JEE Main 2025: Complex Numbers and Quadratic Equations Notes- FREE PDF Download

These are some of the key topics for the JEE Main exams. Looking at the old JEE Main question papers, you'll see the total weightage of questions from this chapter is 7%. Take a look at the table for more details.

Over the last five years of JEE Main exams, about 3 questions were asked from Complex Numbers and Quadratic Equations. Now you know how much weight you have to give to this chapter. The teachers put in dedicated efforts to ensure the Complex Numbers and Quadratic Equations notes align perfectly with the JEE Main syllabus. Get ready to do well – get your Complex Numbers and Quadratic Equations Notes and shine in JEE Main 2025!

JEE Main 2025: Must-ReviseTopics in Complex Numbers and Quadratic Equations

Let's focus on the key topics you should review in Complex Numbers and Quadratic Equations. These important topics can make a real difference in your preparation. Mastering them will give you a strong edge in the upcoming exam. Let's dive into the must-revise topics to help you shine in JEE Main 2025.

Complex Numbers:

• Definition of Complex Numbers

• Complex Plane (Argand Diagram)

• Polar Form and Euler's Formula

• Operations with Complex Numbers (Addition, Subtraction, Multiplication, Division)

• Conjugate of a Complex Number

• Modulus (Magnitude) and Argument of a Complex Number

• De Moivre's Theorem

• Roots of Complex Numbers

• Complex Conjugate Roots Theorem

• Complex Exponential Form

Quadratic Equations:

• Solving Quadratic Equations by Factoring

• Quadratic Formula

• Completing the Square Method

• Discriminant and Nature of Roots

• Sum and Product of Roots

• Quadratic Equations in Real-Life Applications

• Quadratic Inequalities

• Graphs of Quadratic Equations

• Relationship between Roots and Coefficients

• Vieta's Formulas

The Significance of Complex Numbers and Quadratic Equations Notes for JEE Main 2025

Unravel the enigmatic realm of Complex Numbers and Quadratic Equations – a chapter that unveils the captivating interplay between real and imaginary numbers. Get ready to embark on an interactive journey that will not only enhance your mathematical prowess but also leave you mesmerised by the magic of numbers.

Complex Numbers: The Gateway to Imagination

Imagine numbers that defy the boundaries of reality – welcome to the world of complex numbers! With real numbers as companions, complex numbers bring a touch of imagination to mathematics. Expressions like $\sqrt{-1}$ become a reality, giving birth to the enigmatic 'imaginary unit,' denoted as 'i.' With this intriguing building block, you'll explore the complex plane, where numbers reside like stars in the mathematical cosmos.

Key Highlights:

• Imaginary Unit 'i': Discover the wonders of 'i' and its remarkable power to bring the impossible to life.

• Complex Plane Exploration: Navigate the complex plane like an explorer charting new territories.

• Real and Imaginary Components: Witness the synergy of real and imaginary components that form the essence of complex numbers.

Quadratic Equations: Unravelling Mysteries of Second Degree 

Step into the realm of quadratic equations – a world where x², coefficients, and solutions dance in perfect harmony. This chapter demystifies the quadratic equation ax² + bx + c = 0, where you'll decode its secrets, discover its solutions, and even learn how to unveil its hidden roots!

Key Highlights:

• Roots and Solutions: Dive deep into the process of finding roots and solutions of quadratic equations.

• Nature of Roots: Uncover the intriguing possibilities of real and complex roots based on the discriminant.

• Quadratic Formula: Learn the magical quadratic formula that unveils solutions with mathematical elegance.

Interactive Engagements Await! 

• Mathematical Riddles: Challenge yourself with brain-teasing riddles that involve complex numbers and quadratic equations. Unleash your problem-solving skills!

• Visual Explorations: Immerse yourself in interactive visualisations that bring complex numbers and quadratic equations to life, making learning an unforgettable experience.

• Real-World Connections: Discover how these mathematical concepts transcend the classroom and find applications in diverse fields like physics, engineering, and even arts.

Formulaic Support: Boosting Understanding with  Complex Numbers and Quadratic Equations JEE  Mains Notes

If you want to learn well, you can use a clever plan by merging formulas and revision notes. Formulas are like short explanations of important concepts. You get a complete learning method when you use them with detailed revision notes. Revision notes build a strong foundation, and formulas help you grasp concepts and solve problems.This combination makes tough things simpler and improves memory. Following this approach can enhance your understanding and help you perform well in your studies.

