What is the minimum size screw needed to fasten a 3/4" cabinet to a stud covered in 1/2" drywall?

To determine the minimum size screw required for fastening a 3/4" cabinet to a stud that is covered by 1/2" drywall, you need to consider the combined thickness of both the drywall and the cabinet. The 1/2" drywall adds thickness that the screw must penetrate to reach the stud securely, so you'd add this to the 3/4" thickness of the cabinet.

The total material thickness is 3/4" (cabinet) + 1/2" (drywall), which equals 1-1/4". A screw must be long enough to penetrate through the drywall and into the stud securely to ensure that the cabinet is well-mounted and stable.

The selected answer of 2-1/4 inches accounts for both the drywall and cabinet thickness, while also providing additional length to ensure the screw engages adequately with the stud for a solid hold. This extra length helps to avoid the screw coming loose over time or from stresses placed on the cabinet.

The other options, while they may connect the cabinet to the drywall, would not provide enough anchor into the stud to be reliable. Therefore, 2-1/4 inches is the most appropriate choice for this fastening task.