Consider the following statements: (i) The bandwidth of the A.M. wave depends…

The bandwidth of a modulated wave is fixed by how far the modulation spreads the spectrum about the carrier frequency, and never by how much power the carrier itself carries. Two standard results govern the whole item. For conventional amplitude modulation, a message occupying frequencies up to fm produces one upper and one lower sideband, so the transmitted band is BAM = 2fm, exactly twice the bandwidth of the modulating signal; the modulation index μ = Am/Ac only scales the height of those two sidebands and never moves their edges. For frequency modulation, Carson’s rule gives the practical bandwidth BFM ≈ 2(Δf + fm), where the peak deviation Δf = kfAm is set by the amplitude Am of the MODULATING signal. The carrier amplitude Ac appears in neither expression: it fixes radiated power alone.

Applying those two results to the three statements in turn:

• Statement (i), A.M. bandwidth against the bandwidth of the modulating signal: BAM = 2fm is literally twice the message bandwidth, so widening the message from 5 kHz to 15 kHz widens the transmitted band from 10 kHz to 30 kHz. This dependence is real.

• Statement (ii), A.M. bandwidth against the modulation index: raising μ from 0.3 to 0.9 lifts each sideband amplitude from 0.15Ac to 0.45Ac, but both sidebands stay pinned at fc ± fm, so the occupied band is still 2fm. Only the power split between carrier and sidebands changes, so this dependence does not exist.

• Statement (iii), F.M. bandwidth against the amplitude of the carrier: Carson’s rule contains only Δf and fm, and Δf = kfAm tracks the modulating amplitude. Doubling the transmitter’s carrier amplitude Ac quadruples radiated power and leaves the occupied band untouched, so this dependence does not exist either.

Cross-check against broadcast practice: an A.M. station limited to a 5 kHz audio band occupies 10 kHz whether it transmits at 30 % or at 90 % modulation depth, and an F.M. broadcast station with Δf = 75 kHz and fm = 15 kHz occupies 2(75 + 15) = 180 kHz on a 100 W transmitter and the very same 180 kHz on a 10 kW transmitter. Both checks point the same way: bandwidth is decided by the modulating signal’s own bandwidth and amplitude, never by the carrier’s amplitude or by how the modulation index divides the power.

Only the first statement survives, so the correct response is the set that admits statement (i) by itself, with statements (ii) and (iii) both excluded.