Joint Entrance Examination

Graduate Aptitude Test in Engineering

1

MCQ (Single Correct Answer)

If $$z = x - iy$$ and $${z^{{1 \over 3}}} = p + iq$$, then

$${{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}}$$ is equal to

$${{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}}$$ is equal to

A

- 2

B

- 1

C

2

D

1

Given $${z^{{1 \over 3}}} = p + iq$$

$$ \Rightarrow $$ z = (p + iq)^{3}

= p^{3} + (iq)^{3} +3p(iq)(p + iq)

= p^{3} - iq^{3} +3ip^{2}q - 3pq^{2}

= p(p^{2} - 3q^{2}) - iq(q^{2} - 3p^{2})

Given that $$z = x - iy$$

$$\therefore$$ $$x - iy$$ = p(p^{2} - 3q^{2}) - iq(q^{2} - 3p^{2})

By comparing both sides we get,

$${x \over p} = {p^2} - 3{q^2}$$ and $${y \over q} = {q^2} - 3{p^2}$$

$$\therefore$$ $${{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}}$$

= $${{{p^2} - 3{q^2} + {q^2} - 3{p^2}} \over {{p^2} + {q^2}}}$$

= $${{ - 2{q^2} - 2{p^2}} \over {{p^2} + {q^2}}}$$

= $${{ - 2\left( {{q^2} + {p^2}} \right)} \over {{p^2} + {q^2}}}$$

= $$-2$$

$$ \Rightarrow $$ z = (p + iq)

= p

= p

= p(p

Given that $$z = x - iy$$

$$\therefore$$ $$x - iy$$ = p(p

By comparing both sides we get,

$${x \over p} = {p^2} - 3{q^2}$$ and $${y \over q} = {q^2} - 3{p^2}$$

$$\therefore$$ $${{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}}$$

= $${{{p^2} - 3{q^2} + {q^2} - 3{p^2}} \over {{p^2} + {q^2}}}$$

= $${{ - 2{q^2} - 2{p^2}} \over {{p^2} + {q^2}}}$$

= $${{ - 2\left( {{q^2} + {p^2}} \right)} \over {{p^2} + {q^2}}}$$

= $$-2$$

2

MCQ (Single Correct Answer)

Let z and w be complex numbers such that $$\overline z + i\overline w = 0$$ and arg zw = $$\pi $$. Then arg z equals

A

$${{5\pi } \over 4}$$

B

$${{\pi } \over 2}$$

C

$${{3\pi } \over 4}$$

D

$${{\pi } \over 4}$$

Given $$\overline z + i\overline w = 0$$

$$ \Rightarrow \overline z = - i\overline w $$

$$ \Rightarrow \overline{\overline z} = - \overline {i\overline w } $$

$$ \Rightarrow \overline{\overline z} = - \overline i \overline{\overline w} $$

$$ \Rightarrow z = - \overline i w$$

$$ \Rightarrow z = - \left( { - i} \right)w$$

$$ \Rightarrow z = iw$$

Now given that Arg(zw) = $$\pi $$

$$ \Rightarrow $$ Arg(z$$ \times $$$${z \over i}$$) = $$\pi $$

$$ \Rightarrow $$ Arg(z^{2}) - Arg(i) = $$\pi $$

$$ \Rightarrow $$ 2Arg(z) - $${\pi \over 2}$$ = $$\pi $$

[ $$i$$ complex number represent (0, 1) point on imaginary axis and Arg($$i$$) means the angle made by the point (0, 1) with real axis which is $${\pi \over 2}$$]

$$ \Rightarrow $$ 2Arg(z) = $${{3\pi } \over 2}$$

$$ \Rightarrow $$ Arg(z) = $${{3\pi } \over 4}$$

$$ \Rightarrow \overline z = - i\overline w $$

$$ \Rightarrow \overline{\overline z} = - \overline {i\overline w } $$

$$ \Rightarrow \overline{\overline z} = - \overline i \overline{\overline w} $$

$$ \Rightarrow z = - \overline i w$$

$$ \Rightarrow z = - \left( { - i} \right)w$$

$$ \Rightarrow z = iw$$

Now given that Arg(zw) = $$\pi $$

$$ \Rightarrow $$ Arg(z$$ \times $$$${z \over i}$$) = $$\pi $$

$$ \Rightarrow $$ Arg(z

$$ \Rightarrow $$ 2Arg(z) - $${\pi \over 2}$$ = $$\pi $$

[ $$i$$ complex number represent (0, 1) point on imaginary axis and Arg($$i$$) means the angle made by the point (0, 1) with real axis which is $${\pi \over 2}$$]

$$ \Rightarrow $$ 2Arg(z) = $${{3\pi } \over 2}$$

$$ \Rightarrow $$ Arg(z) = $${{3\pi } \over 4}$$

3

MCQ (Single Correct Answer)

