 JEE Mains Previous Years Questions with Solutions

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1

AIEEE 2005

A smooth block is released at rest on a ${45^ \circ }$ incline and then slides a distance $'d'$. The time taken to slide is $'n'$ times as much to slide on rough incline than on a smooth incline. The coefficient of friction is
A
${\mu _k} = \sqrt {1 - {1 \over {{n^2}}}}$
B
${\mu _k} = 1 - {1 \over {{n^2}}}$
C
${\mu _k} = \sqrt {1 - {1 \over {{n^2}}}}$
D
${\mu _s} = 1 - {1 \over {{n^2}}}$

Explanation For smooth surface,

$d = {1 \over 2}\left( {g\,\sin \,\theta } \right)t_1^2,$

${t_1} = \sqrt {{{2d} \over {g\,\sin \,\theta }}} ,$ When surface is rough

$d = {1 \over 2}\left( {g\,\sin \,\theta - \mu g\,\cos \theta } \right)t_2^2$

${t_2} = \sqrt {{{2d} \over {g\,\sin \,\theta - \mu g\,\cos \theta }}}$

According to question, ${t_2} = n{t_1}$

$n\sqrt {{{2d} \over {g\,\sin \,\theta }}} = \sqrt {{{2d} \over {g\,\sin \,\theta - \mu g\,\cos \theta }}}$

$n = {1 \over {\sqrt {1 - {\mu _k}} }}$ ( as $\cos \,{45^ \circ } = \sin \,{45^ \circ } = {1 \over {\sqrt 2 }}$ )

${n^2} = {1 \over {1 - {\mu _k}}}$

or $1 - {\mu _k} = {1 \over {{n^2}}}$

or ${\mu _k} = 1 - {1 \over {{n^2}}}$

2

AIEEE 2005

A parachutist after bailing out falls $50$ $m$ without friction. When parachute opens, it decelerates at $2\,\,m/{s^2}.$ He reaches the ground with a speed of $3$ $m/s$. At what height, did he bail out?
A
$182$ $m$
B
$91$ $m$
C
$111$ $m$
D
$293$ $m$

Explanation The velocity of parachutist when parachute opens at 50 m is

$u = \sqrt {2gh} = \sqrt {2 \times 9.8 \times 50} = \sqrt {980}$

The velocity at ground, $v=3m/s$

$\therefore$ $S = {{{v^2} - {u^2}} \over {2 \times \left( { - 2} \right)}} = {{{3^2} - 980} \over { - 4}} \approx 243\,m$

Initially he has fallen $50$ $m.$

$\therefore$ Total height from where

He bailed out $=243+50=293m$
3

AIEEE 2004

A block rests on a rough inclined plane `making an angle of ${30^ \circ }$ with the horizontal. The coefficient of static friction between the block and the plane is $0.8.$ If the frictionless force on the block is $10$ $N,$ the mass of the block (in $kg$) is $\left( {take\,\,\,g\, = \,10\,\,m/{s^2}} \right)$
A
$1.6$
B
$4.0$
C
$2.0$
D
$2.5$

Explanation For equilibrum of block,

$mg\,\,\sin \theta = {f_s}\,\,$

$\Rightarrow m \times 10 \times \sin {30^ \circ } = 10$

$\Rightarrow m \times 5 = 10$

$\Rightarrow m = 2.0\,\,kg$
4

AIEEE 2004

Two masses ${m_1} = 5kg$ and ${m_2} = 4.8kg$ tied to a string are hanging over a light frictionless pulley. What is the acceleration of the masses when left free to move? $\left( {g = 9.8m/{s^2}} \right)$ A
$5\,\,m/{s^2}$
B
$9.8\,\,m/{s^2}$
C
$0.2\,\,m/{s^2}$
D
$4.8\,\,m/{s^2}$

Explanation For m1 mass :

m1g - T = m1a .........(1)

For m2 mass :

T - m2g = m2a ....... (2)

By solving (1) and (2) we get,

Acceleration a = $\left( {{{{m_1} - {m_2}} \over {{m_1} + {m_2}}}} \right)g$

$= {{\left( {5 - 4.8} \right) \times 9.8} \over {\left( {5 + 4.8} \right)}}m/{s^2}$ = 0.2 m/s2