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 sides of a rhombus ABCD are parallel to the lines, x $$-$$ y + 2 = 0 and 7x $$-$$ y + 3 = 0. If the diagonals of the rhombus intersect P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the coordinate of A is :

A

$${5 \over 2}$$

B

$${7 \over 4}$$

C

2

D

$${7 \over 2}$$

Let the coordinate A be (0, c)

Equations of the given lines are

x $$-$$ y + 2 = 0 and 7x $$-$$ y + 3 = 0

We know that the diagonals of the rhombus will be parallel to the angle bisectors of the two given lines; y = x + 2 and y = 7x + 3

$$\therefore\,\,\,$$ equation of angle bisectors is given as :

$${{x - y + 2} \over {\sqrt 2 }} = \pm {{7x - y + 3} \over {5\sqrt 2 }}$$

5x $$-$$ 5y + 10 = $$ \pm $$ (7x $$-$$ y + 3)

$$\therefore\,\,\,$$ Parallel equations of the diagonals are 2x + 4y $$-$$ 7 = 0

and 12x $$-$$ 6y + 13 = 0

$$\therefore\,\,\,$$ slopes of diagonals are $${{ - 1} \over 2}$$ and 2.

Now, slope of the diagonal from A(0, c) and passing through P(1, 2) is (2 $$-$$ c)

$$\therefore\,\,\,$$ 2 $$-$$ c = 2 $$ \Rightarrow $$ c = 0 (not possible)

$$ \therefore $$$$\,\,\,$$ 2 $$-$$ c = $${{ - 1} \over 2}$$ $$ \Rightarrow $$ c = $${5 \over 2}$$

$$\therefore\,\,\,$$ Coordinate of A is $${5 \over 2}$$.

Equations of the given lines are

x $$-$$ y + 2 = 0 and 7x $$-$$ y + 3 = 0

We know that the diagonals of the rhombus will be parallel to the angle bisectors of the two given lines; y = x + 2 and y = 7x + 3

$$\therefore\,\,\,$$ equation of angle bisectors is given as :

$${{x - y + 2} \over {\sqrt 2 }} = \pm {{7x - y + 3} \over {5\sqrt 2 }}$$

5x $$-$$ 5y + 10 = $$ \pm $$ (7x $$-$$ y + 3)

$$\therefore\,\,\,$$ Parallel equations of the diagonals are 2x + 4y $$-$$ 7 = 0

and 12x $$-$$ 6y + 13 = 0

$$\therefore\,\,\,$$ slopes of diagonals are $${{ - 1} \over 2}$$ and 2.

Now, slope of the diagonal from A(0, c) and passing through P(1, 2) is (2 $$-$$ c)

$$\therefore\,\,\,$$ 2 $$-$$ c = 2 $$ \Rightarrow $$ c = 0 (not possible)

$$ \therefore $$$$\,\,\,$$ 2 $$-$$ c = $${{ - 1} \over 2}$$ $$ \Rightarrow $$ c = $${5 \over 2}$$

$$\therefore\,\,\,$$ Coordinate of A is $${5 \over 2}$$.

2

The foot of the perpendicular drawn from the origin, on the line, 3x + y = $$\lambda $$ ($$\lambda $$ $$ \ne $$ 0) is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is :

A

1 : 3

B

3 : 1

C

1 : 9

D

9 : 1

Equation of the line, which is perpendicular to the line,

3x + y = $$\lambda $$($$\lambda $$ $$ \ne $$0) and passing through origin ,

is given by $${{x - 0} \over 3} = {{y - 0} \over 1} = r$$

For foot of perpendicular

r = $${{ - \left( {\left( {3 \times 0} \right) + \left( {1 \times 0} \right) - \lambda } \right)} \over {{3^2} + {1^2}}}$$ = $${\lambda \over {10}}$$

So, foot of perpendicular P = $$\left( {{{3\lambda } \over {10}},{\lambda \over {10}}} \right)$$

