Maths Complex Numbers and Quadratic Equations Chapter - Maths JEE Main

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

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

Maths Complex Number and Quadratic Equation Important Concepts for JEE Main

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

Addition 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 + b₁b₂a₂2 + b₂2 +ia₂b₁ + a₁b₂a₂2 + b₂2

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 

$\left|z & {z}’ & 1 \\ z_1 & {z_1}’ & 1 \\ z_2 & {z_2}’ & 1 \right| = 0

The general equation of a straight line is given as:

𝑎'𝑧 + 𝑎𝑧' + 𝑏 = 0, 𝑏 is usually a real number.

3. Equation of Circles:

The equation of a circle with its center at the origin and a radius of 'a' is: |𝑧| = 𝑎

The equation of a circle with its center at 𝑧₀ and a radius of 'a' is:|𝑧 - 𝑧₀| = 𝑎

The general equation of a circle is given as: 𝑧𝑧' + 𝑎𝑧' + 𝑎'𝑧 + 𝑏 = 0

Here, 'b' is a purely real number. The center of this circle can be found using the formula: -a, and its radius is $\sqrt{(𝑎𝑎' − 𝑏)}$

What Is a Quadratic Equation?

A quadratic equation is a second-degree equation. The general form of the quadratic equation is ax2+bx+c = 0, where a, b, c are real numbers and a ≠ 0. For example, x2+2x+1 = 0.

An algebraic expression with several terms is called a polynomial. A polynomial of degree two of the form ax2+bx+c (a≠0) is called a quadratic expression in x. When a quadratic polynomial f(x) is equated to zero, we can term it a quadratic equation.

Solution of Quadratic Equation

The quadratic equation ax2 + bx + c = 0 with real coefficients has two roots given by $\dfrac{-b+\sqrt{D}}{2a}$ and $\dfrac{-b-\sqrt{D}}{2a}$, where D = b2 – 4ac, called the discriminant of the equation.

The value of ‘D’ in the equation will tell us what kind of solutions the equation has. Let’s break it down:

Case 1: When D>0,

The equation $ax^2 +bx+c=0$ will have two distinct real roots, which can be found using this formula:

\[x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Case 2: When D=0,

The equation $ax^2 +bx+c=0$ will have two equal real roots, and you can find them using this formula:

\[x = \dfrac{-b}{2a}\]

Case 3: When D < 0, 

The equation $ax^2 +bx+c=0$ will have no real roots and instead will have imaginary roots.

Case 4: When D is the square of a rational number,

The equation $ax^2 +bx+c=0$ will have rational roots.

Case 5: When D is not the square of a rational number,

The equation $ax^2 +bx+c=0$ will have non-rational (irrational) roots in conjugate pairs.

Nature of Roots of Simultaneous Quadratic Equations:

If (D₁ + D₂ ≥ 0), at least one of the equations has real roots.

If (D₁ + D₂ < 0), at least one of the equations has non-real roots.

If (D₁ ⋅ D₂ > 0), both equations have real and distinct roots or both have non-real roots.

If (D₁ ⋅ D₂ < 0), one equation has real and distinct roots while the other has non-real roots.

If (D₁ ⋅ D₂ = 0), one equation has equal roots, and the other can have both real or non-real roots.

Sign of Roots:

• If both roots of a quadratic equation $ax^2 + bx + c = 0$ are positive, 'a' and 'c' have the same sign.

• If both roots are negative, 'a,' 'b,' and 'c' have the same sign.

• If one root is positive while the other is negative, 'a' and 'c' have different signs.

• If the roots have equal magnitude but opposite signs, then b = 0.

• If the roots are reciprocals of each other (one is 1/α and the other is 1/β), then a = c.

• If 'c' is 0, one root must be 0.

• If 'x' is replaced by '1/x', the new roots are 1/α and 1/β.

• If 'x' is replaced by $x^2$ , the new roots are α, −α, β, and −β.

Note: 

Transformation of Quadratic Equations and Condition of Common Roots:

This concept refers to changing or manipulating quadratic equations to determine if they share common roots. It involves altering the equations and observing how their coefficients relate to discover if they have the same solutions.

The Discriminant of Quadratic Equations:

The discriminant is a value derived from the coefficients of a quadratic equation. It's used to determine the nature of the equation's roots. Specifically, it helps us understand whether the roots are real or complex, distinct or equal, based on its positive or negative value.

Note:

(i) When D = 0, roots are real and equal. When D > 0 roots are real and unequal. Further, if a,b, c ∈ Q and D is a perfect square, then the roots of the quadratic equation are real and unequal and if a, b, c ∈ Q and D is not a perfect square, then the roots are irrational and occur in pairs. When D < 0, the roots of the equation are non-real (or complex).

(ii) Let α, β be the roots of quadratic equation ax2 + bx + c = 0, then the sum of roots α + β =$\dfrac{-b}{a}$ and the product of roots αβ = $\dfrac{c}{a}$. 

List of Important Formulas for Maths Complex Number and Quadratic Equations Chapter

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.

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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.