The maximum distance between the transmitting and receiving $TV$ towers is $65 \,km$. If the ratio of the heights of the $TV$ transmitting tower to receiving tower is $36: 49$, the heights of the transmitting and receiving towers respectively are (radius of earth $= 6400 \,km$)

$A$ signal $A \cos \omega t$ is transmitted using $v_0 \sin \omega_0 t$ as a carrier wave. The correct amplitude modulated $(AM)$ signal is:

$A$ diode detector is used to detect an amplitude modulated wave with $60\%$ modulation,using a capacitor of $250 \ pF$ in parallel with a resistor of $100 \ k\Omega$. What is the maximum modulation frequency that can be detected by this circuit?

If multiple waves with modulation indices $n_1, n_2, n_3, ...$ modulate a carrier wave,what is the total modulation index of the resulting wave?

Given below are two statements:
Statement-$I$: $A$ speech signal of $2 \, kHz$ is used to modulate a carrier signal of $1 \, MHz$. The bandwidth requirement for the signal is $4 \, kHz$.
Statement-$II$: The sideband frequencies are $1002 \, kHz$ and $998 \, kHz$.
In the light of the above statements,choose the correct answer from the options given below:

The height of a transmitting antenna at the top of a tower is $25 \; m$ and that of the receiving antenna is $49 \; m$. The maximum distance between them for satisfactory communication in $LOS$ (Line of Sight) is $K \sqrt{5} \times 10^{2} \; m$. The value of $K$ is $\dots$ [Assume radius of Earth is $64 \times 10^{5} \; m$] (Calculate up to the nearest integer value).