1

### 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

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

### 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 ${\omega _c}$ ± ${\omega _m}$.
3

### 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
4

### JEE Main 2017 (Online) 9th April Morning Slot

A signal is to be transmitted through a wave of wavelength $\lambda$, using a linear antenna. The length l of the antenna and effective power radiated Peff will be given respectively as :

(K is a constant of proportionality)
A
$\lambda$, Peff = K $\left( {{1 \over \lambda }} \right)$2
B
${{\lambda \over 8}}$, Peff = K $\left( {{1 \over \lambda }} \right)$
C
${{\lambda \over 16}}$, Peff = K $\left( {{1 \over \lambda }} \right)$3
D
${{\lambda \over 5}}$, Peff = K $\left( {{1 \over \lambda }} \right)$${\left( {{1 \over \lambda }} \right)^{{1 \over 2}}}$