Two of the pairings are near-definitions and they fix the code: (A)-(IV), (B)-(III), (C)-(II), (D)-(I) — option 1.

(A) MODFET and the two-dimensional electron gas. MODFET stands for modulation-doped FET, and the two-dimensional electron gas is precisely what modulation doping creates. A wide-gap layer such as AlGaAs is doped, but the electrons it releases fall into the adjacent undoped GaAs and are trapped in a triangular well at the heterojunction — a sheet of charge one nanometre thick, free to move in the plane. Because the carriers are now spatially separated from the donors that supplied them, ionised-impurity scattering almost vanishes and the mobility becomes very high, which is why the device is used at millimetre-wave frequencies.

(C) Solar cell and fill factor. Fill factor is a term belonging to photovoltaics alone:

\(FF=\dfrac{V_{mp}I_{mp}}{V_{oc}I_{sc}}\)

It measures how nearly rectangular the fourth-quadrant I-V curve is — how close the maximum-power point comes to the product of open-circuit voltage and short-circuit current. Good silicon cells reach 0.75 to 0.85, and the efficiency follows as

\(\eta=\dfrac{V_{oc}I_{sc}FF}{P_{in}}\)

(D) JFET and the voltage-controlled resistor. Below pinch-off the JFET's drain-source path is a resistance set by the gate voltage, and since the gate is a reverse-biased junction drawing no current, the control costs nothing. That is exactly the definition of a voltage-controlled resistor, and it is why JFETs serve as AGC attenuators and analogue switches.

(B) MOSFET and back-to-back p-n junctions. Viewed from source to drain, an n-channel MOSFET on a p substrate presents an n-p junction and then a p-n junction — two diodes in opposition. With no gate voltage one of them is always reverse biased whichever way the drain is driven, which is precisely why the device blocks in both directions until the gate creates an inversion channel bridging them.

The quickest route through the question : MODFET and 2DEG is a definitional pair, and fill factor can belong to nothing but a solar cell. Those two alone leave only option 1.

Hence, the correct code is (A)-(IV), (B)-(III), (C)-(II), (D)-(I).