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

$$\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}$$

A

exists and equals $${1 \over {2\sqrt 2 }}$$

B

exists and equals $${1 \over {4\sqrt 2 }}$$

C

exists and equals $${1 \over {2\sqrt 2 (1 + \sqrt {2)} }}$$

D

does not exists

$$\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}$$

If you put y = 0 at $${{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}$$ it is in $${0 \over 0}$$ form. So we can use L' Hospital's Rule.

= $$\mathop {\lim }\limits_{y \to 0} {{{1 \over {2\sqrt {1 + \sqrt {1 + {y^4}} } }} \times \left( {{1 \over {2\sqrt {1 + {y^4}} }}} \right) \times 4{y^3}} \over {4{y^3}}}$$

= $$\mathop {\lim }\limits_{y \to 0} {1 \over {2\sqrt {1 + \sqrt {1 + {y^4}} } }} \times {1 \over {2\sqrt {1 + {y^4}} }}$$

= $${1 \over {4\sqrt 2 }}$$

If you put y = 0 at $${{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}$$ it is in $${0 \over 0}$$ form. So we can use L' Hospital's Rule.

= $$\mathop {\lim }\limits_{y \to 0} {{{1 \over {2\sqrt {1 + \sqrt {1 + {y^4}} } }} \times \left( {{1 \over {2\sqrt {1 + {y^4}} }}} \right) \times 4{y^3}} \over {4{y^3}}}$$

= $$\mathop {\lim }\limits_{y \to 0} {1 \over {2\sqrt {1 + \sqrt {1 + {y^4}} } }} \times {1 \over {2\sqrt {1 + {y^4}} }}$$

= $${1 \over {4\sqrt 2 }}$$

2

For each x$$ \in $$**R**, let [x] be the greatest integer less than or equal to x.

Then $$\mathop {\lim }\limits_{x \to {0^ - }} \,\,{{x\left( {\left[ x \right] + \left| x \right|} \right)\sin \left[ x \right]} \over {\left| x \right|}}$$ is equal to :

Then $$\mathop {\lim }\limits_{x \to {0^ - }} \,\,{{x\left( {\left[ x \right] + \left| x \right|} \right)\sin \left[ x \right]} \over {\left| x \right|}}$$ is equal to :

A

$$-$$ sin 1

B

1

C

sin 1

D

0

$$\mathop {\lim }\limits_{x \to {0^ - }} {{x\left( {\left[ x \right] + \left| x \right|} \right)\sin \left[ x \right]} \over {\left| x \right|}}$$

$$ = \mathop {\lim }\limits_{h \to 0} {{\left( {0 - h} \right)\left( {\left[ {0 - h} \right] + \left| {0 - h} \right|} \right)\sin \left[ {0 - h} \right]} \over {\left| {0 - h} \right|}}$$

$$ = \mathop {\lim }\limits_{h \to 0} {{ - h\left( { - 1 + h} \right)\sin \left( { - 1} \right)} \over h}$$

$$ = \mathop {\lim }\limits_{h \to 0} \left( { - 1 + h} \right)\sin \left( 1 \right) = - \sin 1$$

$$ = \mathop {\lim }\limits_{h \to 0} {{\left( {0 - h} \right)\left( {\left[ {0 - h} \right] + \left| {0 - h} \right|} \right)\sin \left[ {0 - h} \right]} \over {\left| {0 - h} \right|}}$$

$$ = \mathop {\lim }\limits_{h \to 0} {{ - h\left( { - 1 + h} \right)\sin \left( { - 1} \right)} \over h}$$

$$ = \mathop {\lim }\limits_{h \to 0} \left( { - 1 + h} \right)\sin \left( 1 \right) = - \sin 1$$

3

Let f be a differentiable function from

**R** to **R** such that $$\left| {f\left( x \right) - f\left( y \right)} \right| \le 2{\left| {x - y} \right|^{{3 \over 2}}},$$

for all $$x,y \in $$**R**.

