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

The locus of the point of intersection of the lines, $$\sqrt 2 x - y + 4\sqrt 2 k = 0$$ and $$\sqrt 2 k\,x + k\,y - 4\sqrt 2 = 0$$ (k is any non-zero real parameter), is :

A

an ellipse whose eccentricity is $${1 \over {\sqrt 3 }}.$$

B

an ellipse with length of its major axis $$8\sqrt 2 .$$

C

a hyperbola whose eccentricity is $$\sqrt 3 .$$

D

a hyperbola with length of its transverse axis $$8\sqrt 2 .$$

Here, lines are :

$$\sqrt 2 x$$ $$-$$ y + 4$$\sqrt 2 k$$ = 0

$$ \Rightarrow $$$$\,\,\,$$ $$\sqrt 2 x + 4\sqrt 2 k = y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....$$(i)

and $$\sqrt 2 kx + ky - 4\sqrt 2 = 0\,\,\,\,\,...\left( {ii} \right)$$

Put the value of y from (i) in (ii) we get;

$$ \Rightarrow $$2$$\sqrt 2 $$kx + 4$$\sqrt 2 $$(k^{2} $$-$$ 1) = 0

$$ \Rightarrow $$ x = $${{2\left( {1 - {k^2}} \right)} \over k}$$, y = $${{2\sqrt 2 \left( {1 + {k^2}} \right)} \over k}$$

$$\therefore\,\,\,$$ $${\left( {{y \over {4\sqrt 2 }}} \right)^2} - {\left( {{x \over 4}} \right)^2} = 1$$

$$\therefore\,\,\,$$ length of transverse axis

2a = 2 $$ \times $$ 4$${\sqrt 2 }$$ = 8$${\sqrt 2 }$$

Hence, the locus is a hyperbola with length of its transverse axis equal to 8$${\sqrt 2 }$$

$$\sqrt 2 x$$ $$-$$ y + 4$$\sqrt 2 k$$ = 0

$$ \Rightarrow $$$$\,\,\,$$ $$\sqrt 2 x + 4\sqrt 2 k = y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....$$(i)

and $$\sqrt 2 kx + ky - 4\sqrt 2 = 0\,\,\,\,\,...\left( {ii} \right)$$

Put the value of y from (i) in (ii) we get;

$$ \Rightarrow $$2$$\sqrt 2 $$kx + 4$$\sqrt 2 $$(k

$$ \Rightarrow $$ x = $${{2\left( {1 - {k^2}} \right)} \over k}$$, y = $${{2\sqrt 2 \left( {1 + {k^2}} \right)} \over k}$$

$$\therefore\,\,\,$$ $${\left( {{y \over {4\sqrt 2 }}} \right)^2} - {\left( {{x \over 4}} \right)^2} = 1$$

$$\therefore\,\,\,$$ length of transverse axis

2a = 2 $$ \times $$ 4$${\sqrt 2 }$$ = 8$${\sqrt 2 }$$

Hence, the locus is a hyperbola with length of its transverse axis equal to 8$${\sqrt 2 }$$

2

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus an its nearest vertex on the major axis is $${3 \over 2}$$ units, then its eccentricity is :

A

$${1 \over 2}$$

B

$${1 \over 3}$$

C

$${2 \over 3}$$

D

$${1 \over 9}$$

If the cordinate of focus and vertex are (ae, 0) and (a, 0) respectively,

then distance between focus and vertex,

a $$-$$ ae = $${3 \over 2}$$ (given)

$$ \Rightarrow $$ $$\,\,\,$$ a (1 $$-$$ e) = $${3 \over 2}$$

Length of latus rectum,

$${{2{b^2}} \over a} = 4$$

$$ \Rightarrow $$ $$\,\,\,$$ b^{2} = 2a

$$ \Rightarrow $$ $$\,\,\,$$ a^{2}(1 $$-$$ e^{2}) = 2a [As b^{2} = a^{2} (1 $$-$$ e^{2})]

$$ \Rightarrow $$ $$\,\,\,$$ a (1 $$-$$ e) ( 1 + e) = 2

Putting a (1 $$-$$ e) = $${3 \over 2}$$

$$ \Rightarrow $$ $$\,\,\,$$ $${3 \over 2}$$ (1 + e) = 2

$$ \Rightarrow $$ $$\,\,\,$$ 3 + 3e = 4

$$ \Rightarrow $$ $$\,\,\,$$ e = $${1 \over 3}$$

then distance between focus and vertex,

a $$-$$ ae = $${3 \over 2}$$ (given)

