Joint Entrance Examination

Graduate Aptitude Test in Engineering

4.5 *star* *star* *star* *star* *star* (100k+ *download*)

1

MCQ (Single Correct Answer)

This question has Statement $$1$$ and Statement $$2.$$ Of the four choices given after the Statements, choose the one that best describes the two Statements.

If two springs $${S_1}$$ and $${S_2}$$ of force constants $${k_1}$$ and $${k_2}$$, respectively, are stretched by the same force, it is found that more work is done on spring $${S_1}$$ than on spring $${S_2}$$.
**STATEMENT 1:** If stretched by the same amount work done on $${S_1}$$, Work done on $${S_1}$$ is more than $${S_2}$$
**STATEMENT 2:** $${k_1} < {k_2}$$

A

Statement 1 is false, Statement 2 is true

B

Statement 1 is true, Statement 2 is false

C

Statement 1 is true, Statement 2 is true, Statement 2 is the correct explanation for Statement 1

D

Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1

We know force (F) = kx

$$W = {1 \over 2}k{x^2}$$

$$W =$$ $${{{{\left( {kx} \right)}^2}} \over {2k}}$$ $$\,\,\,$$

$$\therefore$$ $$W = {{{F^2}} \over {2k}}$$ [ as $$F=kx$$ ]

When force is same then,

$$W \propto {1 \over k}$$

Given that, $${W_1} > {W_2}$$

$$\therefore$$ $${k_1} < {k_2}$$

**Statement-2 is true.**

For the same extension, x_{1}
= x_{2}
= x

Work done on spring S_{1} is W_{1} = $${1 \over 2}{k_1}x_1^2 = {1 \over 2}{k_1}{x^2}$$

Work done on spring S_{2} is W_{2} = $${1 \over 2}{k_2}x_2^2 = {1 \over 2}{k_2}{x^2}$$

$$ \therefore $$ $${{{W_1}} \over {{W_2}}} = {{{k_1}} \over {{k_2}}}$$

As $${k_1} < {k_2}$$ then $${W_1} < {W_2}$$

**So, Statement-1 is false.**

$$W = {1 \over 2}k{x^2}$$

$$W =$$ $${{{{\left( {kx} \right)}^2}} \over {2k}}$$ $$\,\,\,$$

$$\therefore$$ $$W = {{{F^2}} \over {2k}}$$ [ as $$F=kx$$ ]

When force is same then,

$$W \propto {1 \over k}$$

Given that, $${W_1} > {W_2}$$

$$\therefore$$ $${k_1} < {k_2}$$

For the same extension, x

Work done on spring S

Work done on spring S

$$ \therefore $$ $${{{W_1}} \over {{W_2}}} = {{{k_1}} \over {{k_2}}}$$

As $${k_1} < {k_2}$$ then $${W_1} < {W_2}$$

2

MCQ (Single Correct Answer)

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $$U\left( x \right) = {a \over {{x^{12}}}} - {b \over {{x^6}}},$$ where $$a$$ and $$b$$ are constants and $$x$$ is the distance between the atoms. If the dissociation energy of the molecule is $$D = \left[ {U\left( {x = \infty } \right) - {U_{at\,\,equilibrium}}} \right],\,\,D$$ is

A

$${{{b^2}} \over {2a}}$$

B

$${{{b^2}} \over {12a}}$$

C

$${{{b^2}} \over {4a}}$$

D

$${{{b^2}} \over {6a}}$$

Given $$U\left( x \right) = {a \over {{x^{12}}}} - {b \over {{x^6}}}$$

$${U\left( {x = \infty } \right)}$$ = 0

We know $$F = - {{dU} \over {dx}} = - \left[ {{{12a} \over {{x^{13}}}} + {{6b} \over {{x^7}}}} \right]$$

