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

A line drawn through the point P(4, 7) cuts the circle x^{2} + y^{2} = 9 at the points A and B. Then PA⋅PB is equal to :

A

53

B

56

C

74

D

65

P(4, 7). Here, x = 4, y = 7

$$ \therefore $$ PA $$ \times $$ PB = PT^{2}

Also; PT = $$\sqrt {{x^2} + {y^2} - {{\left( {x - y} \right)}^2}} $$

$$ \Rightarrow $$ PT = $$\sqrt {16 + 49 - 9} $$ = $$\sqrt {56} $$

$$ \Rightarrow $$ PT^{2} = 56

$$ \therefore $$ PA $$ \times $$ PB = 56

$$ \therefore $$ PA $$ \times $$ PB = PT

Also; PT = $$\sqrt {{x^2} + {y^2} - {{\left( {x - y} \right)}^2}} $$

$$ \Rightarrow $$ PT = $$\sqrt {16 + 49 - 9} $$ = $$\sqrt {56} $$

$$ \Rightarrow $$ PT

$$ \therefore $$ PA $$ \times $$ PB = 56

2

The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60^{o}. If the area of the quadrilateral is $$4\sqrt 3 $$, then the perimeter
of the quadrilateral is :

A

12.5

B

13.2

C

12

D

13

Here; cos$$\theta $$ = $${{{a^2} + {b^2} - {c^2}} \over {2ab}}$$

and $$\theta $$ = 60

$$ \Rightarrow $$ cos 60

$$ \Rightarrow $$ 10 = 29 $$-$$ c

$$ \Rightarrow $$ c

$$ \Rightarrow $$

also; cos$$\theta $$ = $${{{a^2} + {b^2} - {c^2}} \over {2ab}}$$

and $$\theta $$ = 120

$$ \Rightarrow $$ $$-$$ $${1 \over 2}$$ = $${{{a^2} + {b^2} - 19} \over {2ab}}$$

$$ \Rightarrow $$ a

$$ \Rightarrow $$ a

$$ \therefore $$ Area = $${1 \over 2} \times 2 \times 5$$ sin 60 + $${1 \over 2}$$ ab sin 120

$$ \Rightarrow $$ $${{5\sqrt 3 } \over 2} + {{ab\sqrt 3 } \over 4}$$ = $$4\sqrt 3 $$

$$ \Rightarrow $$ $${{ab} \over 4}$$ = 4 $$-$$ $${5 \over 2}$$ = $${3 \over 2}$$

$$ \Rightarrow $$

$$ \therefore $$ a

$$ \Rightarrow $$ a = 2, b = 3

Perimeter = Sum of all sides

= 2 + 5 + 2 + 3 = 12

3

Let the orthocentre and centroid of a triangle be A(-3, 5) and B(3, 3) respectively. If C is the circumcentre
of this triangle, then the radius of the circle having line segment AC as diameter, is :

A

$${{3\sqrt 5 } \over 2}$$

B

$$\sqrt {10} $$

C

$$2\sqrt {10} $$

D

$$3\sqrt {{5 \over 2}} $$

In a triangle, orthocentre, centroid and circumcenter are collinear and centroid divides orthocenter and circumcenter in 2 : 1 ratio.

Here AB = 2 BC

$$AB = \sqrt {{{\left( {3 + 3} \right)}^2} + {{\left( {3 - 5} \right)}^2}} $$

$$ = \sqrt {36 + 4} $$

$$ = \sqrt {40} $$

$$ = 2\sqrt {10} $$

$$\therefore\,\,\,\,$$ BC $$ = \sqrt {10} $$

$$\therefore\,\,\,\,$$ AB + BC

$$ = 2\sqrt {10} + \sqrt {10} $$

$$ = 3\sqrt {10} $$

AC is the diameter of the circle.

$$\therefore\,\,\,$$ Radius $$ = {1 \over 2}\,\,$$ AC

$$ = {1 \over 2} \times 3\sqrt {10} $$

$$ = 3\sqrt {{5 \over 2}} $$

Here AB = 2 BC

$$AB = \sqrt {{{\left( {3 + 3} \right)}^2} + {{\left( {3 - 5} \right)}^2}} $$

$$ = \sqrt {36 + 4} $$

$$ = \sqrt {40} $$

$$ = 2\sqrt {10} $$

$$\therefore\,\,\,\,$$ BC $$ = \sqrt {10} $$

$$\therefore\,\,\,\,$$ AB + BC

$$ = 2\sqrt {10} + \sqrt {10} $$

$$ = 3\sqrt {10} $$

AC is the diameter of the circle.

