The provided parameters give you enough information to calculate wind drag and rolling resistance; are you considering either of those to take into account in your calculations?

The problem seems poorly formed. No information is given about the power output at 2500 RPM so how could one say with any certainty that fourth gear will work at all?
Also, the question states P_{max} is required to maintain the given speed, so there is no way forth gear would work as P_{2500RPM} < P_{max}.

I agree with the OP's conclusion. There's no need to calc. power required to overcome wind drag, rolling drag etc, we are told it's 59.4 HP, gears 2 and 3 meet that requirement. There's an error in the question.

I disagree, the problem statement specifically states "The engine power is seen to be approximately constant at its maximum value Pmax = 59.4hp over the engine speed range 3000-6000rpm." Given a known power output and engine speed, net torque output (and therefore a force balance) can be developed for the vehicle.

The problem is for the student to determine "which of the four gearbox ratios can be used to maintain the speed of 40mph up the incline." There are four gear ratios given, and it has to be determined which would be used based on a vehicle force balance.

The only "error" I see if there is one is that it doesn't specifically state that air drag is to be taken into account. But given the context of the question and the information presented (including coefficient of drag, chassis dimensions, and rolling drag) it seems obvious to me that a force balance must be considered which includes:

Air Drag

Rolling resistance (given as 200N)

Torque required to lift car's weight up the incline (taking into account total mass and the tire diameter)

Given it's to one decimal place I assumed it remained pretty close to that figure.

Either way, the power required for a 1200kg car to go 40mph up a 20deg incline is more than P_{max}, even before accounting for drag & rolling resistance. The car isnt even capable of doing what is claimed! Not surprising really, I've driven up the worlds steepest street (19 deg), there was no way the Suzuki Cultus I was driving (~100hp, ~1 ton) would've done 40 mph up it.
A pic from wiki:

You've calculated the engine speeds at each of the gear ratios which is a good start. Next you need to calculate the torque output of the engine given a known power output and RPM (see here: calculate engine power given torque and speed). Given those calculated torque output values, calculate the net force to the ground at the tires, and compare to the sum of the forces working against it (air drag, rolling drag, and incline drag).

To maintain speed, the net force to the ground from the engine needs to be greater than or equal to the sum of the drag forces.

Good catch, maybe the OP made a mistake transcribing the problem statement? It looks like a 59 hp car could only do 40mph up to an approx 12 deg. incline...

Fourth gear in this vehicle is essentially an overdrive gear, which is usually used for cruising on long stretches of roadway. Unless the vehicle is already at speed when encountering the incline, normal practice is to shift out of overdrive into a lower gear (higher overall gear ratio) to keep the engine in a speed range where it is making maximum power or close to it, during the climb. Otherwise, you are trying to lug the engine while going up the hill, which is not good for keeping the engine in good running condition. Once you get up the hill, then you can shift back into overdrive and resume cruising.

I suspect this isn't actually the answer from an answer book, but is in fact an incorrect answer given on another forum where this question has been cross-posted.

We will require several clarifications from the OP before moving forward; first and foremost, the stated horsepower of the vehicle is not enough to maintain 40mph up a 20 deg. incline.