Complex Numbers And Quadratic Equations

01

PYQ 2013

easy

mathematics ID: jee-main

Let where is any non-zero complex number. The set is equal to:

1

2

3

4

02

PYQ 2013

medium

mathematics ID: jee-main

If is a complex number of unit modulus and argument then is equal to

1

2

3

4

03

PYQ 2013

medium

mathematics ID: jee-main

If the equations and have a common root, then is

1

1:02:03

2

3:02:01

3

1:03:02

4

3:01:02

04

PYQ 2014

medium

mathematics ID: jee-main

If and are the roots of the equation,, then the equation, = 0 has roots :

1

and

2

and

3

and

4

and

05

PYQ 2014

medium

mathematics ID: jee-main

Let be any complex number such that is a purely imaginary number. Then is :

1

0

2

any non-zero real number other than 1

3

any non-zero real number

4

a purely imaginary number

06

PYQ 2014

medium

mathematics ID: jee-main

For all complex numbers z of the form 1 + i, R, if z = x + iy, then :

1

2

3

4

07

PYQ 2015

medium

mathematics ID: jee-main

A complex number is the said to be unimodular if Suppose and are complex number such that is unimodular and is not unimodular. Then the point lies on a :

1

Straight line parallel to x-axis

2

Straight line parallel to y-axis

3

Circle of radius 2

4

Circle of radius

08

PYQ 2016

medium

mathematics ID: jee-main

The sum of all real values of satisfying the equation is

1

3

2

-4

3

6

4

5

09

PYQ 2016

medium

mathematics ID: jee-main

Let be a complex number,, such that is a real number. Then the sum is equal to :

1

2

3

4

10

PYQ 2017

medium

mathematics ID: jee-main

If, for a positive integer n, the quadratic equation, has two consecutive integral solutions, then is equal to :

1

9

2

10

3

11

4

12

11

PYQ 2017

medium

mathematics ID: jee-main

Let, the set of complex numbers. Then the equation, represents :

1

a circle with radius

2

a circle with diameter

3

an ellipse with length of major axis

4

an ellipse with length of minor axis

12

PYQ 2018

medium

mathematics ID: jee-main

If and are the roots of the quadratic equation,, then the value of is :

1

-10

2

10

3

-25

4

25

13

PYQ 2019

medium

mathematics ID: jee-main

Let and be any two non-zero complex numbers such that . If then :

1

2

3

4

14

PYQ 2019

medium

mathematics ID: jee-main

Let be a complex number such that (where i =). Then is equal to :

1

2

3

4

15

PYQ 2019

medium

mathematics ID: jee-main

Let with Im(z) = 10 and it satisfies for some natural number .Then :

1

n = 20 and Re(z) = - 10

2

n = 20 and Re(z) = 10

3

n = 40 and Re(z) = - 10

4

n = 40 and Re(z) = 10

16

PYQ 2019

medium

mathematics ID: jee-main

If is chosen in the quadratic equation such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :

1

2

3

4

17

PYQ 2019

medium

mathematics ID: jee-main

If is a purely imaginary number and, then a value of is :

1

2

3

4

18

PYQ 2019

medium

mathematics ID: jee-main

Let be a root of the quadratic equation, . If, then arg is equal to :

1

2

3

0

4

19

PYQ 2020

medium

mathematics ID: jee-main

If be a complex number satisfying then cannot be :

1

2

3

4

20

PYQ 2021

medium

mathematics ID: jee-main

If, then p and q are roots of the equation :

1

2

3

4

21

PYQ 2021

medium

mathematics ID: jee-main

Let a complex number be w = 1 - \sqrt3 i. Let another complex number z be such that |z w| = 1 and arg(z) - arg(w) = π/2. Then the area of the triangle with vertices origin, z and w is equal to :

1

1/2

2

2

3

1/4

4

4

22

PYQ 2021

medium

mathematics ID: jee-main

Let C be the set of all complex numbers. Let and . Then, the maximum value of for is equal to :

1

2

3

4

23

PYQ 2021

medium

mathematics ID: jee-main

Let and be two complex numbers such that and satisfy the equation . Then the imaginary part of is equal to _________.

24

PYQ 2021

medium

mathematics ID: jee-main

Let be in R. If and are the roots of the equation, and and are the roots of the equation, then is equal to ___________.

25

PYQ 2021

medium

mathematics ID: jee-main

The least positive integer n such that, is a positive integer, is ___________.

