Joint Entrance Examination

Graduate Aptitude Test in Engineering

NEW

New Website Launch

Experience the best way to solve previous year questions with **mock
tests** (very detailed analysis), **bookmark your favourite questions**, **practice** etc...

1

MCQ (Single Correct Answer)

A bird is sitting on the top of a vertical pole $$20$$ m high and its elevation from a point $$O$$ on the ground is $${45^ \circ }$$. It files off horizontally straight away from the point $$O$$. After one second, the elevation of the bird from $$O$$ is reduced to $${30^ \circ }$$. Then the speed (in m/s) of the bird is

A

$$20\sqrt 2 $$

B

$$20\left( {\sqrt 3 - 1} \right)$$

C

$$40\left( {\sqrt 2 - 1} \right)$$

D

$$40\left( {\sqrt 3 - \sqrt 2 } \right)$$

Let the speed be $$y$$ $$m/sec$$.

Let $$AC$$ be the vertical pole of height $$20$$ $$m.$$

Let $$O$$ be the point on the ground such that $$\angle AOC = {45^ \circ }$$

Let $$OC = x$$

Time $$t=1$$ $$s$$

From $$\Delta AOC,\,\,\tan {45^ \circ } = {{20} \over x}\,\,\,\,\,\,\,.....\left( i \right)$$

and from $$\Delta BOD,\,\,\tan {30^ \circ } = {{20} \over {x + y}}...\left( {ii} \right)$$

From $$(i)$$ and $$(ii),$$ we have $$x=20$$

and $${1 \over {\sqrt 3 }} = {{20} \over {x + y}}$$

$$ \Rightarrow {1 \over {\sqrt 3 }} = {{20} \over {20 + y}}$$

$$ \Rightarrow 20 + y = 20\sqrt 3 $$

So, $$y = 20\left( {\sqrt 3 - 1} \right)\,\,i.e.,$$

speed $$ = 20\left( {\sqrt 3 - 1} \right)m/s$$

Let $$AC$$ be the vertical pole of height $$20$$ $$m.$$

Let $$O$$ be the point on the ground such that $$\angle AOC = {45^ \circ }$$

Let $$OC = x$$

Time $$t=1$$ $$s$$

From $$\Delta AOC,\,\,\tan {45^ \circ } = {{20} \over x}\,\,\,\,\,\,\,.....\left( i \right)$$

and from $$\Delta BOD,\,\,\tan {30^ \circ } = {{20} \over {x + y}}...\left( {ii} \right)$$

From $$(i)$$ and $$(ii),$$ we have $$x=20$$

and $${1 \over {\sqrt 3 }} = {{20} \over {x + y}}$$

$$ \Rightarrow {1 \over {\sqrt 3 }} = {{20} \over {20 + y}}$$

$$ \Rightarrow 20 + y = 20\sqrt 3 $$

So, $$y = 20\left( {\sqrt 3 - 1} \right)\,\,i.e.,$$

speed $$ = 20\left( {\sqrt 3 - 1} \right)m/s$$

2

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.

3

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

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

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

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

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

4

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

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

On those following papers in MCQ (Single Correct Answer)

Number in Brackets after Paper Indicates No. of Questions

JEE Main 2021 (Online) 31st August Morning Shift (1)

JEE Main 2021 (Online) 27th August Evening Shift (1)

JEE Main 2021 (Online) 27th August Morning Shift (1)

JEE Main 2021 (Online) 26th August Evening Shift (1)

JEE Main 2021 (Online) 25th July Morning Shift (1)

JEE Main 2021 (Online) 20th July Evening Shift (1)

JEE Main 2021 (Online) 20th July Morning Shift (1)

JEE Main 2021 (Online) 18th March Evening Shift (1)

JEE Main 2021 (Online) 26th February Evening Shift (1)

JEE Main 2021 (Online) 25th February Morning Shift (1)

JEE Main 2021 (Online) 24th February Evening Shift (1)

JEE Main 2021 (Online) 24th February Morning Shift (1)

JEE Main 2020 (Online) 6th September Evening Slot (1)

JEE Main 2020 (Online) 4th September Evening Slot (1)

JEE Main 2020 (Online) 4th September Morning Slot (1)

JEE Main 2019 (Online) 12th April Evening Slot (1)

JEE Main 2019 (Online) 10th April Evening Slot (1)

JEE Main 2019 (Online) 10th April Morning Slot (1)

JEE Main 2019 (Online) 9th April Evening Slot (1)

JEE Main 2019 (Online) 8th April Evening Slot (2)

JEE Main 2019 (Online) 12th January Evening Slot (1)

JEE Main 2019 (Online) 11th January Evening Slot (1)

JEE Main 2019 (Online) 11th January Morning Slot (1)

JEE Main 2019 (Online) 10th January Evening Slot (1)

JEE Main 2019 (Online) 10th January Morning Slot (1)

JEE Main 2018 (Online) 16th April Morning Slot (1)

JEE Main 2018 (Online) 15th April Evening Slot (1)

JEE Main 2018 (Online) 15th April Morning Slot (1)

JEE Main 2016 (Online) 10th April Morning Slot (1)

JEE Main 2015 (Offline) (1)

JEE Main 2014 (Offline) (1)

AIEEE 2010 (1)

AIEEE 2008 (1)

AIEEE 2007 (1)

AIEEE 2005 (2)

AIEEE 2004 (2)

AIEEE 2003 (3)

AIEEE 2002 (2)

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