Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (Single Correct Answer)

The number of solution of $$\tan \,x + \sec \,x = 2\cos \,x$$ in $$\left[ {0,\,2\,\pi } \right]$$ is

A

2

B

3

C

0

D

1

Given equation is $$\tan \,x + \sec \,x = 2\cos \,x$$

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

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

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

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

$$ \Rightarrow $$ $$2{\sin ^2}x + \sin x - 1 = 0$$

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

$$ \Rightarrow $$ $${\sin x = {1 \over 2}}$$ and $${\sin x = - 1}$$

When $${\sin x = {1 \over 2}}$$ then possible $$x$$ = $$30^\circ $$, $$150^\circ $$

When $${\sin x = - 1}$$ then possible $$x$$ = $$270^\circ $$

So three solutions possible.

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

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

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

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

$$ \Rightarrow $$ $$2{\sin ^2}x + \sin x - 1 = 0$$

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

$$ \Rightarrow $$ $${\sin x = {1 \over 2}}$$ and $${\sin x = - 1}$$

When $${\sin x = {1 \over 2}}$$ then possible $$x$$ = $$30^\circ $$, $$150^\circ $$

When $${\sin x = - 1}$$ then possible $$x$$ = $$270^\circ $$

So three solutions possible.

2

MCQ (Single Correct Answer)

The period of $${\sin ^2}\theta $$ is

A

$${\pi ^2}$$

B

$$\pi $$

C

$$2\pi $$

D

$$\pi /2$$

The period of $${\sin ^2}\theta $$ is = $$\pi $$

**Note :**

(1) When $$n$$ is odd then the period of $${\sin ^n}\theta $$, $${\cos ^n}\theta $$, $${\csc ^n}\theta $$, $${\sec ^n}\theta $$ = $$2\pi $$

(2) When $$n$$ is even then the period of $${\sin ^n}\theta $$, $${\cos ^n}\theta $$, $${\csc ^n}\theta $$, $${\sec ^n}\theta $$ = $$\pi $$

(3) When $$n$$ is even/odd then the period of $${\tan ^n}\theta $$, $${\cot ^n}\theta $$ = $$\pi $$

(3) When $$n$$ is even/odd then the period of $$\left| {{{\sin }^n}\theta } \right|$$, $$\left| {{{\cos }^n}\theta } \right|$$, $$\left| {{{\csc }^n}\theta } \right|$$, $$\left| {{{\sec }^n}\theta } \right|$$, $$\left| {{{\tan }^n}\theta } \right|$$, $$\left| {{{\cot }^n}\theta } \right|$$ = $$\pi $$

(1) When $$n$$ is odd then the period of $${\sin ^n}\theta $$, $${\cos ^n}\theta $$, $${\csc ^n}\theta $$, $${\sec ^n}\theta $$ = $$2\pi $$

(2) When $$n$$ is even then the period of $${\sin ^n}\theta $$, $${\cos ^n}\theta $$, $${\csc ^n}\theta $$, $${\sec ^n}\theta $$ = $$\pi $$

(3) When $$n$$ is even/odd then the period of $${\tan ^n}\theta $$, $${\cot ^n}\theta $$ = $$\pi $$

(3) When $$n$$ is even/odd then the period of $$\left| {{{\sin }^n}\theta } \right|$$, $$\left| {{{\cos }^n}\theta } \right|$$, $$\left| {{{\csc }^n}\theta } \right|$$, $$\left| {{{\sec }^n}\theta } \right|$$, $$\left| {{{\tan }^n}\theta } \right|$$, $$\left| {{{\cot }^n}\theta } \right|$$ = $$\pi $$

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