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# Maths Complex Numbers and Quadratic Equations Chapter - Maths JEE Main

Last updated date: 09th Aug 2024
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## Concepts of Maths Complex Numbers and Quadratic Equations for JEE Main Maths

Complex Numbers and Quadratic Equations is a fascinating and essential topic in mathematics. Every year, at least 1 to 3 problems from this chapter appear in IIT JEE and other exams. The concept of this chapter will be used in many other chapters, such as functions and coordinate geometry.

Begin by grasping fundamental ideas such as the definition of a complex number, integral powers of iota, and various representations of a complex number. Then go on to complex number algebra. The Argand plane, modulus, and argument of a complex number, as well as the triangle, are all fundamental concepts. Go over the concepts of solved problems again, and then do the same with the quadratic equation. Refer to the complex numbers and quadratic equations Class 11 solutions provided by Vedantu and download the complex numbers and quadratic equations Class 11 PDF to prepare for the exams.

JEE Main Maths Chapters 2024

 JEE Main Maths Chapters 1 Sets, Relations and Functions 9 Integral Calculus 2 Complex Numbers and Quadratic Equations 10 Differential Equations 3 Matrices and Determinants 11 Co-ordinate Geometry 4 Permutations and Combinations 12 Three Dimensional Geometry 5 Mathematical Induction 13 Vector Algebra 6 Binomial Theorem 14 Statistics and Probability 7 Sequence and Series 15 Trigonometry 8 Limit, Continuity and Differentiability 16 Mathematical Reasoning

### Important Topics of Maths Complex Numbers and Quadratic Equations Chapter

• Algebra of complex numbers

• Properties of complex numbers

• Modulus and conjugate of a complex numbers

• Argument of complex number

• Polar form of complex numbers

• Euler's Formula and De Moiver’s Theorem

• Geometry of Complex Numbers

• Cube root of unity

• Vector representation and rotation of complex numbers

• Nature of roots (in quadratic equations), the relation of coefficient and roots

• Transformation of quadratic equations and condition of common roots

• The discriminant of quadratic equations

### Complex Number

A complex number is one that can be written as p + iq, where p and q are real values and i represents a solution to the x2-1 equation.

$\sqrt{i}=-1$ or, i2 = -1.

Some of the examples of complex numbers are: 8 – 2i, 2 +31i, etc, and the Complex numbers are denoted by ‘z’.

The general form of the complex number is z=p+iq

Here,

• p is known as the real part and is denoted by Re z.

• q is known as the imaginary part and it is denoted by Im z.

• If z = 12 + 35i, then the value Re z = 12 and Im z = 35. If z1 and z2 are two complex numbers such that z1 = p + iq and z2 = r + is , z1 and z2 are equal if p = r and q = s.

### Algebra of Complex Numbers

• (a₁ + ib₁) + (a₂ + ib₂) = (a₁ + a₂) + i(b₁ + b₂).

Subtraction of Complex Numbers

• (a₁ + ib₁) - (a₂ + ib₂) = (a₁ - a₂) + i(b₁ - b₂).

Product of Complex Numbers

• (a₁ + ib₁) (a₂ + ib₂) = (a₁a₂ - b₁b₂) + i(a₁b₂ + b₁a₂).

Division of Complex Numbers

• (a₁ + ib₁)(a₂ + ib₂) = (a₁ + ib₁) (a₂ + ib₂)(a₂ + ib₂) (a₂ - ib₂) = a₁ a₂ + b₁b₂a2 + b2 +ia₂b₁ + a₁b₂a2 + b2

Equality of Complex Numbers

• (a₁ + ib₁) = (a₂ + ib₂), then a₁ = a₂ and b₁ = b₂

### Powers of Iota (i)

Earlier we have discussed that the negative root of unity is called iota. That is i = $\sqrt{-1}$

### Properties of iota.

• i² = i x i = $\sqrt{-1}$ x $\sqrt{-1}$ = -1

• i³ = i x i x i = $\sqrt{-1}$ x $\sqrt{-1}$ x $\sqrt{-1}$ = -1 x $\sqrt{-1}$ = -1 x i = -i

