$A$ $TV$ transmitter has a range of $50 \ km$. The height of the $TV$ transmitter is $......$ (Radius of the earth $R_e = 6.4 \times 10^6 \ m$) (in $m$)

$A$ message signal of $12 kHz$ and peak voltage $20 V$ is used to modulate a carrier wave of frequency $12 MHz$ and peak voltage $30 V$. Then,the modulation index is

In an amplitude modulated wave for an audio frequency of $500 \, \text{cycles/second}$, the appropriate carrier frequency will be ........ $\text{cycles/second}$.

$A$ $TV$ transmission tower has a height of $140\, m$ and the height of the receiving antenna is $40\, m$. What is the maximum distance up to which signals can be broadcasted from this tower in $LOS$ (Line of Sight) mode? (Given: radius of earth $R = 6.4 \times 10^6\, m$)

$A$ radar has a power of $1 \text{ kW}$ and operates at a frequency of $10 \text{ GHz}$. If it is located on top of a mountain at a height of $500 \text{ m}$,what is the maximum distance in $\text{km}$ up to which it can detect objects on the Earth's surface? (Radius of the Earth $R = 6.4 \times 10^{6} \text{ m}$)

For effective transmission of a wave with frequency $3000 \text{ kHz}$,the minimum length of the antenna should be ...... $m$.