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
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
Complex numbers are actually an addition to the real number system
The addition of complex numbers makes a significant difference in mathematics
Complex numbers help in the solution of quadratic equations with the help of the Quadratic formula
When solving any quadratic equations, each complex result will always have his conjugate companion with him— which are known as complex conjugates.
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.
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 Maths Study and Practice Materials
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.
Benefits of Using Vedantu for JEE Main 2024 - Maths Complex Numbers and Quadratic Equations
Embark on a mathematical journey with Vedantu's stellar support for "Complex Numbers and Quadratic Equations" in JEE Main. From interactive learning and instant doubt resolution to personalized guidance, dive into a world where clarity meets convenience, paving the way for confident exam preparation.
Topic-Wise Clarity in "Complex Numbers and Quadratic Equations" for JEE Main: Vedantu offers clear explanations, making complex topics in "Complex Numbers and Quadratic Equations" for JEE Main easily understandable, providing students with topic-wise clarity.
Interactive Learning for Chapter Mastery: Engage in interactive sessions on Vedantu to thoroughly understand each concept in "Complex Numbers and Quadratic Equations," ensuring a strong foundation and mastery of the chapter.
Doubt Resolution in Real-Time: Vedantu ensures instant doubt resolution for "Complex Numbers and Quadratic Equations," addressing uncertainties in real-time during study sessions, enhancing comprehension.
Personalised Guidance for Individual Learning: Benefit from personalised guidance on Vedantu, tailored to your learning pace in "Complex Numbers and Quadratic Equations," facilitating a better understanding of individual concepts.
Access to Varied Practice Materials: Vedantu provides a variety of practice materials for "Complex Numbers and Quadratic Equations," allowing students to reinforce their understanding and improve problem-solving skills.
Regular Assessments for Progress Tracking: Participate in regular assessments on Vedantu to track your progress in understanding "Complex Numbers and Quadratic Equations," identifying areas for improvement and focused revision.
Flexible Learning Schedule for Convenience: Enjoy the flexibility of learning at your own pace on Vedantu, making the study of "Complex Numbers and Quadratic Equations" convenient and effective for JEE Main preparation.
Learning from Experienced Instructors: Learn from experienced instructors on Vedantu, gaining valuable insights that make challenging topics in "Complex Numbers and Quadratic Equations" more accessible and comprehensible.
Effective Exam Preparation for JEE Main: Vedantu aids in effective exam preparation for "Complex Numbers and Quadratic Equations," ensuring that students are well-prepared and confident when facing the JEE Main examination.
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.