• i⁴ = i x i x i x i = $\sqrt{-1}$ x $\sqrt{-1}$ x $\sqrt{-1}$ x $\sqrt{-1}$ = (-1) x (-1) = 1

It means,

• i$^{n}$ = i$^{4k}$  for somen n and k then i$^{n}$ = 1

• i$^{n}$ = i$^{4k+1}$ for some n and k then i$^{n}$ = i

• i$^{n}$ = i$^{4k+2}$  for some n and k then i$^{n}$ = -1

• i$^{n}$ = i$^{4k+3}$ for some n and k then i$^{n}$ = -i

### Conjugate of a Complex Number

The conjugate of a complex number is also a complex number in the opposite imaginary direction on the argand plane. Consider the complex number z = a + ib. Its complex conjugate can be defined as z = a - ib. Arithmetically, we can get the complex conjugate of any complex number by just changing the sign of iota.

For example, consider the complex number z = 12 + i5. It complex conjugate will by z = 12 - i5.

### Modulus of a Complex Number

Consider the complex number z = x + iy. The modulus (absolute values) of z is thus defined as the positive square root of the sum of the squares of the real and imaginary parts, indicated by |z|  i.e. $|\text{z}|=\sqrt{{x^2}+{y^2}}$

It represents the distance of z from the origin in the set of complex number c, where the order relation is not defined

i.e. z1 > z2 or z1 < z2 has no meaning but when |z1| > |z2| or |z1|<|z2| has got its meaning since |z1| and |z2| are the real numbers.

### Argand Plane

Any complex number z = x + iy can be represented geometrically by a point (x, y) in a plane, called the argand plane or gaussian plane. A purely number x, i.e. (x + 0i) is represented by the point (x, 0) on the x-axis. Therefore, the x-axis is called the real axis. A purely imaginary number iy i.e. (0 + iy) is represented by the point (0, y) on the y-axis. Therefore, the y-axis is known as the imaginary axis.

### Argument of a Complex Number

The angle made by a line joining point z to the origin, with the positive direction of the x-axis in an anti-clockwise sense is called the argument or amplitude of a complex number. It is denoted by the symbol arg(z) or amp(z).

arg(z) = θ = $\tan^{-1}\left ( \dfrac{x}{y} \right )$

Image: Argument of complex number

The argument of z is not unique, and its general value of the argument of z is 2nπ + θ, but arg(0) is not defined. The unique value of θ such that -π < θ ≤ π is called the principal value of the amplitude or principal argument.

### Principal Value of Argument

• if x > 0 and y > 0, then arg(z) = θ.

• if x < 0 and y > 0, then arg(z) = π – θ.

• if x < 0 and y < 0, then arg(z) = -(π – θ).

• if x > 0 and y < 0, then arg(z) = -θ.

### Polar Form of a Complex Number

If z = x + iy is a complex number, then  z = |z| (cosθ + i sinθ), where θ = arg(z). This is called polar form. If the general value of the argument is θ, then the polar form of z is z = |z| [cos (2nπ + θ) + i sin(2nπ + θ)], where n is an integer.

### Cube Root of Unity

In mathematics, the cube roots of unity are the solutions to the equation $\omega^3 = 1$. These roots are denoted as 1, $\omega$, and $\omega^2 = 1$, where:

• 1 is the principal cube root of unity, which is simply 1.

• $\omega$ is one of the cube roots of unity, and it is equal to $\dfrac{-1 + \sqrt{3i}}{2}$, where i is the imaginary unit.

• $\omega^2$ is the other cube root of unity, and it is equal to $\dfrac{-1 - \sqrt{3i}}{2}$.

## Complex Number Geometry

### 1. Section formula:

The complex number of a point P, which divides the line between points A(𝑧₁) and B(𝑧₂) into two parts with the ratio of m to n, is:

• $P = \dfrac{mz_1 + nz_2}{m + n}$

Midpoint Formula:

• $M = \dfrac{z_1 + z_2}{2}$

Centroid Formula:

• $\text{Centroid} = \dfrac{z_1 + z_2 + z_3}{3}$

### 2. Equation of Straight Lines:

The equation of a straight line that passes through two points, 𝑧₁ and 𝑧₂, is

### Fun facts

1. Complex numbers are actually an addition to the real number system

2. The addition of complex numbers makes a significant difference in mathematics

3. Complex numbers help in the solution of quadratic equations with the help of the Quadratic formula

4. When solving any quadratic equations, each complex result will always have his conjugate companion with him— which are known as complex conjugates.

