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

If 0 $$ \le $$ x < $${\pi \over 2}$$, then the number of values of x for which sin x $$-$$ sin 2x + sin 3x = 0, is :

A

3

B

1

C

4

D

2

sin x $$-$$ sin 2x + sin 3x = 0 $$x \in \left[ {0,{\pi \over 2}} \right)$$

$$ \Rightarrow $$ (sin3x + sinx) $$-$$ sin2x = 0

$$ \Rightarrow $$ 2sin2x.cos2x $$-$$ sin2x = 0

$$ \Rightarrow $$ sin2x (2cosx $$-$$ 1) = 0

sin 2x = 0

x = 0

and cos x = $${1 \over 2}$$

and x = $${\pi \over 3}$$

two solutions

$$ \Rightarrow $$ (sin3x + sinx) $$-$$ sin2x = 0

$$ \Rightarrow $$ 2sin2x.cos2x $$-$$ sin2x = 0

$$ \Rightarrow $$ sin2x (2cosx $$-$$ 1) = 0

sin 2x = 0

x = 0

and cos x = $${1 \over 2}$$

and x = $${\pi \over 3}$$

two solutions

2

The sum of all values of $$\theta $$ $$ \in $$$$\left( {0,{\pi \over 2}} \right)$$ satisfying

sin^{2} 2$$\theta $$ + cos^{4} 2$$\theta $$ = $${3 \over 4}$$ is -

sin

A

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

B

$${\pi \over 2}$$

C

$$\pi $$

D

$${{3\pi } \over 8}$$

sin^{2}2$$\theta $$ + cos^{4}2$$\theta $$ = $${3 \over 4}, $$$$\theta $$ $$ \in $$ $$\left( {0,{\pi \over 2}} \right)$$

$$ \Rightarrow $$ 1 $$-$$ cos^{2}2$$\theta $$ + cos^{4}2$$\theta $$ = $${3 \over 4}$$

$$ \Rightarrow $$ 4cos2$$\theta $$ $$-$$ 4cos^{2}2$$\theta $$ + 1 = 0

$$ \Rightarrow $$ (2cos^{2}2$$\theta $$ $$-$$ 1)^{2} = 0

$$ \Rightarrow $$ cos^{2}2$$\theta $$ = $${1 \over 2}$$ = cos^{2}$${{\pi \over 4}}$$

$$ \Rightarrow $$ 2$$\theta $$ = n$$\pi $$ $$ \pm $$ $${\pi \over 4}$$, n $$ \in $$ I

$$ \Rightarrow $$ $$\theta $$ = $${{n\pi } \over 2} \pm {\pi \over 8}$$

$$ \Rightarrow $$ $$\theta $$ = $${\pi \over 8},{\pi \over 2} - {\pi \over 8}$$

Sum of solutions $${\pi \over 2}$$

$$ \Rightarrow $$ 1 $$-$$ cos

$$ \Rightarrow $$ 4cos2$$\theta $$ $$-$$ 4cos

$$ \Rightarrow $$ (2cos

$$ \Rightarrow $$ cos

$$ \Rightarrow $$ 2$$\theta $$ = n$$\pi $$ $$ \pm $$ $${\pi \over 4}$$, n $$ \in $$ I

$$ \Rightarrow $$ $$\theta $$ = $${{n\pi } \over 2} \pm {\pi \over 8}$$

$$ \Rightarrow $$ $$\theta $$ = $${\pi \over 8},{\pi \over 2} - {\pi \over 8}$$

Sum of solutions $${\pi \over 2}$$

3

The value of $$\cos {\pi \over {{2^2}}}.\cos {\pi \over {{2^3}}}\,.....\cos {\pi \over {{2^{10}}}}.\sin {\pi \over {{2^{10}}}}$$ is -

A

$${1 \over {256}}$$

B

$${1 \over {2}}$$

C

$${1 \over {1024}}$$

D

$${1 \over {512}}$$

Given $$\cos {\pi \over {{2^2}}}.\cos {\pi \over {{2^3}}}\,.....\cos {\pi \over {{2^{10}}}}.\sin {\pi \over {{2^{10}}}}$$

Let $${\pi \over {{2^{10}}}}\, = \,\theta $$

$$ \therefore $$ $${\pi \over {{2^9}}}\, = \,2\theta $$

$${\pi \over {{2^8}}}\, = \,{2^2}\theta $$

$${\pi \over {{2^7}}}\, = \,{2^3}\theta $$

.

