When the switch is off, the voltmeter and current are all zero;

AB, the voltage indicator changes, indicating that the voltmeter is connected to the power supply, so L is open or R is short, and the current indicator remains unchanged, that is, the indicator is zero, so the fault must be open, so A is correct and B is wrong;

The change of CD and current indication shows that the circuit is a path, so L is short-circuited or R is short-circuited, and the voltage indication remains unchanged, that is, the indication is zero, then the fault must be L short-circuited, so CD is wrong.

So choose a.