Step 1: Picture the swing.
A child on a swing moves like a pendulum. At the highest point the swing stops for a moment, and at the lowest point it moves fastest. The lowest height is $0.75$ m and the highest height is $2$ m above the ground.
Step 2: Use energy idea.
As the swing drops from the top to the bottom, its stored height energy turns into motion energy. Nothing is lost, so the drop in potential energy equals the gain in kinetic energy.
Step 3: Write the energy balance.
\[ \frac{1}{2}mv_{\max}^2 = mg\,(h_{\max} - h_{\min}) \]
The mass $m$ is on both sides, so it cancels.
Step 4: Solve for the speed.
\[ v_{\max} = \sqrt{2g\,(h_{\max} - h_{\min})} \]
Step 5: Find the height drop.
\[ h_{\max} - h_{\min} = 2 - 0.75 = 1.25\ \text{m} \]
Step 6: Put in the numbers.
With $g = 10$ m/s$^2$:
\[ v_{\max} = \sqrt{2 \times 10 \times 1.25} = \sqrt{25} = 5\ \text{m/s} \]
So the greatest speed of the swing is $5$ m/s, which is option (3).
\[ \boxed{v_{\max} = 5\ \text{m/s}} \]