Joint Entrance Examination

Graduate Aptitude Test in Engineering

Geotechnical Engineering

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General Aptitude

1

If the area enclosed between the curves y = kx^{2} and x = ky^{2}, (k > 0), is 1 square unit. Then k is -

A

$$\sqrt 3 $$

B

$${{\sqrt 3 } \over 2}$$

C

$${2 \over {\sqrt 3 }}$$

D

$${1 \over {\sqrt 3 }}$$

Area bounded by

y^{2} = 4ax & x^{2} = 4by, a, b $$ \ne $$ 0

is $$\left| {{{16ab} \over 3}} \right|$$

by using formula :

4a $$=$$ $${1 \over k} = 4b,k > 0$$

Area $$ = \left| {{{16.{1 \over {4k}}.{1 \over {4k}}} \over 3}} \right| = 1$$

$$ \Rightarrow $$ k^{2} $$ = {1 \over 3}$$

$$ \Rightarrow $$ k $$ = {1 \over {\sqrt 3 }}$$

y

is $$\left| {{{16ab} \over 3}} \right|$$

by using formula :

4a $$=$$ $${1 \over k} = 4b,k > 0$$

Area $$ = \left| {{{16.{1 \over {4k}}.{1 \over {4k}}} \over 3}} \right| = 1$$

$$ \Rightarrow $$ k

$$ \Rightarrow $$ k $$ = {1 \over {\sqrt 3 }}$$

2

The value of $$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {\left[ x \right] + \left[ {\sin x} \right] + 4}}} ,$$ where [t] denotes the greatest integer less than or equal to t, is

A

$${1 \over {12}}\left( {7\pi - 5} \right)$$

B

$${1 \over {12}}\left( {7\pi + 5} \right)$$

C

$${3 \over {10}}\left( {4\pi - 3} \right)$$

D

$${3 \over {20}}\left( {4\pi - 3} \right)$$

$${\rm I} = \int\limits_{{{ - \pi } \over 2}}^{{\pi \over 2}} {{{dx} \over {\left[ x \right] + \left[ {\sin x} \right] + 4}}} $$

$$ = \int\limits_{{{ - \pi } \over 2}}^{ - 1} {{{dx} \over { - 2 - 1 + 4}}} + \int\limits_{ - 1}^0 {{{dx} \over { - 1 - 1 + 4}}} $$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \int\limits_0^1 {{{dx} \over {0 + 0 + 4}} + \int\limits_1^{{\pi \over 2}} {{{dx} \over {1 + 0 + 4}}} } $$

$$\int\limits_{{{ - \pi } \over 2}}^{ - 1} {{{dx} \over 1} + \int\limits_{ - 1}^0 {{{dx} \over 2}} } + \int\limits_0^1 {{{dx} \over 4}} + \int\limits_1^{{\pi \over 2}} {{{dx} \over 5}} $$

$$\left( { - 1 + {\pi \over 2}} \right) + {1 \over 2}\left( {0 + 1} \right) + {1 \over 4} + {1 \over 5}\left( {{\pi \over 2} - 1} \right)$$

$$ - 1 + {1 \over 2} + {1 \over 4} - {1 \over 5} + {\pi \over 2} + {\pi \over {10}}$$

$${{ - 20 + 10 + 5 - 4} \over {20}} + {{6\pi } \over {10}}$$

$${{ - 9} \over {20}} + {{3\pi } \over 5}$$

$$ = \int\limits_{{{ - \pi } \over 2}}^{ - 1} {{{dx} \over { - 2 - 1 + 4}}} + \int\limits_{ - 1}^0 {{{dx} \over { - 1 - 1 + 4}}} $$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \int\limits_0^1 {{{dx} \over {0 + 0 + 4}} + \int\limits_1^{{\pi \over 2}} {{{dx} \over {1 + 0 + 4}}} } $$

$$\int\limits_{{{ - \pi } \over 2}}^{ - 1} {{{dx} \over 1} + \int\limits_{ - 1}^0 {{{dx} \over 2}} } + \int\limits_0^1 {{{dx} \over 4}} + \int\limits_1^{{\pi \over 2}} {{{dx} \over 5}} $$

$$\left( { - 1 + {\pi \over 2}} \right) + {1 \over 2}\left( {0 + 1} \right) + {1 \over 4} + {1 \over 5}\left( {{\pi \over 2} - 1} \right)$$

$$ - 1 + {1 \over 2} + {1 \over 4} - {1 \over 5} + {\pi \over 2} + {\pi \over {10}}$$

$${{ - 20 + 10 + 5 - 4} \over {20}} + {{6\pi } \over {10}}$$

$${{ - 9} \over {20}} + {{3\pi } \over 5}$$

3

If $$\int\limits_0^x \, $$f(t) dt = x^{2} + $$\int\limits_x^1 \, $$ t^{2}f(t) dt then f '$$\left( {{1 \over 2}} \right)$$ is -

A

$${{18} \over {25}}$$

B

$${{6} \over {25}}$$

C

$${{24} \over {25}}$$

D

$${{4} \over {5}}$$

$$\int\limits_0^x \, $$f(t) dt = x^{2} + $$\int\limits_x^1 \, $$ t^{2}f(t) dt f '$$\left( {{1 \over 2}} \right)$$ = ?

