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JEE Mains Previous Years Questions with Solutions

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1

JEE Main 2014 (Offline)

MCQ (Single Correct Answer)
A pipe of length $$85$$ $$cm$$ is closed from one end. Find the number of possible natural oscillations of air column in the pipe whose frequencies lie below $$1250$$ $$Hz$$. The velocity of sound in air is $$340$$ $$m/s$$.
A
$$12$$
B
$$8$$
C
$$6$$
D
$$4$$

Explanation

Length of pipe $$=85$$ $$cm$$ $$=0.85m$$

Pipe is closed from one end so it behaves as a closed organ pipe

Frequency of oscillations of air column in closed organ pipe is given by,

$$f = {{\left( {2n - 1} \right)\upsilon } \over {4L}}$$

$$f = {{\left( {2n - 1} \right)\upsilon } \over {4L}} \le 1250$$

$$ \Rightarrow {{\left( {2n - 1} \right) \times 340} \over {0.85 \times 4}} \le 1250$$

$$ \Rightarrow 2n - 1 \le 12.5 \approx 6$$

Possible value of n = 1, 2, 3, 4, 5, 6

So, number of possible natural frequencies lie below 1250 Hz is 6.
2

JEE Main 2013 (Offline)

MCQ (Single Correct Answer)
A sonometer wire of length $$1.5$$ $$m$$ is made of steel. The tension in it produces an elastic strain of $$1\% $$. What is the fundamental frequency of steel if density and elasticity of steel are $$7.7 \times {10^3}\,kg/{m^3}$$ and $$2.2 \times {10^{11}}\,N/{m^2}$$ respectively ?
A
$$188.5$$ $$Hz$$
B
$$178.2$$ $$Hz$$
C
$$200.5$$ $$Hz$$
D
$$770$$ $$Hz$$

Explanation

Fundamental frequency,

$$f = {v \over {2\ell }} = {1 \over {2\ell }}\sqrt {{T \over \mu }} = {1 \over {2\ell }}\sqrt {{T \over {A\rho }}} $$

$$\left[ {\,\,} \right.$$ as $$v = \sqrt {{T \over \mu }} $$ $$\,\,\,\,\,\,$$ and $$\,\,\,\,\,\,$$ $$\left. {\mu = {m \over \ell }\,\,} \right]$$

Also, $$Y = {{T\ell } \over {A\Delta \ell }} \Rightarrow {T \over A} = {{Y\Delta \ell } \over \ell }$$

$$ \Rightarrow f = {1 \over {2\ell }}\sqrt {{{\gamma \Delta \ell } \over {\ell \rho }}} ....\left( i \right)$$

Putting the value of $$\ell ,{{\Delta \ell } \over \ell },\rho $$ $$\,\,\,\,\,\,$$ and

$$\,\,\,\,\,\,$$ $$\gamma $$ in $$e{q^n}.\left( i \right)$$ we get,

$$f = \sqrt {{2 \over 7}} \times {{{{10}^3}} \over 3}$$ $$\,\,\,\,\,\,$$ or, $$\,\,\,\,\,\,$$ $$f \approx 178.2\,Hz$$
3

AIEEE 2012

MCQ (Single Correct Answer)
A cylindrical tube, open at both ends, has a fundamental frequency, $$f,$$ in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now :
A
$$f$$
B
$$f/2$$
C
$$3/4$$
D
$$2f$$

Explanation

The fundamental frequency of open tube

$${v_0} = {v \over {2{l_0}}}\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$

That of closed pipe

$${v_c} = {\upsilon \over {4{l_c}}}\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$$
According to the problem $${l_c} = {{{l_0}} \over 2}$$

Thus $${v_c} = {\upsilon \over {{l_0}/2}} \Rightarrow {v_c}{\upsilon \over {2l}}\,\,\,\,...\left( {iii} \right)$$

From equations $$(i)$$ and $$(iii)$$

$${v_0} = {v_c}$$

Thus, $${v_c} = f$$ $$\,\,\,\left( {\,\,} \right.$$ as $${v_0} = f$$ is given $$\left. {\,\,} \right)$$
4

AIEEE 2011

MCQ (Single Correct Answer)
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 $$ = \sqrt {{b \over a}} $$

Questions Asked from Waves

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