Joint Entrance Examination

Graduate Aptitude Test in Engineering

NEW

New Website Launch

Experience the best way to solve previous year questions with **mock
tests** (very detailed analysis), **bookmark your favourite questions**, **practice** etc...

1

MCQ (Single Correct Answer)

**Statement-2 :** If the line, $$y = mx + {{\sqrt 5 } \over m}\left( {m \ne 0} \right)$$ is their common tangent, then $$m$$ satiesfies $${m^4} - 3{m^2} + 2 = 0$$.

A

Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

B

Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

C

Statement-1 is true; Statement-2 is false.

D

Statement-1 is false Statement-2 is true.

Let common tangent be

$$y = mx + {{\sqrt 5 } \over m}$$

Since, perpendicular distance from center of the circle to

the common tangent is equal to radius of the circle,

therefore $${{{{\sqrt 5 } \over m}} \over {\sqrt {1 + {m^2}} }} = \sqrt {{5 \over 2}} $$

On squaring both the side, we get

$${m^2}\left( {1 + {m^2}} \right) = 2$$

$$ \Rightarrow {m^4} + {m^2} - 2 = 0$$

$$ \Rightarrow \left( {{m^2} + 2} \right)\left( {{m^2} - 1} \right) = 0$$

$$ \Rightarrow m = \pm 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ ( as $$m \ne \pm \sqrt 2 $$ )

$$y = \pm \left( {x + \sqrt 5 } \right),$$ both statements are correct as $$m = \pm 1$$

satisfies the given equation of statement - $$2.$$

$$y = mx + {{\sqrt 5 } \over m}$$

Since, perpendicular distance from center of the circle to

the common tangent is equal to radius of the circle,

therefore $${{{{\sqrt 5 } \over m}} \over {\sqrt {1 + {m^2}} }} = \sqrt {{5 \over 2}} $$

On squaring both the side, we get

$${m^2}\left( {1 + {m^2}} \right) = 2$$

$$ \Rightarrow {m^4} + {m^2} - 2 = 0$$

$$ \Rightarrow \left( {{m^2} + 2} \right)\left( {{m^2} - 1} \right) = 0$$

$$ \Rightarrow m = \pm 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ ( as $$m \ne \pm \sqrt 2 $$ )

$$y = \pm \left( {x + \sqrt 5 } \right),$$ both statements are correct as $$m = \pm 1$$

satisfies the given equation of statement - $$2.$$

2

MCQ (Single Correct Answer)

The equation of the circle passing through the foci of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$, and having centre at $$(0,3)$$ is

A

$${x^2} + {y^2} - 6y - 7 = 0$$

B

$${x^2} + {y^2} - 6y + 7 = 0$$

C

$${x^2} + {y^2} - 6y - 5 = 0$$

D

$${x^2} + {y^2} - 6y + 5 = 0$$

From the given equation of ellipse, we have

$$a = 4,b = 3,e = \sqrt {1 - {9 \over {16}}} $$

$$ \Rightarrow e = {{\sqrt 7 } \over 4}$$

Now, radius of this circle $$ = {a^2} = 16$$

$$ \Rightarrow Focii = \left( { \pm \sqrt 7 ,0} \right)$$

Now equation of circle is

$${\left( {x - 0} \right)^2} + {\left( {y - 3} \right)^2} = 16$$

$${x^2} + y{}^2 - 6y - 7 = 0$$

3

MCQ (Single Correct Answer)

An ellipse is drawn by taking a diameter of thec circle $${\left( {x - 1} \right)^2} + {y^2} = 1$$ as its semi-minor axis and a diameter of the circle $${x^2} + {\left( {y - 2} \right)^2} = 4$$ is semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is:

