### JEE Mains Previous Years Questions with Solutions

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### AIEEE 2011

The transverse displacement $y(x, t)$ of a wave on a string is given by $y\left( {x,t} \right) = {e^{ - \left( {a{x^2} + b{t^2} + 2\sqrt {ab} \,xt} \right)}}.$ This represents $a:$
A
wave moving in $-x$ direction with speed $\sqrt {{b \over a}}$
B
standing wave of frequency $\sqrt b$
C
standing wave of frequency ${1 \over {\sqrt b }}$
D
wave moving in $+x$ direction speed $\sqrt {{a \over b}}$

## Explanation

Given wave equation is

$y\left( {x,t} \right){ = _e}\left( { - a{x^2} + b{t^2} + 2\sqrt {ab} \,xt} \right)$

$= {e^{ - \left[ {{{\left( {\sqrt {ax} } \right)}^2} + {{\left( {\sqrt {bt} } \right)}^2} + 2\sqrt a x.\sqrt b t} \right]}}$

$= {e^{ - {{\left( {\sqrt a x + \sqrt b t} \right)}^2}}}$

$= {e^{ - {{\left( {x + \sqrt {{b \over a}} t} \right)}^2}}}$

It is a function of type $y = f\left( {x + vt} \right)$

$\Rightarrow$ Speed of wave $= \sqrt {{b \over a}}$
2

### AIEEE 2010

The equation of a wave on a string of linear mass density $0.04\,\,kg\,{m^{ - 1}}$ is given by $$y = 0.02\left( m \right)\,\sin \left[ {2\pi \left( {{t \over {0.04\left( s \right)}} - {x \over {0.50\left( m \right)}}} \right)} \right].$$

The tension in the string is

A
$4.0N$
B
$12.5$ $N$
C
$0.5$ $N$
D
$6.25$ $N$

## Explanation

$y = 0.02\left( m \right)\sin \left[ {2\pi \left( {{t \over {0.04\left( s \right)}}} \right) - {x \over {0.50\left( m \right)}}} \right]$

But $y = a\sin \left( {\omega t - kx} \right)$

$\therefore$ $\omega = {{2\pi } \over {0.04}} \Rightarrow v = {1 \over {0.04}} = 25\,Hz$

$k = {{2\pi } \over {0.50}} \Rightarrow \lambda = 0.5m$

$\therefore$ velocity, $v = v\lambda = 25 \times 0.5\,m/s = 12.5\,m/s$

Velocity on a string is given by

$v = \sqrt {{T \over \mu }}$

$\therefore$ $T = {v^2} \times \mu = {\left( {12.5} \right)^2} \times 0.04 = 6.25\,N$
3

### AIEEE 2009

A motor cycle starts from rest and accelerates along a straight path at $2m/{s^2}.$ At the starting point of the motor cycle there is a stationary electric siren. How far has the motor cycle gone when the driver hears the frequency of the siren at $94\%$ of its value when the motor cycle was at rest? (Speed of sound $= 330\,m{s^{ - 1}}$)
A
$98$ $m$
B
$147$ $m$
C
$196\,m$
D
$49$ $m$

## Explanation

$v_m^2 - {u^2} = 2as \Rightarrow v_m^2 = 2 \times 2 \times s$

$\therefore$ ${v_m} = 2\sqrt s$

According to Doppler's effect

$0.94v = v\left[ {{{330 - 2\sqrt s } \over {330}}} \right] \Rightarrow s = 98.01\,m$
4

### AIEEE 2009

Three sound waves of equal amplitudes have frequencies $\left( {v - 1} \right),\,v,\,\left( {v + 1} \right).$ They superpose to give beats. The number of beats produced per second will be :
A
$3$
B
$2$
C
$1$
D
$4$

## Explanation

Maximum number of beats $= \left( {v + 1} \right) - \left( {v - 1} \right) = 2$