volume flow rate $Q = Av$ is constant throughout the flow.
   • Conclusion for Statement - 2: The statement is a fundamental principle of fluid dynamics. Therefore, Statement - 2 is True.

3. Evaluate the relationship between the two statements.

   • In Step 1, we used the principle described in Statement - 2 (the constancy of volume flow rate, $Av = \text{constant}$) to explain the physical phenomena described in Statement - 1.
   • The change in the stream's cross-sectional area is a direct consequence of the change in its velocity (due to gravity) and the conservation of volume flow rate.
   • Therefore, Statement - 2 provides the correct physical explanation for Statement - 1.

4. Final Conclusion:

   • Statement - 1 is True.
   • Statement - 2 is True.
   • Statement - 2 is the correct explanation for Statement - 1.

This corresponds to option A.