At equilibrium: $${{dU\left( x \right)} \over {dx}} = 0$$

$$ \Rightarrow {{ - 12a} \over {{x^{13}}}} = {{ - 6b} \over {{x^7}}} $$

$$\Rightarrow x = {\left( {{{2a} \over h}} \right)^{{1 \over 6}}}$$

$$\therefore$$ $${U_{at\,\,equilibrium\,}} = {a \over {{{\left( {{{2a} \over b}} \right)}^2}}} - {b \over {\left( {{{2a} \over b}} \right)}}$$

$$ = - {{{b^2}} \over {4a}}$$

$$\therefore$$ $$D = 0 - \left( { - {{{b^2}} \over {4a}}} \right) = {{{b^2}} \over {4a}}$$

$${U\left( {x = \infty } \right)}$$ = 0

We know $$F = - {{dU} \over {dx}} = - \left[ {{{12a} \over {{x^{13}}}} + {{6b} \over {{x^7}}}} \right]$$

At equilibrium: $${{dU\left( x \right)} \over {dx}} = 0$$

$$ \Rightarrow {{ - 12a} \over {{x^{13}}}} = {{ - 6b} \over {{x^7}}} $$

$$\Rightarrow x = {\left( {{{2a} \over h}} \right)^{{1 \over 6}}}$$

$$\therefore$$ $${U_{at\,\,equilibrium\,}} = {a \over {{{\left( {{{2a} \over b}} \right)}^2}}} - {b \over {\left( {{{2a} \over b}} \right)}}$$

$$ = - {{{b^2}} \over {4a}}$$

$$\therefore$$ $$D = 0 - \left( { - {{{b^2}} \over {4a}}} \right) = {{{b^2}} \over {4a}}$$

3

MCQ (Single Correct Answer)

A block of mass $$0.50$$ $$kg$$ is moving with a speed of $$2.00$$ $$m{s^{ - 1}}$$ on a smooth surface. It strike another mass of $$1.0$$ $$kg$$ and then they move together as a simple body. The energy loss during the collision is

A

$$0.16J$$

B

$$1.00J$$

C

$$0.67J$$

D

$$0.34$$ $$J$$

Let $$m$$ = 0.50 kg and $$M$$ = 1.0 kg

Initial kinetic energy of the system when 1 kg mass is at rest,

$$K.{E_i} = {1 \over 2}m{u^2} + {1 \over 2}M{\left( 0 \right)^2}$$

$$ = {1 \over 2} \times 0.5 \times 2 \times 2 + 0 = 1J$$

For collision, applying conservation of linear momentum

$$\,\,\,\,\,\,\,\,\,\,\,\,m \times u = \left( {m + M} \right) \times v$$

$$\therefore$$ $$0.5 \times 2 = \left( {0.5 + 1} \right) \times v \Rightarrow v = {2 \over 3}m/s$$

Final kinetic energy of the system is

$$K.{E_f} = {1 \over 2}\left( {m + M} \right){v^2}$$

$$ = {1 \over 2}\left( {0.5 + 1} \right) \times {2 \over 3} \times {2 \over 3} = {1 \over 3}J$$

$$\therefore$$ Energy loss during collision

$$ = \left( {1 - {1 \over 3}} \right)J = 0.67J$$

Initial kinetic energy of the system when 1 kg mass is at rest,

$$K.{E_i} = {1 \over 2}m{u^2} + {1 \over 2}M{\left( 0 \right)^2}$$

$$ = {1 \over 2} \times 0.5 \times 2 \times 2 + 0 = 1J$$

For collision, applying conservation of linear momentum

$$\,\,\,\,\,\,\,\,\,\,\,\,m \times u = \left( {m + M} \right) \times v$$

$$\therefore$$ $$0.5 \times 2 = \left( {0.5 + 1} \right) \times v \Rightarrow v = {2 \over 3}m/s$$

Final kinetic energy of the system is

$$K.{E_f} = {1 \over 2}\left( {m + M} \right){v^2}$$

$$ = {1 \over 2}\left( {0.5 + 1} \right) \times {2 \over 3} \times {2 \over 3} = {1 \over 3}J$$

$$\therefore$$ Energy loss during collision

$$ = \left( {1 - {1 \over 3}} \right)J = 0.67J$$

4

MCQ (Single Correct Answer)