$$\therefore\,\,\,$$ Radius $$ = {1 \over 2}\,\,$$ AC

$$ = {1 \over 2} \times 3\sqrt {10} $$

$$ = 3\sqrt {{5 \over 2}} $$

4

If the tangent at (1, 7) to the curve x^{2} = y - 6

touches the circle x^{2} + y^{2} + 16x + 12y + c = 0, then the value of c is :

touches the circle x

A

95

B

195

C

185

D

85

Equation of tangent at (x

xx

Now equation of tangent at (1, 7) to x

$$x\,.\,1 = 4\,.\,{1 \over 4}\left( {{{y + 7} \over 2}} \right) - 6$$

$$ \Rightarrow 2x = y + 7 - 12$$

$$ \Rightarrow 2x - y + 5 = 0$$

This tangent touches the circle.

So, perpendicular distance from the center of the circle to the tangent is equal to the radius of the circle.

For the circle,

$${x^2} + {y^2} + 16x + 12y + C = 0$$

center is ($$-$$8, $$-$$6)

and radius (r) = $$\sqrt {{8^2} + {6^2} - c} $$

$$ = \sqrt {100 - c} $$

Distance of the tangent from the center of the circle

d = $$\left| {{{2\left( { - 8} \right) - \left( { - 6} \right) + 5} \over {\sqrt {{2^2} + {1^2}} }}} \right|$$

$$ = \left| {{{ - 16 + 6 + 5} \over {\sqrt 5 }}} \right|$$

And we know d = r

$$\therefore\,\,\,$$ $$\left| {{{ - 16 + 11} \over {\sqrt 5 }}} \right| = \sqrt {100 - c} $$

$$ \Rightarrow \left| {{{ - 5} \over {\sqrt 5 }}} \right| = \sqrt {100 - c} $$

$$ \Rightarrow \left| { - \sqrt 5 } \right| = \sqrt {100 - c} $$

$$ \Rightarrow 5 = 100 - c$$

$$ \Rightarrow c = 95$$

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (4) *keyboard_arrow_right*

AIEEE 2003 (2) *keyboard_arrow_right*

AIEEE 2004 (4) *keyboard_arrow_right*

AIEEE 2005 (4) *keyboard_arrow_right*

AIEEE 2006 (2) *keyboard_arrow_right*

AIEEE 2007 (2) *keyboard_arrow_right*

AIEEE 2008 (3) *keyboard_arrow_right*

AIEEE 2009 (1) *keyboard_arrow_right*

AIEEE 2010 (1) *keyboard_arrow_right*

AIEEE 2011 (1) *keyboard_arrow_right*

AIEEE 2012 (1) *keyboard_arrow_right*

JEE Main 2013 (Offline) (1) *keyboard_arrow_right*

JEE Main 2014 (Offline) (1) *keyboard_arrow_right*

JEE Main 2015 (Offline) (2) *keyboard_arrow_right*

JEE Main 2016 (Offline) (2) *keyboard_arrow_right*

JEE Main 2016 (Online) 9th April Morning Slot (1) *keyboard_arrow_right*

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

JEE Main 2017 (Offline) (1) *keyboard_arrow_right*

JEE Main 2017 (Online) 8th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2017 (Online) 9th April Morning Slot (3) *keyboard_arrow_right*

JEE Main 2018 (Offline) (2) *keyboard_arrow_right*

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

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

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

JEE Main 2019 (Online) 9th January Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th January Evening Slot (1) *keyboard_arrow_right*

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

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

JEE Main 2019 (Online) 11th January Morning Slot (3) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th January Morning Slot (2) *keyboard_arrow_right*

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

JEE Main 2019 (Online) 8th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 8th April Evening Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th April Evening Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 10th April Morning Slot (2) *keyboard_arrow_right*

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

JEE Main 2019 (Online) 12th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th April Evening Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 7th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 8th January Evening Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 9th January Morning Slot (1) *keyboard_arrow_right*

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

JEE Main 2020 (Online) 5th September Evening Slot (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 24th February Evening Slot (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th February Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th February Morning Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th February Evening Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 16th March Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 17th March Morning Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 17th March Evening Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 18th March Morning Shift (2) *keyboard_arrow_right*

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

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

JEE Main 2021 (Online) 22th July Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 27th July Evening Shift (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 27th July Morning Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 26th August Morning Shift (1) *keyboard_arrow_right*

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

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

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*

Permutations and Combinations *keyboard_arrow_right*

Mathematical Induction and Binomial Theorem *keyboard_arrow_right*

Sequences and Series *keyboard_arrow_right*

Matrices and Determinants *keyboard_arrow_right*

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*