.

$${\pi \over {{2^2}}}\, = \,{2^8}\theta $$

So given term becomes,

$$\cos {2^8}\theta .\cos {2^7}\theta .....\cos \theta $$$$.\sin {\pi \over {{2^{10}}}}$$

= $$(\cos \theta .\cos 2\theta ......\cos {2^8}\theta )\sin {\pi \over {{2^{10}}}}$$

= $${{\sin {2^9}\theta } \over {{2^9}\sin \theta }}.\sin {\pi \over {{2^{10}}}}$$

= $${{\sin {2^9}\left( {{\pi \over {{2^{10}}}}} \right)} \over {{2^9}\sin {\pi \over {{2^{10}}}}}}.\sin {\pi \over {{2^{10}}}}$$

= $${{\sin \left( {{\pi \over 2}} \right)} \over {{2^9}}}$$

= $${1 \over {{2^9}}}$$ = $${1 \over {512}}$$

**Note :**

$$(\cos \theta .\cos 2\theta ......\cos {2^{n - 1}}\theta )$$ = $${{\sin {2^n}\theta } \over {{2^n}\sin \theta }}$$

Let $${\pi \over {{2^{10}}}}\, = \,\theta $$

$$ \therefore $$ $${\pi \over {{2^9}}}\, = \,2\theta $$

$${\pi \over {{2^8}}}\, = \,{2^2}\theta $$

$${\pi \over {{2^7}}}\, = \,{2^3}\theta $$

.

.

$${\pi \over {{2^2}}}\, = \,{2^8}\theta $$

So given term becomes,

$$\cos {2^8}\theta .\cos {2^7}\theta .....\cos \theta $$$$.\sin {\pi \over {{2^{10}}}}$$

= $$(\cos \theta .\cos 2\theta ......\cos {2^8}\theta )\sin {\pi \over {{2^{10}}}}$$

= $${{\sin {2^9}\theta } \over {{2^9}\sin \theta }}.\sin {\pi \over {{2^{10}}}}$$

= $${{\sin {2^9}\left( {{\pi \over {{2^{10}}}}} \right)} \over {{2^9}\sin {\pi \over {{2^{10}}}}}}.\sin {\pi \over {{2^{10}}}}$$

= $${{\sin \left( {{\pi \over 2}} \right)} \over {{2^9}}}$$

= $${1 \over {{2^9}}}$$ = $${1 \over {512}}$$

$$(\cos \theta .\cos 2\theta ......\cos {2^{n - 1}}\theta )$$ = $${{\sin {2^n}\theta } \over {{2^n}\sin \theta }}$$

4

The maximum value of 3cos$$\theta $$ + 5sin $$\left( {\theta - {\pi \over 6}} \right)$$ for any real value of $$\theta $$ is :

A

$$\sqrt {34} $$

B

$$\sqrt {31} $$

C

$$\sqrt {19} $$

D

$${{\sqrt {79} } \over 2}$$

y = 3cos$$\theta $$ + 5 $$\left( {\sin \theta {{\sqrt 3 } \over 2} - \cos \theta {1 \over 2}} \right)$$

$${{5\sqrt 3 } \over 2}$$ sin$$\theta $$ + $${1 \over 2}$$cos$$\theta $$

y_{max} = $$\sqrt {{{75} \over 4} + {1 \over 4}} $$ = $$\sqrt {19} $$

$${{5\sqrt 3 } \over 2}$$ sin$$\theta $$ + $${1 \over 2}$$cos$$\theta $$

y

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (3) *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*

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Mathematical Induction and Binomial Theorem *keyboard_arrow_right*

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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*