If $$f\left( 0 \right) = 1$$

then $$\int\limits_0^1 {{f^2}} \left( x \right)dx$$ is equal to :

for all $$x,y \in $$

If $$f\left( 0 \right) = 1$$

then $$\int\limits_0^1 {{f^2}} \left( x \right)dx$$ is equal to :

A

1

B

2

C

$${1 \over 2}$$

D

0

$$\left| {f(x) - f(y)} \right| \le 2{\left[ {x - y} \right]^{3/2}}$$

$$\left| {{{f(x) - f(y)} \over {x - y}}} \right| \le 2{\left| {x - y} \right|^{1/2}}$$

$$\mathop {\lim }\limits_{y \to x} \left| {{{f(x) - f(y)} \over {x - y}}} \right| \le \mathop {\lim }\limits_{y \to x} 2{\left| {x - y} \right|^{1/2}}$$

$$ \Rightarrow \left| {f'\left( x \right)} \right| \le 0$$ $$ \Rightarrow f'\left( x \right) = 0$$

$$ \Rightarrow f\left( x \right) = $$ constant

as $$f\left( 0 \right) = 1 \Rightarrow f\left( x \right) = 1$$

$$\int\limits_0^1 {{f^2}} \left( x \right)dx = 1$$

$$\left| {{{f(x) - f(y)} \over {x - y}}} \right| \le 2{\left| {x - y} \right|^{1/2}}$$

$$\mathop {\lim }\limits_{y \to x} \left| {{{f(x) - f(y)} \over {x - y}}} \right| \le \mathop {\lim }\limits_{y \to x} 2{\left| {x - y} \right|^{1/2}}$$

$$ \Rightarrow \left| {f'\left( x \right)} \right| \le 0$$ $$ \Rightarrow f'\left( x \right) = 0$$

$$ \Rightarrow f\left( x \right) = $$ constant

as $$f\left( 0 \right) = 1 \Rightarrow f\left( x \right) = 1$$

$$\int\limits_0^1 {{f^2}} \left( x \right)dx = 1$$

4

Let f : R $$ \to $$ R be a function such that f(x) = x^{3} + x^{2}f'(1) + xf''(2) + f'''(3), x $$ \in $$ R. Then f(2) equals -

A

30

B

$$-$$ 2

C

$$-$$ 4

D

8

f(x) = x^{3} + x^{2}f '(1) + xf ''(2) + f '''(3)

$$ \Rightarrow $$ f '(x) = 3x^{2} + 2xf '(1) + f ''(x) . . . . . (1)

$$ \Rightarrow $$ f ''(x) = 6x + 2f '(1) . . . . . . (2)

$$ \Rightarrow $$ f '''(x) = 6 . . . . . .(3)

put x = 1 in equation (1) :

f '(1) = 3 + 2f '(1) + f ''(2) . . . . .(4)

put x = 2 in equation (2) :

f ''(2) = 12 + 2f '(1) . . . . .(5)

from equation (4) & (5) :

$$-$$3 $$-$$ f '(1) = 12 + 2f'(1)

$$ \Rightarrow $$ 3f '(1) = $$-$$ 15

$$ \Rightarrow $$ f '(1) = $$-$$ 5 $$ \Rightarrow $$ f ''(2) = 2 . . . . .(2)

put x = 3 in equation (3) :

f ''' (3) = 6

$$ \therefore $$ f(x) = x^{3} $$-$$ 5x^{2} + 2x + 6

f(2) = 8 $$-$$ 20 + 4 + 6 = $$-$$ 2

$$ \Rightarrow $$ f '(x) = 3x

$$ \Rightarrow $$ f ''(x) = 6x + 2f '(1) . . . . . . (2)

$$ \Rightarrow $$ f '''(x) = 6 . . . . . .(3)

put x = 1 in equation (1) :

f '(1) = 3 + 2f '(1) + f ''(2) . . . . .(4)

put x = 2 in equation (2) :

f ''(2) = 12 + 2f '(1) . . . . .(5)

from equation (4) & (5) :

$$-$$3 $$-$$ f '(1) = 12 + 2f'(1)

$$ \Rightarrow $$ 3f '(1) = $$-$$ 15

$$ \Rightarrow $$ f '(1) = $$-$$ 5 $$ \Rightarrow $$ f ''(2) = 2 . . . . .(2)

put x = 3 in equation (3) :

f ''' (3) = 6

$$ \therefore $$ f(x) = x

f(2) = 8 $$-$$ 20 + 4 + 6 = $$-$$ 2

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (9) *keyboard_arrow_right*

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Trigonometric Functions & Equations *keyboard_arrow_right*

Properties of Triangle *keyboard_arrow_right*

Inverse Trigonometric Functions *keyboard_arrow_right*

Complex Numbers *keyboard_arrow_right*

Quadratic Equation and Inequalities *keyboard_arrow_right*

Permutations and Combinations *keyboard_arrow_right*

Mathematical Induction and Binomial Theorem *keyboard_arrow_right*

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*

Mathematical Reasoning *keyboard_arrow_right*

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*

Straight Lines and Pair of Straight Lines *keyboard_arrow_right*

Circle *keyboard_arrow_right*

Conic Sections *keyboard_arrow_right*