Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geomatics Engineering Or Surveying

Engineering Mechanics

Hydrology

Transportation Engineering

Strength of Materials Or Solid Mechanics

Reinforced Cement Concrete

Steel Structures

Irrigation

Environmental Engineering

Engineering Mathematics

Structural Analysis

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

General Aptitude

1

A compressive force, F is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by $$\Delta $$T. The net change in its length is zero. Let $$\ell $$ be the length of the rod, A its area of cross-section,Y its Young’s modulus, and $$\alpha $$ its coefficient of linear expansion. Then, F is equal to :

A

$$\ell $$^{2} Y$$\alpha $$ $$\Delta $$T

B

$$\ell $$A Y$$\alpha $$ $$\Delta $$T

C

A Y$$\alpha $$ $$\Delta $$T

D

$${{AY} \over {\alpha \,\Delta T}}$$

Because of thermal expansion, change in length

($$\Delta $$$$\ell $$) = $$\ell $$ $$\alpha $$ $$\Delta $$T . . . . .(1)

Because of compressive force, the compansion is $$\Delta $$$$\ell $$ ' ,

$$\therefore\,\,\,$$ Young's Modulus (y) = $${{{F \over A}} \over {{{\Delta \ell} \over \ell }}}$$

$$ \Rightarrow $$$$\,\,\,$$ F = YA $${{\Delta \ell '} \over \ell }$$

As net change in length is 0 . So,

$$\Delta \ell '$$ = $$\Delta \ell $$ = $$\ell \alpha \,\Delta T$$

$$\therefore\,\,\,$$ F = YA $$ \times $$ $${{\ell \alpha \,\Delta T} \over \ell }$$ = AY$$\alpha $$$$\Delta $$T

($$\Delta $$$$\ell $$) = $$\ell $$ $$\alpha $$ $$\Delta $$T . . . . .(1)

Because of compressive force, the compansion is $$\Delta $$$$\ell $$ ' ,

$$\therefore\,\,\,$$ Young's Modulus (y) = $${{{F \over A}} \over {{{\Delta \ell} \over \ell }}}$$

$$ \Rightarrow $$$$\,\,\,$$ F = YA $${{\Delta \ell '} \over \ell }$$

As net change in length is 0 . So,

$$\Delta \ell '$$ = $$\Delta \ell $$ = $$\ell \alpha \,\Delta T$$

$$\therefore\,\,\,$$ F = YA $$ \times $$ $${{\ell \alpha \,\Delta T} \over \ell }$$ = AY$$\alpha $$$$\Delta $$T

2

A steel rail of length 5 m and area of cross section 40cm^{2}
is prevented from expanding along its length while the temperature rises
by 10^{o}C. If coefficient of linear expansion and Young’s modulus of steel are 1.2×10^{−5} K^{−1} and 2×10^{11} Nm^{−2} respectively, the force developed in the rail is approximately :

A

2 $$ \times $$ 10^{7} N

B

1 $$ \times $$ 10^{5} N

C

2 $$ \times $$ 10^{9} N

D

3 $$ \times $$ 10^{$$-$$5} N

Young's modulus (Y) = $${{{F \over A}} \over {{{\Delta L} \over L}}}$$

as $${{{\Delta L} \over L}}$$ = $$\alpha $$ $$\Delta $$$$\theta $$

$$\therefore\,\,\,$$ Y = $${{F \over {A\alpha \Delta \theta }}}$$

$$ \Rightarrow $$$$\,\,\,$$ F = YA$$\alpha $$$$\Delta $$$$\theta $$

= 2 $$ \times $$ 10^{11} $$ \times $$ 40$$ \times $$10^{$$-$$4} $$ \times $$ 1.2 $$ \times $$ 10^{$$-$$5} $$ \times $$ 10

= 9.6 $$ \times $$ 10^{4} N

$$ \simeq $$ 1 $$ \times $$ 10^{5} N

as $${{{\Delta L} \over L}}$$ = $$\alpha $$ $$\Delta $$$$\theta $$

$$\therefore\,\,\,$$ Y = $${{F \over {A\alpha \Delta \theta }}}$$

$$ \Rightarrow $$$$\,\,\,$$ F = YA$$\alpha $$$$\Delta $$$$\theta $$

= 2 $$ \times $$ 10

= 9.6 $$ \times $$ 10

$$ \simeq $$ 1 $$ \times $$ 10

3

Two tubes of radii r_{1} and r_{2}, and lengths l_{1} and l_{2} , respectively, are connected in series
and a liquid flows through each of them in stream line conditions. P_{1} and P_{2} are pressure differences across the two tubes.