An athlete in the olympic games covers a distance of $$100$$ $$m$$ in $$10$$ $$s.$$ His kinetic energy can be estimated to be in the range

A

$$200J-500J$$

B

$$2 \times {10^5}J - 3 \times {10^5}J$$

C

$$20,000J - 50,000J$$

D

$$2,000J - 5,000J$$

The average speed of the athelete

$$v = {{100} \over {10}} = 10m/s\,\,\,\,$$ $$\therefore$$ $$K.E. = {1 \over 2}m{v^2}$$

If mass of athlete is $$40$$ $$kg$$ then, $$K.E.$$ $$ = {1 \over 2} \times 40 \times {\left( {10} \right)^2} = 2000J$$

If mass of athlete is $$100$$ $$kg$$ then, $$K.E.$$ $$ = {1 \over 2} \times 100 \times {\left( {10} \right)^2} = 5000J$$

His kinetic energy can be in the range = 2000 J to 5000 J.

$$v = {{100} \over {10}} = 10m/s\,\,\,\,$$ $$\therefore$$ $$K.E. = {1 \over 2}m{v^2}$$

If mass of athlete is $$40$$ $$kg$$ then, $$K.E.$$ $$ = {1 \over 2} \times 40 \times {\left( {10} \right)^2} = 2000J$$

If mass of athlete is $$100$$ $$kg$$ then, $$K.E.$$ $$ = {1 \over 2} \times 100 \times {\left( {10} \right)^2} = 5000J$$

His kinetic energy can be in the range = 2000 J to 5000 J.

On those following papers in MCQ (Single Correct Answer)

Number in Brackets after Paper Indicates No. of Questions

JEE Main 2021 (Online) 1st September Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 31st August Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 27th July Evening Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 22th July Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 20th July Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 18th March Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 17th March Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th February Morning Shift (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 6th September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 4th September Evening Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 4th September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 3rd September Evening Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 2nd September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 9th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 7th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 7th January Morning Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th April Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th April Evening Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 8th April Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th January Evening Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th January Morning Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 11th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th January Morning Slot (2) *keyboard_arrow_right*

JEE Main 2018 (Online) 16th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2018 (Online) 15th April Evening Slot (1) *keyboard_arrow_right*

JEE Main 2018 (Offline) (1) *keyboard_arrow_right*

JEE Main 2017 (Online) 9th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2017 (Online) 8th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2017 (Offline) (2) *keyboard_arrow_right*

JEE Main 2016 (Online) 10th April Morning Slot (3) *keyboard_arrow_right*

JEE Main 2016 (Online) 9th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2016 (Offline) (2) *keyboard_arrow_right*

JEE Main 2014 (Offline) (1) *keyboard_arrow_right*

AIEEE 2012 (1) *keyboard_arrow_right*

AIEEE 2010 (1) *keyboard_arrow_right*

AIEEE 2008 (2) *keyboard_arrow_right*

AIEEE 2007 (1) *keyboard_arrow_right*

AIEEE 2006 (4) *keyboard_arrow_right*

AIEEE 2005 (5) *keyboard_arrow_right*

AIEEE 2004 (5) *keyboard_arrow_right*

AIEEE 2003 (4) *keyboard_arrow_right*

AIEEE 2002 (2) *keyboard_arrow_right*

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*

Ray & Wave Optics *keyboard_arrow_right*

Electrostatics *keyboard_arrow_right*

Current Electricity *keyboard_arrow_right*

Magnetics *keyboard_arrow_right*

Alternating Current and Electromagnetic Induction *keyboard_arrow_right*

Dual Nature of Radiation *keyboard_arrow_right*

Atoms and Nuclei *keyboard_arrow_right*

Electronic Devices *keyboard_arrow_right*

Communication Systems *keyboard_arrow_right*

Practical Physics *keyboard_arrow_right*