Given the line meets X-axis where y = 0, so 3x + 0 = $$\lambda $$

$$ \Rightarrow $$ x = $${\lambda \over 3}$$

Hence, coordinates of A = $$\left( {{\lambda \over 3},0} \right)$$ and meets

Y-axis at B = (0, $$\lambda $$)

So, BP = $$\sqrt {{{\left( {{{3\lambda } \over {10}}} \right)}^2} + {{\left( {{\lambda \over {10}} - \lambda } \right)}^2}} $$

$$ \Rightarrow $$ BP = $$\sqrt {{{9{\lambda ^2}} \over {100}} + {{81{\lambda ^2}} \over {100}}} $$

= BP = $$\sqrt {{{90{\lambda ^2}} \over {100}}} $$

Now, PA = $$\sqrt {{{\left( {{\lambda \over 3} - {{3\lambda } \over {10}}} \right)}^2} + {{\left( {0 - {\lambda \over {10}}} \right)}^2}} $$

$$ \Rightarrow $$$$\,\,\,$$ PA = $$\sqrt {{{{\lambda ^2}} \over {900}} + {{{\lambda ^2}} \over {100}}} \Rightarrow PA$$ = $$\sqrt {{{10{\lambda ^2}} \over {900}}} $$

Therefore BP : PA = 9 : 1

3x + y = $$\lambda $$($$\lambda $$ $$ \ne $$0) and passing through origin ,

is given by $${{x - 0} \over 3} = {{y - 0} \over 1} = r$$

For foot of perpendicular

r = $${{ - \left( {\left( {3 \times 0} \right) + \left( {1 \times 0} \right) - \lambda } \right)} \over {{3^2} + {1^2}}}$$ = $${\lambda \over {10}}$$

So, foot of perpendicular P = $$\left( {{{3\lambda } \over {10}},{\lambda \over {10}}} \right)$$

Given the line meets X-axis where y = 0, so 3x + 0 = $$\lambda $$

$$ \Rightarrow $$ x = $${\lambda \over 3}$$

Hence, coordinates of A = $$\left( {{\lambda \over 3},0} \right)$$ and meets

Y-axis at B = (0, $$\lambda $$)

So, BP = $$\sqrt {{{\left( {{{3\lambda } \over {10}}} \right)}^2} + {{\left( {{\lambda \over {10}} - \lambda } \right)}^2}} $$

$$ \Rightarrow $$ BP = $$\sqrt {{{9{\lambda ^2}} \over {100}} + {{81{\lambda ^2}} \over {100}}} $$

= BP = $$\sqrt {{{90{\lambda ^2}} \over {100}}} $$

Now, PA = $$\sqrt {{{\left( {{\lambda \over 3} - {{3\lambda } \over {10}}} \right)}^2} + {{\left( {0 - {\lambda \over {10}}} \right)}^2}} $$

$$ \Rightarrow $$$$\,\,\,$$ PA = $$\sqrt {{{{\lambda ^2}} \over {900}} + {{{\lambda ^2}} \over {100}}} \Rightarrow PA$$ = $$\sqrt {{{10{\lambda ^2}} \over {900}}} $$

Therefore BP : PA = 9 : 1

3

Consider the set of all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following statements
is true ?

A

The lines are not concurrent

B

The lines are concurrent at the point $$\left( {{3 \over 4},{1 \over 2}} \right)$$

C

The lines are all parallel

D

Each line passes through the origin

Equation of lines;

px + qy + r = 0 . . . . . (1)

Also given

3p + 2q + 4r = 0 . . . . . . (2)

divide equation (2) by 4, we get

$${3 \over 4}P + {2 \over 4}q + r = 0$$ . . . . (3)

By comparing (1) and (3) we get,

x = $${3 \over 4}$$ and y = $${2 \over 4}$$ = $${1 \over 2}$$

For any value of p,q and r, the equation of set of lines will pan through $$\left( {{3 \over 4},{1 \over 2}} \right)$$

px + qy + r = 0 . . . . . (1)