Differentiate w.r.t. 'x'

f(x) = 2x + 0 $$-$$ x^{2} f(x)

f(x) = $${{2x} \over {1 + {x^2}}}$$ $$ \Rightarrow $$ f '(x) = $${{\left( {1 + {x^2}} \right)2 - 2x\left( {2x} \right)} \over {{{\left( {1 + {x^2}} \right)}^2}}}$$

f '(x) = $${{2{x^2} - 4{x^2} + 2} \over {{{\left( {1 + {x^2}} \right)}^2}}}$$

f '$$\left( {{1 \over 2}} \right) = {{2 - 2\left( {{1 \over 4}} \right)} \over {{{\left( {1 + {1 \over 4}} \right)}^2}}} = {{\left( {{3 \over 2}} \right)} \over {{{25} \over {16}}}} = {{48} \over {50}} = {{24} \over {25}}$$

Differentiate w.r.t. 'x'

f(x) = 2x + 0 $$-$$ x

f(x) = $${{2x} \over {1 + {x^2}}}$$ $$ \Rightarrow $$ f '(x) = $${{\left( {1 + {x^2}} \right)2 - 2x\left( {2x} \right)} \over {{{\left( {1 + {x^2}} \right)}^2}}}$$

f '(x) = $${{2{x^2} - 4{x^2} + 2} \over {{{\left( {1 + {x^2}} \right)}^2}}}$$

f '$$\left( {{1 \over 2}} \right) = {{2 - 2\left( {{1 \over 4}} \right)} \over {{{\left( {1 + {1 \over 4}} \right)}^2}}} = {{\left( {{3 \over 2}} \right)} \over {{{25} \over {16}}}} = {{48} \over {50}} = {{24} \over {25}}$$

4

The value of the integral $$\int\limits_{ - 2}^2 {{{{{\sin }^2}x} \over { \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \,dx$$ (where [x] denotes the greatest integer less than or equal to x) is

A

0

B

4

C

4$$-$$ sin 4

D

sin 4

I $$=$$ $$\int\limits_{ - 2}^2 {{{{{\sin }^2}x} \over { \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \,dx$$

$${\rm I} = \int\limits_0^2 {\left( {{{{{\sin }^2}x} \over {\left[ {{x \over \pi }} \right] + {1 \over 2}}} + {{{{\sin }^2}\left( { - x} \right)} \over {\left[ { - {x \over \pi }} \right] + {1 \over 2}}}} \right)dx} $$

$$\left( {\left[ {{x \over \pi }} \right] + \left[ { - {x \over \pi }} \right] = - 1\,\,} \right.$$ as $$\left. {\matrix{ \, \cr \, \cr } x \ne n\pi } \right)$$

$${\rm I} = \int\limits_0^2 {\left( {{{{{\sin }^2}x} \over {\left[ {{x \over \pi }} \right] + {1 \over 2}}} + {{{{\sin }^2}x} \over { - 1 - \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \right)dx = 0} $$

$${\rm I} = \int\limits_0^2 {\left( {{{{{\sin }^2}x} \over {\left[ {{x \over \pi }} \right] + {1 \over 2}}} + {{{{\sin }^2}\left( { - x} \right)} \over {\left[ { - {x \over \pi }} \right] + {1 \over 2}}}} \right)dx} $$

$$\left( {\left[ {{x \over \pi }} \right] + \left[ { - {x \over \pi }} \right] = - 1\,\,} \right.$$ as $$\left. {\matrix{ \, \cr \, \cr } x \ne n\pi } \right)$$

$${\rm I} = \int\limits_0^2 {\left( {{{{{\sin }^2}x} \over {\left[ {{x \over \pi }} \right] + {1 \over 2}}} + {{{{\sin }^2}x} \over { - 1 - \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \right)dx = 0} $$

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Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

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Complex Numbers *keyboard_arrow_right*

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Sequences and Series *keyboard_arrow_right*

Matrices and Determinants *keyboard_arrow_right*

Vector Algebra and 3D Geometry *keyboard_arrow_right*

Probability *keyboard_arrow_right*

Statistics *keyboard_arrow_right*

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Trigonometric Functions & Equations *keyboard_arrow_right*

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Functions *keyboard_arrow_right*

Limits, Continuity and Differentiability *keyboard_arrow_right*

Differentiation *keyboard_arrow_right*

Application of Derivatives *keyboard_arrow_right*

Indefinite Integrals *keyboard_arrow_right*

Definite Integrals and Applications of Integrals *keyboard_arrow_right*

Differential Equations *keyboard_arrow_right*