5. Despite the fact that complex numbers have an imaginary part, there are actually a number of real-life applications of these "imaginary" numbers, such as wavering springs and voltaic electronics.

6. Imaginary numbers are used by the engineers to explain electric current and its intensity in the real world.

## JEE Main Maths - Complex Numbers and Quadratic Equations Study Materials

Here, you'll find a comprehensive collection of study resources for Complex Numbers and Quadratic Equations designed to help you excel in your JEE Main preparation. These materials cover various topics, providing you with a range of valuable content to support your studies. Simply click on the links below to access the study materials of Complex Numbers and Quadratic Equations and enhance your preparation for this challenging exam.

 JEE Main Complex Numbers and Quadratic Equations Study Materials JEE Main Complex Numbers and Quadratic Equations Notes JEE Main Complex Numbers and Quadratic Equations Important Questions JEE Main Complex Numbers and Quadratic Equations Practice Paper

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Explore an array of resources in the JEE Main Maths Study and Practice Materials section. Our practice materials offer a wide variety of questions, comprehensive solutions, and a realistic test experience to elevate your preparation for the JEE Main exam. These tools are indispensable for self-assessment, boosting confidence, and refining problem-solving abilities, guaranteeing your readiness for the test. Explore the links below to enrich your Maths preparation.

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## Conclusion

In this article, we have elaborated on concepts and solutions of Complex Numbers and Quadratic Equations for your JEE Main preparation. Students will explore complex numbers, how to add, subtract, multiply, and divide them. Plus, clear breakdown of quadratic equations, covering their solutions and roots. Everything you're looking for is available in a single location. Students can carefully read through the concepts, definitions and questions in the PDFs, which are also free to download and understand the concepts used to solve these questions. This will be extremely beneficial to the students in their exams.

## FAQs on Maths Complex Numbers and Quadratic Equations Chapter - Maths JEE Main

1. What are the contributions of the chapter Complex Numbers and Quadratic Equation?

Every year, in JEE Main and other exams, at least 1 to 3 problems from this chapter appear, and the concept of this chapter can be used in many other chapters, such as functions and coordinate geometry. One of the most significant and fundamental chapters in the preparation of competitive admission tests is Complex Numbers and Quadratic Equations.

2. How difficult is the chapter Complex Numbers and Quadratic Equation?

Complex numbers might be difficult for some pupils to comprehend and solve issues with at first. However, as you solve more problems involving complex numbers, you will get more comfortable with the subject. After that, the questions will seem simple to you. The Quadratic section is a little easier to understand and correlate than the difficult part; the notion of the Quadratic part is quite simple and logically easy to comprehend and correlate. You can refer to the NCERT solutions for Class 11 maths Chapter 5 and complex numbers and quadratic equations class 11 solutions available on Vedantu’s website for practice. Head over to our website and download the complex numbers and quadratic equations Class 11 PDF.

3. What is the meaning of modulus?

Modulus is the factor that is multiplied by a logarithm of a number to one base to get the logarithm of the number to a new base.

4. Why is the "Complex Numbers and Quadratic Equations Chapter" crucial for JEE Main 2024 in Maths?

This chapter forms the foundation for understanding complex numbers and quadratic equations, which are frequently tested in the JEE Main exam.

5. How does Vedantu assist in learning "Maths Complex Numbers and Quadratic Equations Chapter" for JEE Main 2024?

Vedantu provides clear explanations, interactive sessions, and personalized guidance, helping students tackle complex topics effectively.

6. Can Vedantu help with doubt resolution and exam preparation specifically for "Complex Numbers and Quadratic Equations Chapter - Maths JEE Main2024"?

Absolutely! Vedantu offers instant doubt resolution, flexible learning schedules, and comprehensive exam preparation, ensuring students are well-equipped for success in this critical JEE Main chapter.

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