For the output of an $AM$ detector circuit to follow the envelope of the carrier wave,which condition must be satisfied? ($T_c$ is the time period of the carrier wave.)

The heights of the transmitting and receiving antennas are $33.8 \ m$ and $64.8 \ m$ respectively. The maximum distance between the antennas for satisfactory communication in a line of sight mode is (Radius of the earth $= 6400 \ km$) (in $km$)

The modulation index used in amplitude modulation of a message signal of peak voltage $10 \,V$ is $0.5$. By how much should the peak voltage of the carrier signal be changed, so as to obtain the modulation index of $0.8$, keeping the peak voltage of the message signal the same as before?

$A$ transmitting antenna has a height of $20 \,m$. What will be the height of the receiving antenna if the maximum distance between them for satisfactory communication in line-of-sight $(LOS)$ mode is $40 \,km$ (in $\,m$)? (The radius of the Earth is $6400 \,km$.)

$A$ message signal of frequency $3\,kHz$ is used to modulate a carrier signal of frequency $1.5\,MHz$. The bandwidth of the amplitude modulated wave is

The frequency range for commercial $FM$ radio broadcasting is .......