 JEE Mains Previous Years Questions with Solutions

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1

AIEEE 2008

This question contains Statement - $1$ and Statement - $2$. of the four choices given after the statements, choose the one that best describes the two statements.

Statement - $1$:

For a mass $M$ kept at the center of a cube of side $'a'$, the flux of gravitational field passing through its sides $4\,\pi \,GM.$

Statement - 2:

If the direction of a field due to a point source is radial and its dependence on the distance $'r'$ from the source is given as ${1 \over {{r^2}}},$ its flux through a closed surface depends only on the strength of the source enclosed by the surface and not on the size or shape of the surface.
A
Statement - $1$ is false, Statement - $2$ is true
B
Statement - $1$ is true, Statement - $2$ is true; Statement - $2$ is a correct explanation for Statement - $1$
C
Statement - $1$ is true, Statement - $2$ is true; Statement - $2$ is not a correct explanation for Statement - $1$
D
Statement - $1$ is true, Statement - $2$ is false

Explanation

Gravitational field $\overrightarrow g$ = $- {{GM} \over {{r^2}}}$

where, $M=$ mass enclosed in the closed surface

Gravitational flux through a closed surface is given by

${\left| {\overrightarrow g .d\overrightarrow S } \right|}$ = $4\pi {r^2}.{{GM} \over {{r^2}}}$ = $4\pi GM$

So Statement - 1 is correct.

Statement - 2 is also correct because when the shape of the earth is spherical, area of the Gaussian surface is $4\pi {r^2}$. This proves inverse square law.
2

AIEEE 2005

The change in the value of $g$ at a height $h$ above the surface of the earth is the same as at a depth $d$ below the surface of earth. When both $d$ and $h$ are much smaller than the radius of earth, then which one of the following is correct?
A
$d = {{3h} \over 2}$
B
$d = {h \over 2}$
C
$d = h$
D
$d = 2\,h$

Explanation

At height h acceleration due to gravity, ${g_h} = g\left[ {1 - {{2h} \over R}} \right];$

At depth d acceleration due to gravity, ${g_d} = g\left[ {1 - {d \over R}} \right]$

According to the question,

${g_h} = {g_d}$

$\therefore$ $g\left[ {1 - {{2h} \over R}} \right]$ = $g\left[ {1 - {d \over R}} \right]$

$\Rightarrow {{2hg} \over R} = {{dg} \over R}$

$\Rightarrow$ $d=2h$
3

AIEEE 2005

Average density of the earth
A
is a complex function of $g$
B
does not depend on $g$
C
is inversely proportional to $g$
D
is directly proportional to $g$

Explanation

Mass of earth = Volume $\times$ Density of earth($\rho$)

$\therefore$ M = ${{4 \over 3}\pi {R^3}} $$\times \rho We know, g = {{GM} \over {{R^2}}} \Rightarrow g = {{G \times \rho \times {4 \over 3}\pi {R^3}} \over {{R^2}}} \Rightarrow g = {4 \over 3}\rho \pi GR \therefore g \propto \rho or \rho \propto g 4 AIEEE 2005 MCQ (Single Correct Answer) A particle of mass 10 g is kept on the surface of a uniform sphere of mass 100 kg and radius 10 cm. Find the work to be done against the gravitational force between them to take the particle far away from the sphere (you may take G = 6.67 \times {10^{ - 11}}\,\,N{m^2}/k{g^2}) A 3.33 \times {10^{ - 10}}\,J B 13.34 \times {10^{ - 10}}\,J C 6.67 \times {10^{ - 10}}\,J D 6.67 \times {10^{ - 9}}\,J Explanation We know, Work done = Difference in potential energy \therefore W = \Delta U = {U_f} - {U_i} = 0 - \left[ {{{ - GMm} \over R}} \right] \Rightarrow$$W = {{6.67 \times {{10}^{ - 11}} \times 100} \over {0.1}} \times {{10} \over {1000}}$

$= 6.67 \times {10^{ - 10}}J$