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

Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then :

A

a, b, c are in A.P.

B

$$\sqrt a ,\sqrt b ,\sqrt c $$ are in A.P

C

$${1 \over {\sqrt b }} + {1 \over {\sqrt c }}$$ = $${1 \over {\sqrt a }}$$

D

$${1 \over {\sqrt b }} = {1 \over {\sqrt a }} + {1 \over {\sqrt c }}$$

AB = AC + CB

$$\sqrt {{{\left( {b + c} \right)}^2} - {{\left( {c - b} \right)}^2}} $$ = $$\sqrt {{{\left( {b + a} \right)}^2} - {{\left( {b - a} \right)}^2}} $$

+ $$\sqrt {{{\left( {c + a} \right)}^2} - {{\left( {c - a} \right)}^2}} $$

$$ \Rightarrow $$ $$\sqrt {2bc} $$ = $$\sqrt {2ac} $$ + $$\sqrt {2ab} $$

Dividing by $$\sqrt {abc} $$ we get.

$$ \Rightarrow $$ $${1 \over {\sqrt a }}$$ = $${1 \over {\sqrt b }}$$ + $${1 \over {\sqrt c }}$$

2

If the circles

x^{2} + y^{2} $$-$$ 16x $$-$$ 20y + 164 = r^{2}

and (x $$-$$ 4)^{2} + (y $$-$$ 7)^{2} = 36

intersect at two distinct points, then :

x

and (x $$-$$ 4)

intersect at two distinct points, then :

A

r > 11

B

0 < r < 1

C

r = 11

D

1 < r < 11

Circles are x^{2} + y^{2} $$-$$ 16x $$-$$ 20y + 164 = r^{2} $$ \Rightarrow $$ c_{1} (8, 10)

and (x $$-$$ 4)^{2} + (y $$-$$ 7)^{2} = 36

they intersect at two distinct points

$$\left| {{r_1} - {r_2}} \right| < {c_1}{c_2} < {r_1} + {r_2}\left\{ {{c_1}{c_2} = \sqrt {16 + 9} = 5} \right\}$$

Now $$\left| {r - 6} \right| < 5 < r + 6$$

$$\left| {r - 6} \right| < 5$$

$$ \Rightarrow $$ $$ - 5 < r - 6 < 5$$

$$ \Rightarrow $$ $$1 < r < 11\,\,\,\,\,\,\,\,\,...(i)$$

$$5 < r + 6$$

$$ - 1 < r\,\,\,\,\,\,\,\,\,\,\,\,\,...(ii)$$

from (i) and (ii)

r $$ \in $$ (1, 11)

and (x $$-$$ 4)

they intersect at two distinct points

$$\left| {{r_1} - {r_2}} \right| < {c_1}{c_2} < {r_1} + {r_2}\left\{ {{c_1}{c_2} = \sqrt {16 + 9} = 5} \right\}$$

Now $$\left| {r - 6} \right| < 5 < r + 6$$

$$\left| {r - 6} \right| < 5$$

$$ \Rightarrow $$ $$ - 5 < r - 6 < 5$$

$$ \Rightarrow $$ $$1 < r < 11\,\,\,\,\,\,\,\,\,...(i)$$

$$5 < r + 6$$

$$ - 1 < r\,\,\,\,\,\,\,\,\,\,\,\,\,...(ii)$$

from (i) and (ii)

r $$ \in $$ (1, 11)

3

If a circle C passing through the point (4, 0) touches the circle x^{2} + y^{2} + 4x – 6y = 12 externally at the point (1, – 1), then the radius of C is -

A

5

B

2$$\sqrt {5} $$

C

4

D

$$\sqrt {37} $$

x^{2} + y^{2} + 4x $$-$$ 6y $$-$$ 12 = 0

Equation of tangent at (1, $$-$$ 1)

x $$-$$ y + 2(x + 1) $$-$$ 3(y $$-$$ 1) $$-$$ 12 = 0

3x $$-$$ 4y $$-$$ 7 = 0

$$ \therefore $$ Equation of circle is

(x^{2} + y^{2} + 4x $$-$$ 6y $$-$$ 12) + $$\lambda $$ (3x $$-$$ 4y $$-$$ 7) = 0

It passes through (4, 0) :

(16 + 16 $$-$$ 12) + $$\lambda $$ (12 $$-$$ 7) = 0

$$ \Rightarrow $$ 20 + $$\lambda $$(5) = 0

$$ \Rightarrow $$ $$\lambda $$ = $$-$$ 4

$$ \therefore $$ (x^{2} + y^{2} + 4x $$-$$ 6y $$-$$ 12) $$-$$ 4(3x $$-$$ 4y $$-$$ 7) = 0

or x^{2} + y^{2} $$-$$ 8x + 10y + 16 = 0

Radius = $$\sqrt {16 + 25 - 16} = 5$$

Equation of tangent at (1, $$-$$ 1)

x $$-$$ y + 2(x + 1) $$-$$ 3(y $$-$$ 1) $$-$$ 12 = 0

3x $$-$$ 4y $$-$$ 7 = 0

$$ \therefore $$ Equation of circle is

(x

It passes through (4, 0) :

(16 + 16 $$-$$ 12) + $$\lambda $$ (12 $$-$$ 7) = 0

$$ \Rightarrow $$ 20 + $$\lambda $$(5) = 0

$$ \Rightarrow $$ $$\lambda $$ = $$-$$ 4

$$ \therefore $$ (x

or x

Radius = $$\sqrt {16 + 25 - 16} = 5$$

4

If the area of an equilateral triangle inscribed in the circle x^{2} + y^{2}
+ 10x + 12y + c = 0 is $$27\sqrt 3 $$ sq units then c is equal to

A

20

B

25

C

$$-$$ 25

D

13

$$3\left( {{1 \over 2}{r^2}.\sin {{120}^o}} \right) = 27\sqrt 3 $$

$${{{r^2}} \over 2}{{\sqrt 3 } \over 2} = {{27\sqrt 3 } \over 3}$$

$${r^2} = {{108} \over 3} = 36$$

Radius $$ = \sqrt {25 + 36 - C} = \sqrt {36} $$

$$C = 25$$

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*

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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*

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

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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*

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

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

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

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

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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*

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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*