A

$$4{x^2} + {y^2} = 4$$

B

$${x^2} + 4{y^2} = 8$$

C

$$4{x^2} + {y^2} = 8$$

D

$${x^2} + 4{y^2} = 16$$

Equation of circle is $${\left( {x - 1} \right)^2} + {y^2} = 1$$

$$ \Rightarrow $$ radius $$=1$$ and diameter $$=2$$

$$\therefore$$ Length of semi-minor axis is $$2.$$

Equation of circle is $${x^2} + {\left( {y - 2} \right)^2} = 4 = {\left( 2 \right)^2}$$

$$ \Rightarrow $$ radius $$=2$$ and diameter $$=4$$

$$\therefore$$ Length of semi major axis is $$4$$

We know, equation of ellipse is given by

$${{{x^2}} \over {\left( {Major\,\,\,axi{s^{\,\,2}}} \right)}} + {{{y^2}} \over {\left( {Minor\,\,\,axi{s^{\,\,2}}} \right)}} = 1$$

$$ \Rightarrow {{{x^2}} \over {{{\left( 4 \right)}^2}}} + {{{y^2}} \over {{{\left( 2 \right)}^2}}} = 1$$

$$ \Rightarrow {{{x^2}} \over {16}} + {{{y^2}} \over 4} = 1$$

$$ \Rightarrow {x^2} + 4{y^2} = 16$$

$$ \Rightarrow $$ radius $$=1$$ and diameter $$=2$$

$$\therefore$$ Length of semi-minor axis is $$2.$$

Equation of circle is $${x^2} + {\left( {y - 2} \right)^2} = 4 = {\left( 2 \right)^2}$$

$$ \Rightarrow $$ radius $$=2$$ and diameter $$=4$$

$$\therefore$$ Length of semi major axis is $$4$$

We know, equation of ellipse is given by

$${{{x^2}} \over {\left( {Major\,\,\,axi{s^{\,\,2}}} \right)}} + {{{y^2}} \over {\left( {Minor\,\,\,axi{s^{\,\,2}}} \right)}} = 1$$

$$ \Rightarrow {{{x^2}} \over {{{\left( 4 \right)}^2}}} + {{{y^2}} \over {{{\left( 2 \right)}^2}}} = 1$$

$$ \Rightarrow {{{x^2}} \over {16}} + {{{y^2}} \over 4} = 1$$

$$ \Rightarrow {x^2} + 4{y^2} = 16$$

4

MCQ (Single Correct Answer)

**STATEMENT-2 :**If line $$y = mx + {{4\sqrt 3 } \over m},\left( {m \ne 0} \right)$$ is a common tangent to the parabola $${y^2} = 16\sqrt {3x} $$and the ellipse $$2{x^2} + {y^2} = 4$$, then $$m$$ satisfies $${m^4} + 2{m^2} = 24$$

A

Statement-1 is false, Statement-2 is true.

B

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

C

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

D

Statement-1 is true, Statement-2 is false.

Given equation of ellipse is $$2{x^2} + {y^2} = 4$$

$$ \Rightarrow {{2{x^2}} \over 4} + {{{y^2}} \over 4} = 1 \Rightarrow {{{x_2}} \over 2} + {{{y^2}} \over 4} = 1$$

Equation of tangent to the ellipse $${{{x^2}} \over 2} + {{{y^2}} \over 4} = 1$$ is

$$y = mx \pm \sqrt {2{m^2} + 4} \,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

( as equation of tangent to the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$

is $$y=mx+c$$ where $$c = \pm \sqrt {{a^2}{m^2} + {b^2}} $$ )

Now, Equation of tangent to the parabola

$${y^2} = 16\sqrt 3 x$$ is $$y = mx + {{4\sqrt 3 } \over m}\,\,\,\,\,\,\,\,...\left( 2 \right)$$

( as equation of tangent to the parabola

$${y^2} = 4ax$$ is $$y = mx + {a \over m}$$ )

On comparing $$(1)$$ and $$(2),$$ we get

$${{4\sqrt 3 } \over m} = \pm \sqrt {2{m^2} + 4} $$

Squaring on both the sides, we get

$$16\left( 3 \right) = \left( {2{m^2} + 4} \right){m^2}$$

$$ \Rightarrow 48 = {m^2}\left( {2{m^2} + 4} \right) \Rightarrow 2{m^4} + 4{m^2} - 48 = 0$$

$$ \Rightarrow {m^4} + 2{m^2} - 24 = 0 \Rightarrow \left( {{m^2} + 6} \right)\left( {{m^2} - 4} \right) = 0$$

$$ \Rightarrow {m^2} = 4$$ ( as $${m^2} \ne - 6$$ ) $$ \Rightarrow m = \pm 2$$

$$ \Rightarrow $$ Equation of common tangents are $$y = \pm 2x \pm 2\sqrt 3 $$

Thus, statement - $$1$$ is true.

