Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (Single Correct Answer)

The equation $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$ has:

A

infinite number of real roots

B

no real roots

C

exactly one real root

D

exactly four real roots

Given equation is $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$

Put $${e^{{\mathop{\rm sinx}\nolimits} \,}} = t$$ in the given equation,

we get $${t^2} - 4t - 1 = 0$$

$$ \Rightarrow t = {{4 \pm \sqrt {16 + 4} } \over 2}$$

$$\,\,\,\,\,\,\,\,\,\,\, = {{4 \pm \sqrt {20} } \over 2}$$

$$\,\,\,\,\,\,\,\,\,\,\, = {{4 \pm 2\sqrt 5 } \over 2}$$

$$\,\,\,\,\,\,\,\,\,\,\, = 2 \pm \sqrt 5 $$

$$ \Rightarrow {e^{\sin x}} = 2 \pm \sqrt 5 $$ $$\,\,\,\,\,$$ (as $$t = {e^{\sin x}}$$)

$$ \Rightarrow {e^{\sin x}} = 2 - \sqrt 5 $$ and

$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $${e^{\sin x}} = 2 + \sqrt 5 $$

$$ \Rightarrow {e^{\sin x}} = 2 - \sqrt 5 < 0$$

and $$\,\,\,\,\,\,\sin x = \ln \left( {2 + \sqrt 5 } \right) > 1$$ So, rejected

Hence given equation has no solution.

$$\therefore$$ The equation has no real roots.

Put $${e^{{\mathop{\rm sinx}\nolimits} \,}} = t$$ in the given equation,

we get $${t^2} - 4t - 1 = 0$$

$$ \Rightarrow t = {{4 \pm \sqrt {16 + 4} } \over 2}$$

$$\,\,\,\,\,\,\,\,\,\,\, = {{4 \pm \sqrt {20} } \over 2}$$

$$\,\,\,\,\,\,\,\,\,\,\, = {{4 \pm 2\sqrt 5 } \over 2}$$

$$\,\,\,\,\,\,\,\,\,\,\, = 2 \pm \sqrt 5 $$

$$ \Rightarrow {e^{\sin x}} = 2 \pm \sqrt 5 $$ $$\,\,\,\,\,$$ (as $$t = {e^{\sin x}}$$)

$$ \Rightarrow {e^{\sin x}} = 2 - \sqrt 5 $$ and

$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $${e^{\sin x}} = 2 + \sqrt 5 $$

$$ \Rightarrow {e^{\sin x}} = 2 - \sqrt 5 < 0$$

and $$\,\,\,\,\,\,\sin x = \ln \left( {2 + \sqrt 5 } \right) > 1$$ So, rejected

Hence given equation has no solution.

$$\therefore$$ The equation has no real roots.

2

MCQ (Single Correct Answer)

If $$\alpha $$ and $$\beta $$ are the roots of the equation $${x^2} - x + 1 = 0,$$ then $${\alpha ^{2009}} + {\beta ^{2009}} = $$

A

$$\, - 1$$

B

$$\, 1$$

C

$$\, 2$$

D

$$\, - 2$$

$${x^2} - x + 1 = 0$$

$$ \Rightarrow x = {{1 \pm \sqrt {1 - 4} } \over 2}$$

$$x = {{1 \pm \sqrt 3 i} \over 2}$$

$$\alpha = {1 \over 2} + i{{\sqrt 3 } \over 2} = - {\omega ^2}$$

$$\beta = {1 \over 2} - {{i\sqrt 3 } \over 2} = - \omega $$

$${\alpha ^{2009}} + {\beta ^{2009}} = {\left( { - {\omega ^2}} \right)^{2009}} + {\left( { - \omega } \right)^{2009}}$$

$$ = - {\omega ^2} - \omega = 1$$

$$ \Rightarrow x = {{1 \pm \sqrt {1 - 4} } \over 2}$$

$$x = {{1 \pm \sqrt 3 i} \over 2}$$

$$\alpha = {1 \over 2} + i{{\sqrt 3 } \over 2} = - {\omega ^2}$$

$$\beta = {1 \over 2} - {{i\sqrt 3 } \over 2} = - \omega $$

$${\alpha ^{2009}} + {\beta ^{2009}} = {\left( { - {\omega ^2}} \right)^{2009}} + {\left( { - \omega } \right)^{2009}}$$

$$ = - {\omega ^2} - \omega = 1$$

3

MCQ (Single Correct Answer)

If the roots of the equation $$b{x^2} + cx + a = 0$$ imaginary, then for all real values of $$x$$, the expression $$3{b^2}{x^2} + 6bcx + 2{c^2}$$ is :

A

less than $$4ab$$

B

greater than $$-4ab$$

C

less than $$-4ab$$

D

greater than $$4ab$$

Given that roots of the equation

$$b{x^2} + cx + a = 0$$ are imaginary

$$\therefore$$ $${c^2} - 4ab < 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$

Let $$y = 3{b^2}{x^2} + 6bc\,x + 2{c^2}$$

$$ \Rightarrow 3{b^2}{x^2} + 6bc\,x + 2{c^2} - y = 0$$

As $$x$$ is real, $$D \ge 0$$

$$ \Rightarrow 36{b^2}{c^2} - 12{b^2}\left( {2{c^2} - y} \right) \ge 0$$

$$ \Rightarrow 12{b^2}\left( {3{c^2} - 2{c^2} + y} \right) \ge 0$$

$$ \Rightarrow {c^2} + y \ge 0$$

$$ \Rightarrow y \ge - {c^2}$$

But from eqn. $$(i),$$ $${c^2} < 4ab$$

or $$ - {c^2} > - 4ab$$

$$\therefore$$ we get $$y \ge - {c^2} > - 4ab$$

$$y > - 4ab$$

$$b{x^2} + cx + a = 0$$ are imaginary

$$\therefore$$ $${c^2} - 4ab < 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$