Also given

3p + 2q + 4r = 0 . . . . . . (2)

divide equation (2) by 4, we get

$${3 \over 4}P + {2 \over 4}q + r = 0$$ . . . . (3)

By comparing (1) and (3) we get,

x = $${3 \over 4}$$ and y = $${2 \over 4}$$ = $${1 \over 2}$$

For any value of p,q and r, the equation of set of lines will pan through $$\left( {{3 \over 4},{1 \over 2}} \right)$$

4

Let the equations of two sides of a triangle be 3x $$-$$ 2y + 6 = 0 and 4x + 5y $$-$$ 20 = 0. If the orthocentre of this triangle is at (1, 1), then the equation of its third side is :

A

122y $$-$$ 26x $$-$$ 1675 = 0

B

122y + 26x + 1675 = 0

C

26x + 61y + 1675 = 0

D

26x $$-$$ 122y $$-$$ 1675 = 0

4x + 5y $$-$$ 20 = 0 . . .(1)

3x $$-$$ 2y + 6 = 0 . . . (2)

orthocentre is (1, 1)

line perpendicular to 4x + 5y $$-$$ 20 = 0

and passes through (1, 1) is

(y $$-$$ 1) = $${5 \over 4}$$(x $$-$$ 1)

$$ \Rightarrow $$ 5x $$-$$ 4y = 1 . . .(3)

and line $$ \bot $$ to 3x $$-$$ 2y + 6 = 0

and passes through (1, 1)

y $$-$$ 1 = $$-$$ $${2 \over 3}$$ (x $$-$$ 1)

$$ \Rightarrow $$ 2x + 3y = 5 . . .(4)

Solving (1) and (4) we get C$$\left( {{{35} \over 2}, - 10} \right)$$

Solving (2) and (3) we get A $$\left( { - 13,{{ - 33} \over 2}} \right)$$

Side BC is y + 10 = $${{{{ - 33} \over 2} + 10} \over { - 13 - {{35} \over 2}}}\left( {x - {{35} \over 2}} \right)$$

$$ \Rightarrow $$ y + 10 = $${{13} \over {61}}\left( {x - {{35} \over 2}} \right)$$

$$ \Rightarrow $$ 26x $$-$$ 122y $$-$$ 1675 = 0

Number in Brackets after Paper Name Indicates No of Questions

AIEEE 2002 (4) *keyboard_arrow_right*

AIEEE 2003 (5) *keyboard_arrow_right*

AIEEE 2004 (4) *keyboard_arrow_right*

AIEEE 2005 (2) *keyboard_arrow_right*

AIEEE 2006 (2) *keyboard_arrow_right*

AIEEE 2007 (3) *keyboard_arrow_right*

AIEEE 2008 (1) *keyboard_arrow_right*

AIEEE 2009 (3) *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) (2) *keyboard_arrow_right*

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

JEE Main 2015 (Offline) (1) *keyboard_arrow_right*

JEE Main 2016 (Offline) (1) *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) (1) *keyboard_arrow_right*

JEE Main 2017 (Online) 9th April Morning Slot (1) *keyboard_arrow_right*

JEE Main 2018 (Offline) (1) *keyboard_arrow_right*

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

JEE Main 2018 (Online) 15th April Evening Slot (2) *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 (3) *keyboard_arrow_right*

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

JEE Main 2020 (Online) 3rd September Evening Slot (1) *keyboard_arrow_right*

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

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

JEE Main 2020 (Online) 6th September Morning Slot (1) *keyboard_arrow_right*

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

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

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

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

JEE Main 2021 (Online) 17th March Morning Shift (1) *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) 27th July Evening Shift (2) *keyboard_arrow_right*

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

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

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

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

JEE Main 2021 (Online) 31st August Morning 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*