Which one of the following is the correct set-up for studying the dependence of the current on the potential difference across a resistor and why?"

In the given diagram the correct set-up for studying the dependence of the current on the potential difference across a resistor is option (c).

It is correct because in this experiment of determining the dependence of the current on the potential difference across a resistor, the voltmeter is connected in parallel to the resistor and the ammeter is connected in series with the resistor. Also, the polarities of the ammeter and voltmeter are marked correctly.