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

If tangents are drawn to the ellipse x2^{} + 2y^{2} = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve :

A

$${{{x^2}} \over 2} + {{{y^2}} \over 4} = 1$$

B

$${1 \over {2{x^2}}} + {1 \over {4{y^2}}} = 1$$

C

$${1 \over {4{x^2}}} + {1 \over {2{y^2}}} = 1$$

D

$${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$$

Equation of general tangent on ellipse

$${x \over {a\,\sec \theta }} + {y \over {b\cos ec\theta }} = 1$$

$$a = \sqrt 2 ,\,\,b = 1$$

$$ \Rightarrow {x \over {\sqrt 2 \sec \theta }} + {y \over {\cos ec\theta }} = 1$$

Let the midpoint be (h, k)

$$h = {{\sqrt 2 \sec \theta } \over 2} \Rightarrow \cos \theta = {1 \over {\sqrt 2 h}}$$

and $$k = {{\cos ec\theta } \over 2} \Rightarrow \sin \theta = {1 \over {2k}}$$

$$ \because $$ $${\sin ^2}\theta + {\cos ^2}\theta = 1$$

$$ \Rightarrow $$ $${1 \over {2{h^2}}} + {1 \over {4{k^2}}} = 1$$

$$ \Rightarrow $$ $${1 \over {2{x^2}}} + {1 \over {4{y^2}}} = 1$$

$${x \over {a\,\sec \theta }} + {y \over {b\cos ec\theta }} = 1$$

$$a = \sqrt 2 ,\,\,b = 1$$

$$ \Rightarrow {x \over {\sqrt 2 \sec \theta }} + {y \over {\cos ec\theta }} = 1$$

Let the midpoint be (h, k)

$$h = {{\sqrt 2 \sec \theta } \over 2} \Rightarrow \cos \theta = {1 \over {\sqrt 2 h}}$$

and $$k = {{\cos ec\theta } \over 2} \Rightarrow \sin \theta = {1 \over {2k}}$$

$$ \because $$ $${\sin ^2}\theta + {\cos ^2}\theta = 1$$

$$ \Rightarrow $$ $${1 \over {2{h^2}}} + {1 \over {4{k^2}}} = 1$$

$$ \Rightarrow $$ $${1 \over {2{x^2}}} + {1 \over {4{y^2}}} = 1$$

2

Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it ?

A

$$\left( {4\sqrt 2 ,2\sqrt 3 } \right)$$

B

$$\left( {4\sqrt 3 ,2\sqrt 3 } \right)$$

C

$$\left( {4\sqrt 3 ,2\sqrt 2 } \right)$$

D

$$\left( {4\sqrt 2 ,2\sqrt 2 } \right)$$

$${{2{b^2}} \over a} = 8$$ and 2ae $$=$$ 2b

$$ \Rightarrow $$ $${b \over a}$$ = e and 1 $$-$$ e^{2} = e^{2} $$ \Rightarrow $$ e $$=$$ $${1 \over {\sqrt 2 }}$$

$$ \Rightarrow $$ b = 4$$\sqrt 2 $$ and a $$=$$ 8

So equation of ellipse is $${{{x^2}} \over {64}} + {{{y^2}} \over {32}} = 1$$

$$ \Rightarrow $$ $${b \over a}$$ = e and 1 $$-$$ e

$$ \Rightarrow $$ b = 4$$\sqrt 2 $$ and a $$=$$ 8

So equation of ellipse is $${{{x^2}} \over {64}} + {{{y^2}} \over {32}} = 1$$

3

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the
eccentricity of the hyperbola is :

A

$${{13} \over 6}$$

B

2

C

$${{13} \over 12}$$

D

$${{13} \over 8}$$

2b = 5 and 2ae = 13

b^{2} = a^{2}(e^{2} $$-$$ 1) $$ \Rightarrow $$ $${{25} \over 4}$$ = $${{169} \over 4}$$ $$-$$ a^{2}

$$ \Rightarrow $$ a $$=$$ 6 $$ \Rightarrow $$ e $$=$$ $${{13} \over {12}}$$

b

$$ \Rightarrow $$ a $$=$$ 6 $$ \Rightarrow $$ e $$=$$ $${{13} \over {12}}$$

4

If the area of the triangle whose one vertex is at the vertex of the parabola, y^{2} + 4(x – a^{2}) = 0 and the othertwo vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is

