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JEE Main 201610 AprilOnline
Exam
JEE Main
Year
2016
Date
10 April
Mode
Online
Subject
Physics
Chapter
Communication System

JEE Main 2016 · 10 April · Online · Physics · Communication System

A modulated signal C _ m (t) has the form C _ m (t) = 30 300 + 10(...

A modulated signal Cm(t){C}_{m}\left(t\right) has the form Cm(t)=30sin⁡300\pit+10(cos⁡200\pit−cos⁡400\pit).{C}_{m}\left(t\right) = 30\sin 300\pit + 10\left(\cos 200\pit - \cos 400\pit\right) . The carrier frequency fc{f}_{c} , the modulating frequency (message frequency) fm,{f}_{m ,}and the modulation index μ\mu are respectively given by :

Options

  1. Afc=200Hz;fm=50Hz;μ=12{f}_{c} = 200Hz; {f}_{m} = 50Hz; \mu = \dfrac{1}{2}
  2. Bfc=150Hz;fm=50Hz;μ=23{f}_{c} = 150Hz; {f}_{m} = 50Hz; \mu = \dfrac{2}{3}
  3. Cfc=150Hz;fm=30Hz;μ=13{f}_{c} = 150Hz; {f}_{m} = 30Hz; \mu = \dfrac{1}{3}
  4. Dfc=200Hz;fm=30Hz;μ=12{f}_{c} = 200Hz; {f}_{m} = 30Hz; \mu = \dfrac{1}{2}

Correct answer

B. fc=150Hz;fm=50Hz;μ=23{f}_{c} = 150Hz; {f}_{m} = 50Hz; \mu = \dfrac{2}{3}

Step-by-step solution

Compare with

Cm(t)=Acsin⁡(wct)+μAc2[cos⁡(wc−wm)t−cos⁡(wc+wm)]{C}_{m} ( t ) = {A}_{c} \sin ( {w}_{c} t ) + \dfrac{\mu{A}_{c}}{2} [ \cos ( {w}_{c} - {w}_{m} ) t - \cos ( {w}_{c} + {w}_{m} ) ]

μAc2=10\dfrac{\mu{A}_{c}}{2} = 10

Ac=30{A}_{c} = 30

μ=23\mu = \dfrac{2}{3}

wc=2πfc=300π{w}_{c} = 2\pi{f}_{c} = 300\pi⇒fc=150Hz⇒ {f}_{c} = 150Hz

wc−wm=200π{w}_{c} - {w}_{m} = 200\pi

wm=100π{w}_{m} = 100\pi

fm=50Hz{f}_{m} = 50Hz

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