The correct answer is option 3: the difference in the densities of the object and liquid.

When an object is placed in a liquid, whether it will float or sink depends on the relative densities of the object and the liquid. Density is a measure of how much mass is packed into a given volume.

Option 1, mass of the object only, is not sufficient to determine if it will float or sink. A small object with a high mass can still float if its density is lower than that of the liquid it is placed in.

Option 2, mass of the object and density of the liquid, is partially correct. The density of the liquid does play a role in determining if the object will float or sink. However, it is not solely dependent on these two factors.

Option 4, mass and shape of the object only, is also not completely accurate. While the mass of the object does affect its buoyancy, the shape of the object also has an impact. Objects with a larger surface area tend to experience more buoyant force, which can affect whether they float or sink.

Therefore, option 3, the difference in the densities of the object and liquid, accurately captures the main factor that determines whether an object will float or sink in a liquid