If P_{2} is 4P_{1} and l_{2}
is $${{{1_1}} \over 4}$$, then the radius r_{2} will be equal to :

If P

A

r_{1}

B

2r_{1}

C

4r_{1}

D

$${{{r_1}} \over 2}$$

We know,

Rate of flow of liquid through narrow tube,

$${{dv} \over {dt}}$$ = $${{\pi {{\Pr }^4}} \over {8\eta l}}$$

Both tubes are connected in series so rate of flow of liquid is same.

$$\therefore\,\,\,$$ $${{\pi {P_1}r_1^4} \over {8\eta {l_1}}}$$ = $${{\pi {P_2}r_2^4} \over {8\eta {l_2}}}$$

$$ \Rightarrow $$$$\,\,\,$$ $${{{P_1}{r_1}^4} \over {{l_1}}}$$ = $${{{P_2}{r_2}^4} \over {{l_2}}}$$

Given,

P_{2} = 4P_{1} and $${l_2}$$ = $${{{l_1}} \over 4}$$

$$ \Rightarrow $$$$\,\,\,$$ $${{{P_1}{r_1}^4} \over {{l_1}}}$$ = $${{4{P_1}{r_2}^4} \over {{{{l_1}} \over 4}}}$$

$$ \Rightarrow $$$$\,\,\,$$ $${r_2}^4$$ = $${{{r_1}^4} \over {16}}$$

$$ \Rightarrow $$$$\,\,\,$$ r_{2} = $${{{r_1}} \over 2}$$

Rate of flow of liquid through narrow tube,

$${{dv} \over {dt}}$$ = $${{\pi {{\Pr }^4}} \over {8\eta l}}$$

Both tubes are connected in series so rate of flow of liquid is same.

$$\therefore\,\,\,$$ $${{\pi {P_1}r_1^4} \over {8\eta {l_1}}}$$ = $${{\pi {P_2}r_2^4} \over {8\eta {l_2}}}$$

$$ \Rightarrow $$$$\,\,\,$$ $${{{P_1}{r_1}^4} \over {{l_1}}}$$ = $${{{P_2}{r_2}^4} \over {{l_2}}}$$

Given,

P

$$ \Rightarrow $$$$\,\,\,$$ $${{{P_1}{r_1}^4} \over {{l_1}}}$$ = $${{4{P_1}{r_2}^4} \over {{{{l_1}} \over 4}}}$$

$$ \Rightarrow $$$$\,\,\,$$ $${r_2}^4$$ = $${{{r_1}^4} \over {16}}$$

$$ \Rightarrow $$$$\,\,\,$$ r

4

A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a
cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross
section of cylindrical container. When a mass m is placed on the surface of the piston to compress the
liquid, the fractional decrement in the radius of the sphere, $$\left( {{dr \over r}} \right)$$ is:

A

$${{mg} \over {Ka}}$$

B

$${{Ka} \over {mg}}$$

C

$${{Ka} \over {3mg}}$$

D

$${{mg} \over {3Ka}}$$

Because of m mass the extra pressure created is,

$$\Delta $$P = $${{mg} \over a}$$

And Bulk modulus, $$\beta $$ = $${{\Delta P} \over {{{\Delta V} \over V}}}$$

Given $$\beta $$ = K

$$\therefore\,\,\,$$ K = $${{{{mg} \over a}} \over {{{\Delta V} \over V}}}$$

We know volume of sphere,

V = $${4 \over 3}\pi {r^3}$$

$$\therefore\,\,\,$$ $${{dV} \over V}$$ = 3 $${{dr} \over r}$$

$$\therefore\,\,\,$$ K = $${{{{mg} \over a}} \over {3{{dr} \over r}}}$$

$$ \Rightarrow $$$$\,\,\,$$ $${{dr} \over r}$$ = $${{mg} \over {3Ka}}$$

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (1) *keyboard_arrow_right*

AIEEE 2004 (4) *keyboard_arrow_right*

AIEEE 2005 (2) *keyboard_arrow_right*

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AIEEE 2008 (3) *keyboard_arrow_right*

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AIEEE 2012 (1) *keyboard_arrow_right*

JEE Main 2013 (Offline) (2) *keyboard_arrow_right*

JEE Main 2014 (Offline) (2) *keyboard_arrow_right*

JEE Main 2016 (Online) 9th April Morning Slot (4) *keyboard_arrow_right*

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Units & Measurements *keyboard_arrow_right*

Motion *keyboard_arrow_right*

Laws of Motion *keyboard_arrow_right*

Work Power & Energy *keyboard_arrow_right*

Simple Harmonic Motion *keyboard_arrow_right*

Impulse & Momentum *keyboard_arrow_right*

Rotational Motion *keyboard_arrow_right*

Gravitation *keyboard_arrow_right*

Properties of Matter *keyboard_arrow_right*

Heat and Thermodynamics *keyboard_arrow_right*

Waves *keyboard_arrow_right*

Vector Algebra *keyboard_arrow_right*

Dual Nature of Radiation *keyboard_arrow_right*

Electronic Devices *keyboard_arrow_right*

Practical Physics *keyboard_arrow_right*

Atoms and Nuclei *keyboard_arrow_right*

Communication Systems *keyboard_arrow_right*

Electrostatics *keyboard_arrow_right*

Current Electricity *keyboard_arrow_right*

Magnetics *keyboard_arrow_right*

Alternating Current and Electromagnetic Induction *keyboard_arrow_right*

Ray & Wave Optics *keyboard_arrow_right*