Statement - $$2$$ is obviously true.

$$ \Rightarrow {{2{x^2}} \over 4} + {{{y^2}} \over 4} = 1 \Rightarrow {{{x_2}} \over 2} + {{{y^2}} \over 4} = 1$$

Equation of tangent to the ellipse $${{{x^2}} \over 2} + {{{y^2}} \over 4} = 1$$ is

$$y = mx \pm \sqrt {2{m^2} + 4} \,\,\,\,\,\,\,\,\,\,...\left( 1 \right)$$

( as equation of tangent to the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$

is $$y=mx+c$$ where $$c = \pm \sqrt {{a^2}{m^2} + {b^2}} $$ )

Now, Equation of tangent to the parabola

$${y^2} = 16\sqrt 3 x$$ is $$y = mx + {{4\sqrt 3 } \over m}\,\,\,\,\,\,\,\,...\left( 2 \right)$$

( as equation of tangent to the parabola

$${y^2} = 4ax$$ is $$y = mx + {a \over m}$$ )

On comparing $$(1)$$ and $$(2),$$ we get

$${{4\sqrt 3 } \over m} = \pm \sqrt {2{m^2} + 4} $$

Squaring on both the sides, we get

$$16\left( 3 \right) = \left( {2{m^2} + 4} \right){m^2}$$

$$ \Rightarrow 48 = {m^2}\left( {2{m^2} + 4} \right) \Rightarrow 2{m^4} + 4{m^2} - 48 = 0$$

$$ \Rightarrow {m^4} + 2{m^2} - 24 = 0 \Rightarrow \left( {{m^2} + 6} \right)\left( {{m^2} - 4} \right) = 0$$

$$ \Rightarrow {m^2} = 4$$ ( as $${m^2} \ne - 6$$ ) $$ \Rightarrow m = \pm 2$$

$$ \Rightarrow $$ Equation of common tangents are $$y = \pm 2x \pm 2\sqrt 3 $$

Thus, statement - $$1$$ is true.

Statement - $$2$$ is obviously true.

On those following papers in MCQ (Single Correct Answer)

Number in Brackets after Paper Indicates No. of Questions

JEE Main 2021 (Online) 1st September Evening Shift (2)

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

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

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

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

JEE Main 2021 (Online) 26th August Evening Shift (2)

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

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

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

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

JEE Main 2021 (Online) 22th July Evening Shift (2)

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

JEE Main 2021 (Online) 20th July Morning Shift (1)

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

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

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

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

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

JEE Main 2021 (Online) 25th February Morning Shift (1)

JEE Main 2021 (Online) 24th February Evening Shift (1)

JEE Main 2021 (Online) 24th February Morning Shift (1)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

JEE Main 2019 (Online) 10th April Evening Slot (3)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

JEE Main 2018 (Offline) (2)

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

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

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

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

JEE Main 2017 (Offline) (2)

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

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

JEE Main 2016 (Offline) (2)

JEE Main 2015 (Offline) (3)

JEE Main 2014 (Offline) (2)

JEE Main 2013 (Offline) (2)

AIEEE 2012 (2)

AIEEE 2011 (1)

AIEEE 2010 (1)

AIEEE 2009 (1)

AIEEE 2008 (2)

AIEEE 2007 (3)

AIEEE 2006 (3)

AIEEE 2005 (3)

AIEEE 2004 (2)

AIEEE 2003 (2)

AIEEE 2002 (1)

Complex Numbers

Quadratic Equation and Inequalities

Permutations and Combinations

Mathematical Induction and Binomial Theorem

Sequences and Series

Matrices and Determinants

Vector Algebra and 3D Geometry

Probability

Statistics

Mathematical Reasoning

Trigonometric Functions & Equations

Properties of Triangle

Inverse Trigonometric Functions

Straight Lines and Pair of Straight Lines

Circle

Conic Sections

Functions

Limits, Continuity and Differentiability

Differentiation

Application of Derivatives

Indefinite Integrals

Definite Integrals and Applications of Integrals

Differential Equations