The maximum distance up to which $TV$ transmission from a $TV$ tower of height $h$ can be received is proportional to

Given the electric field of a complete amplitude modulated wave as $\vec E = \hat i E_c (1 + \frac{E_m}{E_c} \cos \omega_m t) \cos \omega_c t$,where the subscript $c$ stands for the carrier wave and $m$ for the modulating signal. The frequencies present in the modulated wave are:

$A$ transmitting and receiving antenna have a height of $d$ meters each. The maximum distance between them for satisfactory communication in Line-of-Sight $(LOS)$ mode is $2d$ kilometers. If the radius of the Earth is $6400 \text{ km}$, then the value of $d$ is: (in $\text{ m}$)

An amplitude modulated wave is represented by $10[1+0.6 \sin(40 \times 10^3 t)] \sin(4 \times 10^6 t)$ volt,where $t$ is in seconds. The ratio of the upper sideband frequency to the lower sideband frequency is:

Match List-$I$ with List-$II$:

List-$I$ (Layer of atmosphere)	List-$II$ (Approximate height over earth's surface)
$(A)$ $F_1$-Layer	$(I)$ $10\,km$
$(B)$ $D$-Layer	$(II)$ $170-190\,km$
$(C)$ Troposphere	$(III)$ $100\,km$
$(D)$ $E$-Layer	$(IV)$ $65-75\,km$

Choose the correct answer from the options given below:

If a signal of frequency $3 \ kHz$ is to be transmitted via amplitude modulation,which of the following should be chosen as the carrier wave frequency?