If $${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$$ then

A

x = 2n + 1, where n is any positive integer

B

x = 4n , where n is any positive integer

C

x = 2n, where n is any positive integer

D

x = 4n + 1, where n is any positive integer.

$${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$$

$$ \Rightarrow $$ $${\left[ {{{\left( {1 + i} \right)\left( {1 + i} \right)} \over {\left( {1 - i} \right)\left( {1 + i} \right)}}} \right]^x} = 1$$

$$ \Rightarrow $$ $${\left[ {{{{{\left( {1 + i} \right)}^2}} \over {1 - {i^2}}}} \right]^x} = 1$$

$$ \Rightarrow $$ $${\left[ {{{1 + 2i + {i^2}} \over {1 + 1}}} \right]^x} = 1$$

$$ \Rightarrow $$ $${\left[ {{{1 + 2i - 1} \over 2}} \right]^x} = 1$$

$$ \Rightarrow {\left( i \right)^x} = 1$$

We know $$i = \sqrt { - 1} $$

$$\therefore$$ $${i^2} = - 1$$

$$ \Rightarrow $$ $${i^3} = - 1 \times i = - i$$

$$ \Rightarrow $$ $${i^4} = - i \times i = - {i^2} = - \left( { - 1} \right) = 1$$

So when power of $$i$$ is 4 or multiple of 4 then it's value is = 1

$$\therefore$$ $${\left( i \right)^x} = 1$$ $$ = {\left( i \right)^{4n}}$$ where n is a positive integer.

$$ \Rightarrow $$ $${\left[ {{{\left( {1 + i} \right)\left( {1 + i} \right)} \over {\left( {1 - i} \right)\left( {1 + i} \right)}}} \right]^x} = 1$$

$$ \Rightarrow $$ $${\left[ {{{{{\left( {1 + i} \right)}^2}} \over {1 - {i^2}}}} \right]^x} = 1$$

$$ \Rightarrow $$ $${\left[ {{{1 + 2i + {i^2}} \over {1 + 1}}} \right]^x} = 1$$

$$ \Rightarrow $$ $${\left[ {{{1 + 2i - 1} \over 2}} \right]^x} = 1$$

$$ \Rightarrow {\left( i \right)^x} = 1$$

We know $$i = \sqrt { - 1} $$

$$\therefore$$ $${i^2} = - 1$$

$$ \Rightarrow $$ $${i^3} = - 1 \times i = - i$$

$$ \Rightarrow $$ $${i^4} = - i \times i = - {i^2} = - \left( { - 1} \right) = 1$$

So when power of $$i$$ is 4 or multiple of 4 then it's value is = 1

$$\therefore$$ $${\left( i \right)^x} = 1$$ $$ = {\left( i \right)^{4n}}$$ where n is a positive integer.

4

MCQ (Single Correct Answer)

Let $${Z_1}$$ and $${Z_2}$$ be two roots of the equation $${Z^2} + aZ + b = 0$$, Z being complex. Further , assume that the origin, $${Z_1}$$ and $${Z_2}$$ form an equilateral triangle. Then

A

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

B

$${a^2} = b$$

C

$${a^2} = 2b$$

D

$${a^2} = 3b$$

Given quadratic equation,

$${Z^2} + aZ + b = 0$$

and two roots are $${Z_1}$$ and $${Z_2}$$.

$$\therefore$$ $${Z_1}$$ + $${Z_2}$$ = $$-a$$ and $${Z_1}$$$${Z_2}$$ = $$b$$

Question says,

There are three complex numbers:

1. Origin (0)

2. $${Z_1}$$

3. $${Z_2}$$

and they form an equilateral triangle. So They are the vertices of the triangle.

[**Important Point :** If $${Z_1}$$, $${Z_2}$$ and $${Z_3}$$ are the vertices of an equilateral triangle then -

$$Z_1^2$$ + $$Z_2^2$$ + $$Z_3^2$$ = $${Z_1}{Z_2}$$ + $${Z_2}{Z_3}$$ + $${Z_3}{Z_1}$$ ]

