Each circuit is identified by the one element that gives its function away.

a → iv. Amplitude detector. A series diode followed by a parallel R–C is the classic envelope detector. The diode rectifies the modulated carrier and the capacitor holds the peak between cycles, so the output follows the envelope — i.e. the message. The time constant must satisfy

\(\dfrac{1}{f_c}\ll RC\ll\dfrac{1}{f_m}\)

fast enough to follow the modulation but slow enough to smooth the carrier; too large an RC produces diagonal clipping.

b → i. Frequency modulated detector. The giveaway is the tuned transformer feeding the diode. An FM detector must first convert frequency variations into amplitude variations, and a tuned circuit operated on the slope of its response does exactly that; the diode and RC then recover the resulting envelope. This slope/ratio-detector arrangement is the standard FM demodulator.

c → iii. Amplitude modulator. Here the diode is driven from a resistive divider and its output feeds an L–C tuned tank. The non-linear diode mixes the carrier and modulating signals, generating sum and difference frequencies, and the tank selects the wanted band around the carrier — producing AM rather than detecting it. Non-linearity plus a tuned output is always a modulator.

d → ii. Pre-emphasis. The audio enters a transistor whose collector load is an inductor in series with a resistor. Since \(X_L=\omega L\) rises with frequency, so does the stage gain: the circuit deliberately boosts the treble before modulation. At the receiver a matching de-emphasis network restores the balance and, in doing so, attenuates the high-frequency noise that FM detection produces — improving the output signal-to-noise ratio. The standard time constant is 75 μs.

Recognition rule: diode + RC → detector; tuned transformer ahead of the diode → FM detector; non-linear device with a tuned output → modulator; a series L in the load → a deliberate high-frequency boost, i.e. pre-emphasis.

Hence, the correct matching is a-iv, b-i, c-iii, d-ii.