Change the direction: "Shall I take the cars", he asked.

He said he will take the car.

He asked he would take the car.

He asked if he should take the car.

None of these

He asked if he should take the car.

When changing the direction of a yes/no question like "Shall I...", we use "asked if" in reported speech.

"Shall I" changes to "should he", maintaining the sense of offering or suggestion.