26

PYQ 2021

medium

mathematics ID: jee-main

If, then :

1

S contains only one element

2

S contains exactly two elements

3

S is a straight line in the complex plane

4

S is a circle in the complex plane

27

PYQ 2021

medium

mathematics ID: jee-main

A point moves in the complex plane such that, then the minimum value of is equal to _________

28

PYQ 2021

medium

mathematics ID: jee-main

If is a complex number such that is purely imaginary, then the minimum value of is :

1

2

3

4

29

PYQ 2021

medium

mathematics ID: jee-main

If the real part of the complex number, is zero, then the value of is equal to ________.

30

PYQ 2021

medium

mathematics ID: jee-main

Let i = \sqrt-1. If, and . Then is equal to __________.

31

PYQ 2021

medium

mathematics ID: jee-main

If are such that 1 2i (here = 1) is a root of z + z+ =0, then () is equal to:

1

3

2

-3

3

7

4

-7

32

PYQ 2021

medium

mathematics ID: jee-main

If the equation a|z|² + āz + a z̄ + d = 0 represents a circle where a, d are real constants, then which of the following condition is correct ?

1

|a|² - a d \geq 0 and a \in R

2

|a|² - a d>0 and a \in R - {0}

3

|a|² - a d \neq 0

4

α = 0, a, d \in R⁺

33

PYQ 2021

medium

mathematics ID: jee-main

Let z₁, z₂ be the roots of the equation z² + a z + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, the value of |a| is ________.

34

PYQ 2021

medium

mathematics ID: jee-main

Let C be the set of all complex numbers. Let, and . Then the number of elements in is equal to :

1

0

2

1

3

2

4

Infinite

35

PYQ 2021

medium

mathematics ID: jee-main

If the least and the largest real values of, for which the equation (and) has a solution, are and respectively; then is equal to _________

36

PYQ 2022

medium

mathematics ID: jee-main

The number of distinct real roots of the equation is

1

5

2

7

3

1

4

3

37

PYQ 2022

medium

mathematics ID: jee-main

Let a circle C in complex plane pass through the points, and . If is a point on C such that the line through and is perpendicular to the line through and, then is equal to:

1

2

3

4

38

PYQ 2023

medium

mathematics ID: jee-main

For if the minimum value of is, then a value of is ______

1

3

2

3

4

4

39

PYQ 2023

easy

mathematics ID: jee-main

Let and Then and are roots of the equation

1

2

3

4

40

PYQ 2023

easy

mathematics ID: jee-main

If the ellipse meets the line on the -axis and the line on the -axis, then the eccentricity of the ellipse is

1

2

3

4

41

PYQ 2023

easy

mathematics ID: jee-main

The remainder when is divided by is

1

1

2

4

3

6

4

8

42

PYQ 2023

hard

mathematics ID: jee-main

If the shortest between the lines and is, then the square of sum of all possible values of is

43

PYQ 2023

hard

mathematics ID: jee-main

If and x, y, z are non-negative integers, then find the number of integral solutions is

1

136

2

171

3

90

4

130

44

PYQ 2023

hard

mathematics ID: jee-main

Let be the roots of the equation, then is equal to:

1

-256

2

-128

3

4

45

PYQ 2023

hard

mathematics ID: jee-main

Let the complex number be such that is purely imaginary. If, then is equal to

1

2

3

4

46

PYQ 2023

medium

mathematics ID: jee-main

For two non-zero complex numbers z 1 and z 2, if Re(z 1 z 2) = 0 and Re(z 1 + z 2), then which of the following are possible? A. Im(z 1) > 0 and Im(z 2) > 0 | B. Im(z 1) < 0 and Im(z 2) > 0 C. Im(z 1) > 0 and Im(z 2) < 0 D. Im(z 1) < 0 and Im(z 2) < 0 Choose the correct answer from the options given below

1

B and C

2

B and D

3

A and B

4

A and C

47

PYQ 2024

hard

mathematics ID: jee-main

The minimum value of is,

1

2

3

4

48

PYQ 2026

hard

mathematics ID: jee-main

The number of distinct real solutions of the equation is

1

2

2

0

3

3

4

1

49

PYQ 2026

medium

mathematics ID: jee-main

Let, where . If, then is equal to

50

PYQ 2026

hard

mathematics ID: jee-main

Let and . Find the value of .

51

PYQ 2026

hard

mathematics ID: jee-main

Let and, where . If then the value of is ___________.

52

PYQ 2026

medium

mathematics ID: jee-main

If where & then value of is ________

53

PYQ 2026

medium

mathematics ID: jee-main

If, then number of ordered pairs, where is ______.

High-Yield Trend

Chapter Questions 53 MCQs

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Step 1: Representing complex numbers.

Step 2: Using the given conditions.

Step 3: Analyzing the conditions.

Step 4: Conclusion.

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Chapter Questions
53 MCQs