Joint Entrance Examination

Graduate Aptitude Test in Engineering

Strength of Materials Or Solid Mechanics

Structural Analysis

Construction Material and Management

Reinforced Cement Concrete

Steel Structures

Geotechnical Engineering

Fluid Mechanics and Hydraulic Machines

Hydrology

Irrigation

Geomatics Engineering Or Surveying

Environmental Engineering

Transportation Engineering

Engineering Mathematics

General Aptitude

1

The position co-ordinates of a particle moving in a 3-D coordinate system is given by

x = a cos$$\omega $$t

y = a sin$$\omega $$t and

z = a$$\omega $$t

The speed of the particle is :

x = a cos$$\omega $$t

y = a sin$$\omega $$t and

z = a$$\omega $$t

The speed of the particle is :

A

$$\sqrt 2 \,a\omega $$

B

$$a\omega $$

C

$$\sqrt 3 \,a\omega $$

D

2a$$\omega $$

Given that,

x = a cos $$\omega $$t

y = a sin $$\omega $$t

z = a $$\omega $$t

Velocity in x-direction,

V_{x} = $${{dx} \over {dt}} = - a\omega \sin \omega t$$

Velocity in y-direction,

V_{y} = $${{dy} \over {dt}}$$ = a $$\omega $$cos $$\omega $$t

Velocity in z-direction,

V_{z}_{} = $${{dz} \over {dt}}$$ = a$$\omega $$

Net velocity,

$$\overrightarrow V $$ = V_{x}$$\widehat i$$ + V_{y}$$\widehat j$$ + V_{z}$$\widehat k$$

Speed = $$\left| {\overrightarrow V } \right| = \sqrt {V_x^2 + V_y^2 + V_z^2} $$

$$ = \sqrt {{a^2}{\omega ^2}{{\sin }^2}\omega t + {a^2}{\omega ^2}{{\cos }^2}\omega t + {a^2}{\omega ^2}} $$

$$ = \sqrt {{a^2}{\omega ^2}\left( {{{\sin }^2}\omega t + {{\cos }^2}\omega t} \right) + {a^2}{\omega ^2}} $$

$$ = \sqrt {2{a^2}{\omega ^2}} $$

$$ = \sqrt 2 a\omega $$

x = a cos $$\omega $$t

y = a sin $$\omega $$t

z = a $$\omega $$t

Velocity in x-direction,

V

Velocity in y-direction,

V

Velocity in z-direction,

V

Net velocity,

$$\overrightarrow V $$ = V

Speed = $$\left| {\overrightarrow V } \right| = \sqrt {V_x^2 + V_y^2 + V_z^2} $$

$$ = \sqrt {{a^2}{\omega ^2}{{\sin }^2}\omega t + {a^2}{\omega ^2}{{\cos }^2}\omega t + {a^2}{\omega ^2}} $$

$$ = \sqrt {{a^2}{\omega ^2}\left( {{{\sin }^2}\omega t + {{\cos }^2}\omega t} \right) + {a^2}{\omega ^2}} $$

$$ = \sqrt {2{a^2}{\omega ^2}} $$

$$ = \sqrt 2 a\omega $$

2

Two guns A and B can fire bullets at speeds 1 km/s and 2 km/s respectively. From a point on a horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by the bullets fired by the two guns, on the ground is -

A

1 : 16

B

1 : 8

C

1 : 2

D

1 : 4

$$R = {{{u^2}\sin 2\theta } \over g}$$

$$A = \pi \,{R^2}$$

$$A \propto {R^2}$$

$$A \propto {u^4}$$

$${{{A_1}} \over {{A_2}}} = {{u_1^4} \over {u_2^4}} = {\left[ {{1 \over 2}} \right]^4} = {1 \over {16}}$$

3

In the cube of side ‘a’ shown in the figure, the vector from the central point of the face ABOD to the central point of the face BEFO will be -

A

$${1 \over 2}a\left( {\widehat k - \widehat i} \right)$$

B

$${1 \over 2}a\left( {\widehat j - \widehat i} \right)$$

C

$${1 \over 2}a\left( {\widehat j - \widehat k} \right)$$

D

$${1 \over 2}a\left( {\widehat i - \widehat k} \right)$$

$$\overrightarrow {{r_g}} = {a \over 2}\widehat i + {a \over 2}\widehat k$$

$$\overrightarrow {{r_H}} = {a \over 2}\widehat i + {a \over 2}\widehat k$$

$$\overrightarrow {{r_H}} - \overrightarrow {{r_g}} = {a \over 2}\left( {\widehat j - \widehat i} \right)$$

$$\overrightarrow {{r_H}} = {a \over 2}\widehat i + {a \over 2}\widehat k$$

$$\overrightarrow {{r_H}} - \overrightarrow {{r_g}} = {a \over 2}\left( {\widehat j - \widehat i} \right)$$

4

Two vectors $$\overrightarrow A $$ and $$\overrightarrow B $$ have equal magnitudes. The magnitude of $$\left( {\overrightarrow A + \overrightarrow B } \right)$$ is 'n' times the magnitude of $$\left( {\overrightarrow A - \overrightarrow B } \right)$$ . The angle between $${\overrightarrow A }$$ and $${\overrightarrow B }$$ is -

A

$${\sin ^{ - 1}}\left[ {{{n - 1} \over {n + 1}}} \right]$$

B

$${\sin ^{ - 1}}\left[ {{{{n^2} - 1} \over {{n^2} + 1}}} \right]$$

C

$${\cos ^{ - 1}}\left[ {{{{n^2} - 1} \over {{n^2} + 1}}} \right]$$

D

$${\cos ^{ - 1}}\left[ {{{n - 1} \over {n + 1}}} \right]$$

$$\left| {\overrightarrow A + \overrightarrow B } \right| = 2a\cos \theta /2$$ . . . (1)

$$\left| {\overrightarrow A - \overrightarrow B } \right| = 2a\cos {{\left( {\pi - \theta } \right)} \over 2} = 2a\sin \theta /2$$ . . . (2)

$$ \Rightarrow \,\,\,n\left( {2a\cos {\theta \over 2}} \right) = 2a{{\sin \theta } \over 2}$$

$$ \Rightarrow \,\,\,\tan {\theta \over 2} = n$$

$$\left| {\overrightarrow A - \overrightarrow B } \right| = 2a\cos {{\left( {\pi - \theta } \right)} \over 2} = 2a\sin \theta /2$$ . . . (2)

$$ \Rightarrow \,\,\,n\left( {2a\cos {\theta \over 2}} \right) = 2a{{\sin \theta } \over 2}$$

$$ \Rightarrow \,\,\,\tan {\theta \over 2} = n$$

Number in Brackets after Paper Name Indicates No of Questions

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Atoms and Nuclei *keyboard_arrow_right*

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Rotational Motion *keyboard_arrow_right*

Gravitation *keyboard_arrow_right*

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