1

### JEE Main 2016 (Online) 9th April Morning Slot

An audio signal consists of two distinct sounds : one a human speech signal in the frequency band of 200 Hz to 2700 Hz, while the other is a high frequency music signal in the frequency band of 10200 Hz to 15200 Hz. The ratio of the AM signal bandwidth required to send both the signals together to the AM signal bandwidth required to send just the human speech is :
A
3
B
5
C
6
D
2

## Explanation

The ratio = ${{15200 - 200} \over {2700 - 200}}$

=    ${{15000} \over {2500}}$

=    6
2

### JEE Main 2016 (Online) 10th April Morning Slot

A modulated signal Cm(t) has the form Cm(t) = 30 sin 300 $\pi$t + 10 (cos 200 $\pi$t −cos 400 $\pi$t). The carrier frequency c, the modulating frequency (message frequency) f$\omega$, and the modulation index $\mu$ are respectively given by :
A
fc = 200 Hz; f$\omega$ = 50 Hz; $\mu$ = ${1 \over 2}$
B
fc = 150 Hz; f$\omega$ = 50 Hz; $\mu$ = ${2 \over 3}$
C
fc = 150 Hz; f$\omega$ = 30 Hz; $\mu$ = ${1 \over 3}$
D
fc = 200 Hz; f$\omega$ = 30 Hz; $\mu$ = ${1 \over 2}$

## Explanation

Given,

Cm(t) = 30 sin 300$\pi$t + 10 (cos 200 $\pi$t $-$ cos 400 $\pi$t)

Standard equation of amplitude modulated wave,

Cm(t) = Ac sin ($\omega$ct) $-$ ${{\mu {A_c}} \over 2}$ cos $\left( {{\omega _c} + {\omega _m}} \right)$ t + ${{\mu {A_c}} \over 2}$ cos $\left( {{\omega _c} - {\omega _m}} \right)$ t

By comparing we get,

Ac = 30 V

$\omega$c = 300 $\pi$

$\Rightarrow$   2$\pi$fc = 300 $\pi$

$\Rightarrow$   fc = 150 Hz

$\omega$c $-$ $\omega$s = 200 $\pi$

$\Rightarrow$   2$\pi$ (fc $-$ f$\omega$) = 200 $\pi$

$\Rightarrow$   fc $-$ f$\omega$ = 100 Hz

$\therefore$   f$\omega$ = 150 $-$ 100 = 50 Hz

and ${{\mu {A_c}} \over 2}$ = 10

$\Rightarrow$   $\mu$ $\times$ ${{30} \over 2}$ = 10

$\Rightarrow$   $\mu$ = ${{10} \over {15}}$ = ${2 \over 3}$
3

### JEE Main 2017 (Offline)

In amplitude modulation, sinusoidal carrier frequency used is denoted by ${\omega _c}$ and the signal frequency is denoted by ${\omega _m}$. The bandwidth ($\Delta {\omega _m}$) of the signal is such that $\Delta {\omega _m}$ < < $\omega _c$. Which of the following frequencies is not contained in the modulated wave?
A
${\omega _m}$
B
${\omega _c}$
C
${\omega _m}$ + ${\omega _c}$
D
${\omega _c}$ - ${\omega _m}$

## Explanation

Modulated carrier wave contains frequency ${\omega _c}$ and ${\omega _c}$ ± ${\omega _m}$.
4

### JEE Main 2017 (Online) 8th April Morning Slot

A signal of frequency 20 kHz and peak voltage of 5 Volt is used to modulate a carrier wave of frequency 1.2 MHz and peak voltage 25 Volts. Choose the correct statement.
A
Modulation index = 5, side frequency bands are at 1400 kHz and 1000 kHz
B
Modulation index = 5, side frequency bands are at 21.2 kHz and 18.8 kHz
C
Modulation index = 0.8, side frequency bands are at 1180 kHz and 1220 kHz
D
Modulation index = 0.2, side frequency bands are at 1220 kHz and 1180 kHz

## Explanation

Modulation index (m) = ${{{V_m}} \over {{V_0}}} = {5 \over {25}} = 0.2$

Given, carrier wave,

Fc = 1.2 $\times$ 106 Hz = 1200 kHz

and frequency of modulate wave,

Fm = 20 kHz

$\therefore\,\,\,$ Side frequency bands are

F1 = 1200 + 20 = 1220 kHz

F2 = 1200 $-$ 80 = 1180 kHz