Joint Entrance Examination

Graduate Aptitude Test in Engineering

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1

MCQ (Single Correct Answer)

For a regular polygon, let $$r$$ and $$R$$ be the radii of the inscribed and the circumscribed circles. A $$false$$ statement among the following is

A

There is a regular polygon with $${r \over R} = {1 \over {\sqrt 2 }}$$

B

There is a regular polygon with $${r \over R} = {2 \over 3}$$

C

There is a regular polygon with $${r \over R} = {{\sqrt 3 } \over 2}$$

D

There is a regular polygon with $${r \over R} = {1 \over 2}$$

If $$O$$ is center of polygon and

$$AB$$ is one of the side, then by figure

$$\cos {\pi \over n} = {r \over R}$$

$$ \Rightarrow {r \over R} = {1 \over 2},{1 \over {\sqrt 2 }},{{\sqrt 3 } \over 2}\,\,for$$

$$n = 3,4,6$$ respectively.

2

MCQ (Single Correct Answer)

$$AB$$ is a vertical pole with $$B$$ at the ground level and $$A$$ at the top. $$A$$ man finds that the angle of elevation of the point $$A$$ from a certain point $$C$$ on the ground is $${60^ \circ }$$. He moves away from the pole along the line $$BC$$ to a point $$D$$ such that $$CD=7$$ m. From $$D$$ the angle of elevation of the point $$A$$ is $${45^ \circ }$$. Then the height of the pole is

A

$${{7\sqrt 3 } \over 2} {1 \over {\sqrt {3 - 1} }}m$$

B

$${{7\sqrt 3 } \over 2}\left( {\sqrt {3 + 1} } \right)m$$

C

$${{7\sqrt 3 } \over 2}\left( {\sqrt {3 - 1} } \right)m$$

D

$${{7\sqrt 3 } \over 2} {1 \over {\sqrt {3 + 1} }}m$$

In $$\Delta ABC$$

$${h \over x} = \tan {60^ \circ } = \sqrt 3 $$

$$ \Rightarrow x = {h \over {\sqrt 3 }}$$

In $$\Delta ABD{h \over {x + 7}}$$

$$ = \tan {45^ \circ } = 1$$

$$ \Rightarrow h = x + 7 \Rightarrow h - {h \over {\sqrt 3 }} = 7$$

$$ \Rightarrow h = {{7\sqrt 3 } \over {\sqrt 3 - 1}} \times {{\sqrt 3 + 1} \over {\sqrt 3 + 1}}$$

$$ \Rightarrow h = {{7\sqrt 3 } \over 2}\left( {\sqrt 3 + 1\,m} \right)$$

3

MCQ (Single Correct Answer)

A tower stands at the centre of a circular park. $$A$$ and $$B$$ are two points on the boundary of the park such that $$AB(=a)$$ subtends an angle of $${60^ \circ }$$ at the foot of the tower, and the angle of elevation of the top of the tower from $$A$$ or $$B$$ is $${30^ \circ }$$. The height of the tower is

A

$$a/\sqrt 3 $$

B

$$a\sqrt 3 $$

C

$$2a/\sqrt 3 $$

D

$$2a\sqrt 3 $$

In the $$\Delta AOB,\,\,\angle AOB = {60^ \circ },$$ and

$$\angle OBA = \angle OAB$$

(since $$OA=OB=AB$$ radius of same circle).

$$\therefore$$ $$\Delta AOB$$ is a equilateral triangle.

Let the height of tower is $$h$$

$$m.$$ Given distance between two points $$A$$ & $$B$$ lie on boundary of

circular park, subtends an angle of $${60^ \circ }$$ at the foot of the tower

is $$AB$$ i.e. $$AB$$$$=a.$$ A tower $$OC$$ stands at the center of a circular

park. Angle of elevation of the top of the tower from $$A$$ and $$B$$ is $${30^ \circ }$$ .

In $$\Delta OAC\,\,\tan {30^ \circ } = {h \over a}$$

$$ \Rightarrow {1 \over {\sqrt 3 }} = {h \over a} \Rightarrow h = {a \over {\sqrt 3 }}$$

$$\angle OBA = \angle OAB$$

(since $$OA=OB=AB$$ radius of same circle).

$$\therefore$$ $$\Delta AOB$$ is a equilateral triangle.

Let the height of tower is $$h$$

$$m.$$ Given distance between two points $$A$$ & $$B$$ lie on boundary of

circular park, subtends an angle of $${60^ \circ }$$ at the foot of the tower

is $$AB$$ i.e. $$AB$$$$=a.$$ A tower $$OC$$ stands at the center of a circular

park. Angle of elevation of the top of the tower from $$A$$ and $$B$$ is $${30^ \circ }$$ .

In $$\Delta OAC\,\,\tan {30^ \circ } = {h \over a}$$

$$ \Rightarrow {1 \over {\sqrt 3 }} = {h \over a} \Rightarrow h = {a \over {\sqrt 3 }}$$

4

MCQ (Single Correct Answer)

In a triangle $$ABC$$, let $$\angle C = {\pi \over 2}$$. If $$r$$ is the inradius and $$R$$ is the circumradius of the triangle $$ABC$$, then $$2(r+R)$$ equals

A

$$b+c$$

B

$$a+b$$

C

$$a+b+c$$

D

$$c+a$$

We know by sin c rule

$${c \over {\sin C}} = 2R \Rightarrow c = 2R\sin C$$

$$ \Rightarrow c = 2R$$

$$\left( \, \right.$$as $$\,\,\,\angle C = {90^ \circ }$$ $$\left. \, \right)$$

Also $$\tan {C \over 2} = {r \over {s - c}}$$

$$ \Rightarrow \tan {\pi \over 4} = {r \over {s - c}}$$

$$\left( \, \right.$$as $$\,\,\,\angle C = {90^ \circ }$$ $$\left. \, \right)$$

$$ \Rightarrow r = s - c = {{a + b - c} \over 2}$$

$$ \Rightarrow 2r + c = a + b$$

$$ \Rightarrow 2r + 2R = a + b$$

(using $$c=2R$$)

$${c \over {\sin C}} = 2R \Rightarrow c = 2R\sin C$$

$$ \Rightarrow c = 2R$$

$$\left( \, \right.$$as $$\,\,\,\angle C = {90^ \circ }$$ $$\left. \, \right)$$

Also $$\tan {C \over 2} = {r \over {s - c}}$$

$$ \Rightarrow \tan {\pi \over 4} = {r \over {s - c}}$$

$$\left( \, \right.$$as $$\,\,\,\angle C = {90^ \circ }$$ $$\left. \, \right)$$

$$ \Rightarrow r = s - c = {{a + b - c} \over 2}$$

$$ \Rightarrow 2r + c = a + b$$

$$ \Rightarrow 2r + 2R = a + b$$

(using $$c=2R$$)

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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

Sets and Relations

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