A

$$5\sqrt 5 $$

B

$${\left( {10} \right)^{2/3}}$$

C

$$5\left( {{2^{1/3}}} \right)$$

D

5

Vertex is (a^{2}, 0)

y^{2} $$=$$ $$-$$(x $$-$$ a^{2}) and x $$=$$ 0 $$ \Rightarrow $$ (0, $$ \pm $$ 2a)

Area of triangle is $$ = {1 \over 2}.$$4a.(a^{2}) = 250

$$ \Rightarrow $$ a^{3} = 125 or a = 5

y

Area of triangle is $$ = {1 \over 2}.$$4a.(a

$$ \Rightarrow $$ a

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (1) *keyboard_arrow_right*

AIEEE 2003 (2) *keyboard_arrow_right*

AIEEE 2004 (2) *keyboard_arrow_right*

AIEEE 2005 (3) *keyboard_arrow_right*

AIEEE 2006 (3) *keyboard_arrow_right*

AIEEE 2007 (3) *keyboard_arrow_right*

AIEEE 2008 (2) *keyboard_arrow_right*

AIEEE 2009 (1) *keyboard_arrow_right*

AIEEE 2010 (1) *keyboard_arrow_right*

AIEEE 2011 (1) *keyboard_arrow_right*

AIEEE 2012 (2) *keyboard_arrow_right*

JEE Main 2013 (Offline) (2) *keyboard_arrow_right*

JEE Main 2014 (Offline) (2) *keyboard_arrow_right*

JEE Main 2015 (Offline) (3) *keyboard_arrow_right*

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

JEE Main 2016 (Online) 9th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2016 (Online) 10th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2017 (Offline) (2) *keyboard_arrow_right*

JEE Main 2017 (Online) 8th April Morning Slot (5) *keyboard_arrow_right*

JEE Main 2017 (Online) 9th April Morning Slot (2) *keyboard_arrow_right*

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

JEE Main 2018 (Online) 15th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2018 (Online) 15th April Evening Slot (2) *keyboard_arrow_right*

JEE Main 2018 (Online) 16th April Morning Slot (3) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th January Morning Slot (4) *keyboard_arrow_right*

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

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

JEE Main 2019 (Online) 10th January Evening Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 11th January Morning Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 11th January Evening Slot (4) *keyboard_arrow_right*

JEE Main 2019 (Online) 12th January Morning Slot (3) *keyboard_arrow_right*

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

JEE Main 2019 (Online) 8th April Morning Slot (2) *keyboard_arrow_right*

JEE Main 2019 (Online) 8th April Evening Slot (3) *keyboard_arrow_right*

JEE Main 2019 (Online) 9th April Morning Slot (2) *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 (3) *keyboard_arrow_right*

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

JEE Main 2019 (Online) 12th April Evening Slot (3) *keyboard_arrow_right*

JEE Main 2020 (Online) 7th January Morning Slot (2) *keyboard_arrow_right*

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

JEE Main 2020 (Online) 8th January Morning Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 8th January Evening Slot (2) *keyboard_arrow_right*

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

JEE Main 2020 (Online) 9th January Evening Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 2nd September Morning Slot (1) *keyboard_arrow_right*

JEE Main 2020 (Online) 2nd September Evening Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 3rd September Morning Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 3rd September Evening Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 4th September Morning Slot (2) *keyboard_arrow_right*

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

JEE Main 2020 (Online) 5th September Morning Slot (2) *keyboard_arrow_right*

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

JEE Main 2020 (Online) 6th September Morning Slot (2) *keyboard_arrow_right*

JEE Main 2020 (Online) 6th September Evening Slot (2) *keyboard_arrow_right*

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

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

JEE Main 2021 (Online) 25th February Morning Slot (1) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th February Evening Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 16th March Morning Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 16th March Evening Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 17th March Evening Shift (1) *keyboard_arrow_right*

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

JEE Main 2021 (Online) 20th July Morning 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 (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th July Morning Shift (3) *keyboard_arrow_right*

JEE Main 2021 (Online) 25th 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 (2) *keyboard_arrow_right*

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

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

JEE Main 2021 (Online) 31st August Morning Shift (2) *keyboard_arrow_right*

JEE Main 2021 (Online) 31st August Evening Shift (2) *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*