Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

MCQ (Single Correct Answer)

Let **A** and **B** denote the statements

**A**: $$\cos \alpha + \cos \beta + \cos \gamma = 0$$

**B**: $$\sin \alpha + \sin \beta + \sin \gamma = 0$$

If $$\cos \left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left( {\alpha - \beta } \right) = - {3 \over 2},$$ then:

A

B

both **A** and **B** are true

C

both **A** and **B** are false

D

2

MCQ (Single Correct Answer)

If $$0 < x < \pi $$ and $$\cos x + \sin x = {1 \over 2},$$ then $$\tan x$$ is

A

$${{\left( {1 - \sqrt 7 } \right)} \over 4}$$

B

$${{\left( {4 - \sqrt 7 } \right)} \over 3}$$

C

$$ - {{\left( {4 + \sqrt 7 } \right)} \over 3}$$

D

$${{\left( {1 + \sqrt 7 } \right)} \over 4}$$

$$\cos x + \sin x = {1 \over 2}$$

$$ \Rightarrow {\left( {\cos x + {\mathop{\rm sinx}\nolimits} } \right)^2} = {1 \over 4}$$

$$ \Rightarrow {\cos ^2}x + {\sin ^2}x + 2\cos x\sin x = {1 \over 4}$$

$$\left[ \because {{{\cos }^2}x + {{\sin }^2}x = 1\, \,and \,\,2\cos x\sin x = \sin 2x} \right]$$

$$ \Rightarrow 1 + \sin 2x = {1 \over 4}$$

$$ \Rightarrow \sin 2x = - {3 \over 4},$$ so $$x$$ is obtuse and

$${{2\tan x} \over {1 + {{\tan }^2}x}} = - {3 \over 4}$$

$$ \Rightarrow 3{\tan ^2}x + 8\tan x + 3 = 0$$

$$\therefore$$ $$\tan x = {{ - 8 \pm \sqrt {64 - 36} } \over 6}$$

$$ = {{ - 4 \pm \sqrt 7 } \over 3}$$

as $$\tan x < 0\,$$

$$\therefore$$ $$\tan x = {{ - 4 - \sqrt 7 } \over 3}$$

$$ \Rightarrow {\left( {\cos x + {\mathop{\rm sinx}\nolimits} } \right)^2} = {1 \over 4}$$

$$ \Rightarrow {\cos ^2}x + {\sin ^2}x + 2\cos x\sin x = {1 \over 4}$$

$$\left[ \because {{{\cos }^2}x + {{\sin }^2}x = 1\, \,and \,\,2\cos x\sin x = \sin 2x} \right]$$

$$ \Rightarrow 1 + \sin 2x = {1 \over 4}$$

$$ \Rightarrow \sin 2x = - {3 \over 4},$$ so $$x$$ is obtuse and

$${{2\tan x} \over {1 + {{\tan }^2}x}} = - {3 \over 4}$$

$$ \Rightarrow 3{\tan ^2}x + 8\tan x + 3 = 0$$

$$\therefore$$ $$\tan x = {{ - 8 \pm \sqrt {64 - 36} } \over 6}$$

$$ = {{ - 4 \pm \sqrt 7 } \over 3}$$

as $$\tan x < 0\,$$

$$\therefore$$ $$\tan x = {{ - 4 - \sqrt 7 } \over 3}$$

3

MCQ (Single Correct Answer)

The number of values of $$x$$ in the interval $$\left[ {0,3\pi } \right]\,$$ satisfying the equation $$2{\sin ^2}x + 5\sin x - 3 = 0$$ is

A

4

B

6

C

1

D

2

$$2{\sin ^2}x + 5\sin x - 3 = 0$$

$$ \Rightarrow \left( {\sin x + 3} \right)\left( {2\sin x - 1} \right) = 0$$

$$\sin x = {1 \over 2}$$ and $$\,\,\sin x \ne - 3$$

Given that $$x \in \left[ {0,3\pi } \right]$$

So possible values of x are $$30^\circ $$, $$150^\circ $$, $$390^\circ $$, $$510^\circ $$. That means x have 4 values.

4

MCQ (Single Correct Answer)

A line makes the same angle $$\theta $$, with each of the $$x$$ and $$z$$ axis.

If the angle $$\beta \,$$, which it makes with y-axis, is such that $$\,{\sin ^2}\beta = 3{\sin ^2}\theta ,$$ then $${\cos ^2}\theta $$ equals

If the angle $$\beta \,$$, which it makes with y-axis, is such that $$\,{\sin ^2}\beta = 3{\sin ^2}\theta ,$$ then $${\cos ^2}\theta $$ equals

A

$${2 \over 5}$$

B

$${1 \over 5}$$

C

$${3 \over 5}$$

D

$${2 \over 3}$$

In this question given that the line makes angle θ with x and z-axis and β with y−axis.

$$\therefore\: cos^2\theta+cos^2\beta+cos^2\theta=1$$

$$\Rightarrow\:2cos^2\theta=1-cos^2\beta$$

$$ \Rightarrow 2{\cos ^2}\theta = {\sin ^2}\beta $$

But given that $$sin^2\beta=3sin^2\theta$$

$$\therefore$$ $$2{\cos ^2}\theta = 3{\sin ^2}\theta $$

$$ \Rightarrow 2{\cos ^2}\theta = 3\left( {1 - {{\cos }^2}\theta } \right)$$

$$ \Rightarrow 2{\cos ^2}\theta = 3 - 3{\cos ^2}\theta $$

$$ \Rightarrow 5{\cos ^2}\theta = 3$$

$$ \Rightarrow {\cos ^2}\theta = {3 \over 5}$$

On those following papers in MCQ (Single Correct Answer)

Number in Brackets after Paper Indicates No. of Questions

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Complex Numbers

Quadratic Equation and Inequalities

Permutations and Combinations

Mathematical Induction and Binomial Theorem

Sequences and Series

Matrices and Determinants

Vector Algebra and 3D Geometry

Probability

Statistics

Mathematical Reasoning

Trigonometric Functions & Equations

Properties of Triangle

Inverse Trigonometric Functions

Straight Lines and Pair of Straight Lines

Circle

Conic Sections

Functions

Limits, Continuity and Differentiability

Differentiation

Application of Derivatives

Indefinite Integrals

Definite Integrals and Applications of Integrals

Differential Equations