In this question,

$${Z_1}$$ = 0, $${Z_2}$$ = $${Z_1}$$ and $${Z_3}$$ = $${Z_2}$$

By putting those values in the equation we get,

$${0^2}$$ + $$Z_1^2$$ + $$Z_2^2$$ = $$0$$ + $${Z_1}{Z_2}$$ + 0

$$ \Rightarrow $$ $$Z_1^2$$ + $$Z_2^2$$ = $${Z_1}{Z_2}$$

$$ \Rightarrow $$ $$Z_1^2$$ + $$Z_2^2$$ = $$b$$ [ as $${Z_1}$$$${Z_2}$$ = $$b$$ ]

$$ \Rightarrow $$ $${\left( {{Z_1} + {Z_2}} \right)^2}$$ - $$2{Z_1}{Z_2}$$ = $$b$$

$$ \Rightarrow $$ $${\left( {{Z_1} + {Z_2}} \right)^2}$$ - $$2b$$ = $$b$$

$$ \Rightarrow $$ $${\left( {{Z_1} + {Z_2}} \right)^2}$$ = $$3b$$

$$ \Rightarrow $$ $${\left( { - a} \right)^2}$$ = $$3b$$

$$ \Rightarrow $$ $${a^2}$$ = $$3b$$

So Option (D) is correct.

[**Note :** This question is asked to check if you know the following formula -

"If $${Z_1}$$, $${Z_2}$$ and $${Z_3}$$ are the vertices of an equilateral triangle then -

$$Z_1^2$$ + $$Z_2^2$$ + $$Z_3^2$$ = $${Z_1}{Z_2}$$ + $${Z_2}{Z_3}$$ + $${Z_3}{Z_1}$$" ]

$${Z^2} + aZ + b = 0$$

and two roots are $${Z_1}$$ and $${Z_2}$$.

$$\therefore$$ $${Z_1}$$ + $${Z_2}$$ = $$-a$$ and $${Z_1}$$$${Z_2}$$ = $$b$$

Question says,

There are three complex numbers:

1. Origin (0)

2. $${Z_1}$$

3. $${Z_2}$$

and they form an equilateral triangle. So They are the vertices of the triangle.

[

$$Z_1^2$$ + $$Z_2^2$$ + $$Z_3^2$$ = $${Z_1}{Z_2}$$ + $${Z_2}{Z_3}$$ + $${Z_3}{Z_1}$$ ]

In this question,

$${Z_1}$$ = 0, $${Z_2}$$ = $${Z_1}$$ and $${Z_3}$$ = $${Z_2}$$

By putting those values in the equation we get,

$${0^2}$$ + $$Z_1^2$$ + $$Z_2^2$$ = $$0$$ + $${Z_1}{Z_2}$$ + 0

$$ \Rightarrow $$ $$Z_1^2$$ + $$Z_2^2$$ = $${Z_1}{Z_2}$$

$$ \Rightarrow $$ $$Z_1^2$$ + $$Z_2^2$$ = $$b$$ [ as $${Z_1}$$$${Z_2}$$ = $$b$$ ]

$$ \Rightarrow $$ $${\left( {{Z_1} + {Z_2}} \right)^2}$$ - $$2{Z_1}{Z_2}$$ = $$b$$

$$ \Rightarrow $$ $${\left( {{Z_1} + {Z_2}} \right)^2}$$ - $$2b$$ = $$b$$

$$ \Rightarrow $$ $${\left( {{Z_1} + {Z_2}} \right)^2}$$ = $$3b$$

$$ \Rightarrow $$ $${\left( { - a} \right)^2}$$ = $$3b$$

$$ \Rightarrow $$ $${a^2}$$ = $$3b$$

So Option (D) is correct.

[

"If $${Z_1}$$, $${Z_2}$$ and $${Z_3}$$ are the vertices of an equilateral triangle then -

$$Z_1^2$$ + $$Z_2^2$$ + $$Z_3^2$$ = $${Z_1}{Z_2}$$ + $${Z_2}{Z_3}$$ + $${Z_3}{Z_1}$$" ]

On those following papers in MCQ (Single Correct Answer)

Number in Brackets after Paper Indicates No. of Questions

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

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

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

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

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

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

JEE Main 2021 (Online) 22th July Evening Shift (1)

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

JEE Main 2021 (Online) 18th March Evening Shift (1)

JEE Main 2021 (Online) 18th March Morning Shift (1)

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

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

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

JEE Main 2021 (Online) 16th March Morning Shift (1)

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

JEE Main 2021 (Online) 25th 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 (1)

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

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) 12th April Morning Slot (1)

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

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

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 (1)

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

JEE Main 2018 (Online) 16th April Morning Slot (1)

JEE Main 2018 (Offline) (1)

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

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

JEE Main 2017 (Offline) (1)

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

JEE Main 2016 (Offline) (1)

JEE Main 2015 (Offline) (1)

JEE Main 2014 (Offline) (1)

JEE Main 2013 (Offline) (1)

AIEEE 2012 (1)

AIEEE 2011 (2)

AIEEE 2010 (1)

AIEEE 2008 (2)

AIEEE 2007 (1)

AIEEE 2006 (2)

AIEEE 2005 (3)

AIEEE 2004 (3)

AIEEE 2003 (3)

AIEEE 2002 (3)

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