$$ \Rightarrow $$ $$\,\,\,$$ a (1 $$-$$ e) = $${3 \over 2}$$

Length of latus rectum,

$${{2{b^2}} \over a} = 4$$

$$ \Rightarrow $$ $$\,\,\,$$ b

$$ \Rightarrow $$ $$\,\,\,$$ a

$$ \Rightarrow $$ $$\,\,\,$$ a (1 $$-$$ e) ( 1 + e) = 2

Putting a (1 $$-$$ e) = $${3 \over 2}$$

$$ \Rightarrow $$ $$\,\,\,$$ $${3 \over 2}$$ (1 + e) = 2

$$ \Rightarrow $$ $$\,\,\,$$ 3 + 3e = 4

$$ \Rightarrow $$ $$\,\,\,$$ e = $${1 \over 3}$$

3

Let P be a point on the parabola, x^{2} = 4y. If the distance of P from the center of the circle, x^{2} + y^{2} + 6x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P, is :

A

x + 4y $$-$$ 2 = 0

B

x $$-$$ y + 3 = 0

C

x + y +1 = 0

D

x + 2y = 0

Let P(2t, t^{2}) be any point on the parabola.

Center of the given circle C = ($$-$$ g, $$-$$f) = ($$-$$3, 0)

For PC to be minimum, it must be the normal to the parabola at P.

Slope of line PC = $${{{y_2} - {y_1}} \over {{x_2} - {x_1}}}$$ = $${{{t^2} - 0} \over {2t + 3}}$$

Also, slope of tangent to parabola at P = $${{dy} \over {dx}}$$ = $${x \over 2}$$ = t

$$ \therefore $$ Slope of normal = $${{ - 1} \over t}$$

$$ \therefore $$ $${{{t^2} - 0} \over {2t + 3}}$$ = $${{ - 1} \over t}$$

$$ \Rightarrow $$ t^{3} + 2t + 3 = 0

$$ \Rightarrow $$ (t+1) (t^{2} $$-$$ t + 3) = 0

$$\therefore\,\,\,$$ Real roots of above equation is

t = $$-$$ 1

Coordinate of P = (2t, t^{2}) = ($$-$$2, 1)

Slope of tangent to parabola at P = t = $$-$$ 1

Therefore, equation of tangent is :

(y $$-$$ 1) = ($$-$$ 1) (x + 2)

$$ \Rightarrow $$ x + y + 1 = 0

Center of the given circle C = ($$-$$ g, $$-$$f) = ($$-$$3, 0)

For PC to be minimum, it must be the normal to the parabola at P.

Slope of line PC = $${{{y_2} - {y_1}} \over {{x_2} - {x_1}}}$$ = $${{{t^2} - 0} \over {2t + 3}}$$

Also, slope of tangent to parabola at P = $${{dy} \over {dx}}$$ = $${x \over 2}$$ = t

$$ \therefore $$ Slope of normal = $${{ - 1} \over t}$$

$$ \therefore $$ $${{{t^2} - 0} \over {2t + 3}}$$ = $${{ - 1} \over t}$$

$$ \Rightarrow $$ t

$$ \Rightarrow $$ (t+1) (t

$$\therefore\,\,\,$$ Real roots of above equation is

t = $$-$$ 1

Coordinate of P = (2t, t

Slope of tangent to parabola at P = t = $$-$$ 1

Therefore, equation of tangent is :

(y $$-$$ 1) = ($$-$$ 1) (x + 2)

$$ \Rightarrow $$ x + y + 1 = 0

4

Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the
origin, on the positive x-axis then which of the following points does not lie on it ?

A

(5, 2$$\sqrt 6$$)

B

(6, 4$$\sqrt 2$$)

C

(8, 6)

D

(4, -4)

So the equation of the parabola,

$${\left( {y - 0} \right)^2} = 4.a\left( {x - 2} \right)$$

$$ \Rightarrow $$ y

$$ \Rightarrow $$ y

By checking each options you can see. point (8, 6) does not lie on the parabola.

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*