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

Charge is distributed within a sphere of radius R with a volume charge density $$\rho \left( r \right) = {A \over {{r^2}}}{e^{ - {{2r} \over s}}},$$ where A and a are constants. If Q is the total charge of this charge distribution, the radius R is :

A

a log $$\left( {1 - {Q \over {2\pi aA}}} \right)$$

B

$${a \over 2}$$ log $$\left( {{1 \over {1 - {Q \over {2\pi aA}}}}} \right)$$

C

a log $$\left( {{1 \over {1 - {Q \over {2\pi aA}}}}} \right)$$

D

$${a \over 2}$$ log $$\left( {1 - {Q \over {2\pi aA}}} \right)$$

Volume of this spherical layer,

dv = (4$$\pi $$r

charge present in this layer,

dq = $$\rho $$ (4$$\pi $$r

= $${A \over {{r^2}}}{e^{ - {{2r} \over a}}}\,\,\left( {4\pi {r^2}dr} \right)$$

= $$A\,{e^{ - {{2r} \over a}}}\left( {4\pi dr} \right)$$

$$ \therefore $$ Total charge in the sphere,

Q= $$\int\limits_0^R {4\pi A{e^{ - {{2r} \over a}}}} \,dr$$

= 4$$\pi $$A$$\int\limits_0^R {{e^{ - {{2r} \over a}}}} \,dr$$

= 4$$\pi $$A$$\left[ {{{{e^{ - {{2r} \over a}}}} \over { - {2 \over a}}}} \right]_0^R$$

= 4$$\pi $$A $$\left( { - {a \over 2}} \right)\left( {{e^{ - {{2R} \over a}}} - 1} \right)$$

$$ \therefore $$ Q = 2$$\pi $$aA $$\left( {1 - {e^{ - {{2R} \over a}}}} \right)$$

$$ \Rightarrow $$ $${1 - {e^{ - {{2R} \over a}}}}$$ = $${Q \over {2\pi aA}}$$

$$ \Rightarrow $$ $${{e^{ - {{2R} \over a}}}}$$ = 1 $$-$$ $${Q \over {2\pi aA}}$$

$$ \Rightarrow $$ $${e^{{{2R} \over a}}}$$ = $${1 \over {1 - {Q \over {2\pi aA}}}}$$

$$ \Rightarrow $$ $${{2R} \over a} = \log \left( {{1 \over {1 - {Q \over {2\pi aA}}}}} \right)$$

$$ \Rightarrow $$ R = $${a \over 2}$$ log $$\left( {{1 \over {1 - {Q \over {2\pi aA}}}}} \right)$$

2

In the given circuit the the internal resistance of the 18 V cell is negligible. If R_{1} = 400 $$\Omega $$, R_{3} = 100 $$\Omega $$ and R_{4} = 500 $$\Omega $$ and the reading of an ideal voltmeter across R_{4} is 5V, then the value of R_{2} will be :

A

300 $$\Omega $$

B

450 $$\Omega $$

C

550 $$\Omega $$

D

230 $$\Omega $$

Voltage accross resistance R_{4} = 5 V

$$ \therefore $$ IR_{4} = 5 V

$$ \Rightarrow $$ 500 $$ \times $$ I = 5

$$ \Rightarrow $$ I = $${1 \over {100}}$$ A

$$ \therefore $$ Voltage across resistor R_{3} = $${1 \over {100}}\left( {100} \right)$$ = 1 A

$$ \therefore $$ Total drop in resistance R_{3} and R_{4} = 5 + 1 = 6V

So, voltage accross R_{2} resistance is also 6V as R_{3}, R_{4} and R_{2} are in parallel

$$ \therefore $$ Voltage accross R_{1} resistor R_{1} resistor = 18 $$-$$ 6 = 12 V

$$ \therefore $$ Current through R_{1} resistor = $${{12} \over {400}}$$ = $${3 \over {100}}$$ A

$$ \therefore $$ Current through R_{2} resistor

= $${3 \over {100}} - {1 \over {100}}$$

= $${2 \over {100}}$$ A

$$ \therefore $$ $$\left( {{2 \over {100}}} \right)$$ R_{2} = 6

$$ \Rightarrow $$ R_{2} = 300 $$\Omega $$

$$ \therefore $$ IR

$$ \Rightarrow $$ 500 $$ \times $$ I = 5

$$ \Rightarrow $$ I = $${1 \over {100}}$$ A

$$ \therefore $$ Voltage across resistor R

$$ \therefore $$ Total drop in resistance R

So, voltage accross R

$$ \therefore $$ Voltage accross R

$$ \therefore $$ Current through R

$$ \therefore $$ Current through R

= $${3 \over {100}} - {1 \over {100}}$$

= $${2 \over {100}}$$ A

$$ \therefore $$ $$\left( {{2 \over {100}}} \right)$$ R

$$ \Rightarrow $$ R

3

Two electric dipoles, A, B with respective dipole moments $${\overrightarrow d _A} = - 4qai$$ and $${\overrightarrow d _B} = - 2qai$$ are placed on the x-axis with a separation R, as shown in the figure. The distance from A at which both of them produce the same potential is -

A

$${{\sqrt 2 R} \over {\sqrt 2 + 1}}$$

B

$${R \over {\sqrt 2 + 1}}$$

C

$${{\sqrt 2 R} \over {\sqrt 2 - 1}}$$

D

$${R \over {\sqrt 2 - 1}}$$

V $$ = {{4qa} \over {\left( {R + x} \right)}} = {{2qa} \over {\left( {{x^2}} \right)}}$$

$$\sqrt 2 x = R + x$$

$$x = {R \over {\sqrt 2 - 1}}$$

dist $$ = {R \over {\sqrt 2 - 1}} + R = {{\sqrt 2 R} \over {\sqrt 2 - 1}}$$

$$\sqrt 2 x = R + x$$

$$x = {R \over {\sqrt 2 - 1}}$$

dist $$ = {R \over {\sqrt 2 - 1}} + R = {{\sqrt 2 R} \over {\sqrt 2 - 1}}$$

4

Charges –q and +q located at A and B, respectively, constitude an electric dipole. Distance AB = 2a, O is the mid point of the dipole and OP is perpendicular to AB. A charge Q is placed at P where OP = y and y >> 2a. The charge Q experiences an electrostatic force F. If Q is now moved along the equatorial line to P' such that OP' = $$\left( {{y \over 3}} \right)$$, the force on Q will be close to - $$\left( {{y \over 3} > > 2a} \right)$$

A

9F

B

3F

C

F/3

D

27F

Electric field of equitorial plane of dipole

$$ = - {{K\overrightarrow P } \over {{r^3}}}$$

$$ \therefore $$ At P, F $$ = - {{K\overrightarrow P } \over {{r^3}}}$$Q.

At P^{1} , F^{1} $$ = - {{K\overrightarrow P Q} \over {{{\left( {r/3} \right)}^3}}} = 27F.$$

$$ = - {{K\overrightarrow P } \over {{r^3}}}$$

$$ \therefore $$ At P, F $$ = - {{K\overrightarrow P } \over {{r^3}}}$$Q.

At P

Number in Brackets after Paper Name Indicates No of Questions

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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*

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*

Atoms and Nuclei *keyboard_arrow_right*

Electronic Devices *keyboard_arrow_right*

Communication Systems *keyboard_arrow_right*

Practical Physics *keyboard_arrow_right*

Dual Nature of Radiation *keyboard_arrow_right*