Download Vedantu's JEE Main Maths Formulas for FREE. Blend it with JEE Main Notes, and you'll notice positive score changes.

Here are the important formulas for Complex Numbers and Quadratic Equations relevant to JEE Mains:

Complex Numbers

1. Standard Form: 

A complex number is written as $ z = a + ib $, where $ a $ is the real part ($ \text{Re}(z) $) and $ b $ is the imaginary part ($ \text{Im}(z) $).

2. Modulus of $ z = a + ib $:

\[|z| = \sqrt{a^2 + b^2}\]

3. Conjugate of $ z = a + ib $:

\[\overline{z} = a - ib\]

4. Polar Form:

A complex number $ z = r (\cos \theta + i \sin \theta) = r e^{i \theta} $, where $ r = |z| $ and $ \theta = \arg(z) $.

5. Euler's Formula:

\[e^{i \theta} = \cos \theta + i \sin \theta\]

6. Multiplication and Division:

• If $ z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) $ and $ z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) $:

• Multiplication: $ z_1 \cdot z_2 = r_1 r_2 \left( \cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2) \right) $

• Division: $ \dfrac{z_1}{z_2} = \dfrac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right) $

7. De Moivre’s Theorem:

\[(r (\cos \theta + i \sin \theta))^n = r^n (\cos n\theta + i \sin n\theta)\]

8. Roots of Unity: For $ z^n = 1 $, the $ n $ Distinct Roots are Given by:

\[z_k = e^{i \dfrac{2\pi k}{n}} = \cos \dfrac{2\pi k}{n} + i \sin \dfrac{2\pi k}{n}, \quad k = 0, 1, 2, \ldots, n-1\]

9. Square Root of a Complex Number $ z = a + ib $:

\[\sqrt{z} = \pm \left( \sqrt{\dfrac{|z| + a}{2}} + i \, \text{sign}(b) \sqrt{\dfrac{|z| - a}{2}} \right)\]

Quadratic Equations

1. Standard Form:

A quadratic equation is of the form $ ax^2 + bx + c = 0 $.

2. Roots of a Quadratic Equation:

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

3. Discriminant (D):

The discriminant is $ D = b^2 - 4ac $.

• If $ D > 0 $: Two distinct real roots.

• If $ D = 0 $: Two equal real roots.

• If $ D < 0 $: Two complex roots.

4. Sum and Product of Roots:

• If $ \alpha $ and $ \beta $ are roots of $ ax^2 + bx + c = 0 $:

• Sum of Roots: $ \alpha + \beta = -\dfrac{b}{a} $

• Product of Roots: $ \alpha \beta = \dfrac{c}{a} $

5. Nature of Roots Using Discriminant:

• Real and distinct if $ D > 0 $

• Real and equal if $ D = 0 $

• Complex conjugates if $ D < 0 $

6. Quadratic Expressions and Inequalities:

• For $ ax^2 + bx + c $:

• Minimum value (if $ a > 0 $): $ f(x) = \dfrac{4ac - b^2}{4a} $

• Maximum value (if $ a < 0 $): $ f(x) = \dfrac{4ac - b^2}{4a} $

Maths Notes: Chapter-wise links 2025

Download JEE Main Maths Chapter-wise Notes - Free PDF from the table below:

Further Resources for JEE Main 2025 Complex Numbers and Quadratic Equations

Ready to take your JEE Main 2025 Complex Numbers and Quadratic Equations preparation up a notch? Delve into these crucial resources:

Download Subject-wise Notes for JEE Main 

Get the links to download the subject-wise JEE Main Notes PDF for FREE from the table below.

Get Ahead in your Exams with Vedantu’s JEE Main Notes

In your journey towards exam success, Vedantu's Complex Numbers and Quadratic Equations JEE Main Notes are your trusted companion. Our unique approach, combining expert guidance and student-friendly resources, sets us apart. With comprehensive revision notes at your fingertips, complex concepts become clear and preparation becomes more efficient. What's more, these JEE Complex Numbers and Quadratic Equations Notes are available to you as FREE PDFs. Trust Vedantu to help you stride confidently towards acing your exams. Elevate your learning experience, simplify your studies, and embrace excellence with Vedantu.

Important Study Materials Links for JEE Exams