Let $$y = 3{b^2}{x^2} + 6bc\,x + 2{c^2}$$

$$ \Rightarrow 3{b^2}{x^2} + 6bc\,x + 2{c^2} - y = 0$$

As $$x$$ is real, $$D \ge 0$$

$$ \Rightarrow 36{b^2}{c^2} - 12{b^2}\left( {2{c^2} - y} \right) \ge 0$$

$$ \Rightarrow 12{b^2}\left( {3{c^2} - 2{c^2} + y} \right) \ge 0$$

$$ \Rightarrow {c^2} + y \ge 0$$

$$ \Rightarrow y \ge - {c^2}$$

But from eqn. $$(i),$$ $${c^2} < 4ab$$

or $$ - {c^2} > - 4ab$$

$$\therefore$$ we get $$y \ge - {c^2} > - 4ab$$

$$y > - 4ab$$

4

MCQ (Single Correct Answer)

If $$\,\left| {z - {4 \over z}} \right| = 2,$$ then the maximum value of $$\,\left| z \right|$$ is equal to :

A

$$\sqrt 5 + 1$$

B

2

C

$$2 + \sqrt 2 $$

D

$$\sqrt 3 + 1$$

Given that $$\left| {z - {4 \over z}} \right| = 2$$

Now $$\left| z \right| = \left| {z - {4 \over z} + {4 \over { - z}}} \right| \le \left| {z - {4 \over z}} \right| + {4 \over {\left| z \right|}}$$

$$ \Rightarrow \left| z \right| \le 2 + {4 \over {\left| z \right|}}$$

$$ \Rightarrow {\left| z \right|^2} - 2\left| z \right| - 4 \le 0$$

$$ \Rightarrow \left( {\left| z \right| - {{2 + \sqrt {20} } \over 2}} \right)\left( {\left| z \right| - {{2 - \sqrt {20} } \over 2}} \right) \le 0$$

$$\left( {\left| z \right| - \left( {1 + \sqrt 5 } \right)} \right)\left( {\left| z \right| - \left( {1 - \sqrt 5 } \right)} \right) \le 0$$

$$ \Rightarrow \left( { - \sqrt 5 + 1} \right) \le \left| z \right| \le \left( {\sqrt 5 + 1} \right)$$

$$ \Rightarrow {\left| z \right|_{\max }} = \sqrt 5 + 1$$

Now $$\left| z \right| = \left| {z - {4 \over z} + {4 \over { - z}}} \right| \le \left| {z - {4 \over z}} \right| + {4 \over {\left| z \right|}}$$

$$ \Rightarrow \left| z \right| \le 2 + {4 \over {\left| z \right|}}$$

$$ \Rightarrow {\left| z \right|^2} - 2\left| z \right| - 4 \le 0$$

$$ \Rightarrow \left( {\left| z \right| - {{2 + \sqrt {20} } \over 2}} \right)\left( {\left| z \right| - {{2 - \sqrt {20} } \over 2}} \right) \le 0$$

$$\left( {\left| z \right| - \left( {1 + \sqrt 5 } \right)} \right)\left( {\left| z \right| - \left( {1 - \sqrt 5 } \right)} \right) \le 0$$

$$ \Rightarrow \left( { - \sqrt 5 + 1} \right) \le \left| z \right| \le \left( {\sqrt 5 + 1} \right)$$

$$ \Rightarrow {\left| z \right|_{\max }} = \sqrt 5 + 1$$

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 (1)

JEE Main 2021 (Online) 31st August Evening Shift (1)

JEE Main 2021 (Online) 31st August Morning Shift (1)

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

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

JEE Main 2021 (Online) 27th July Morning Shift (1)

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

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

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

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

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

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

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

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

JEE Main 2020 (Online) 6th September Evening Slot (1)

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

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

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

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 (1)

JEE Main 2020 (Online) 3rd September Morning Slot (1)

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

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

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

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

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

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

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

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

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

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

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

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

JEE Main 2019 (Online) 8th April Evening Slot (1)

JEE Main 2019 (Online) 8th April Morning Slot (1)

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

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

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

JEE Main 2019 (Online) 11th January Morning Slot (1)

JEE Main 2019 (Online) 10th January Evening Slot (1)

JEE Main 2019 (Online) 10th January Morning Slot (1)

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

JEE Main 2018 (Online) 16th April Morning Slot (2)

JEE Main 2018 (Offline) (1)

JEE Main 2018 (Online) 15th April Evening Slot (1)

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

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

JEE Main 2017 (Online) 8th April Morning Slot (1)

JEE Main 2017 (Offline) (1)

JEE Main 2016 (Online) 10th April Morning Slot (1)

JEE Main 2016 (Online) 9th April Morning Slot (1)

JEE Main 2016 (Offline) (1)

JEE Main 2015 (Offline) (1)

JEE Main 2014 (Offline) (2)

JEE Main 2013 (Offline) (3)

AIEEE 2012 (1)

AIEEE 2010 (1)

AIEEE 2009 (3)

AIEEE 2008 (1)

AIEEE 2007 (1)

AIEEE 2006 (3)

AIEEE 2005 (4)

AIEEE 2004 (3)

AIEEE 2003 (4)

AIEEE 2002 (5)

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