Complex Numbers And Quadratic Equations Class 11th Notes - Free NCERT Class 11 Maths Chapter 5 notes - Download PDF

Have you ever tried solving quadratic equations like $x^2+1=0$? Then you will find $x^2=-1$. Is it possible that the square of a real number is $-1$? The answer lies in this chapter. From NCERT Class 12 Maths, the chapter Complex Numbers and Quadratic Equations contains the concepts of Complex Numbers, Algebra of Complex Numbers, The Modulus and the Conjugate of a Complex Number, Argand Plane and Polar Representation, etc. These concepts will help the students grasp more advanced complex number topics easily and will also enhance their problem-solving ability in real-world applications.

This article on NCERT notes Class 12 Maths Chapter 4 Complex Numbers and Quadratic Equations offers well-structured NCERT notes to help the students grasp the concepts of Complex Numbers easily. Students who want to revise the key topics of Complex Numbers and Quadratic Equations quickly will find this article very useful. It will also boost the exam preparation of the students by many folds. These notes of NCERT Class 12 Maths Chapter 4 Complex Numbers and Quadratic Equations are made by the Subject Matter Experts according to the latest CBSE syllabus, ensuring that students can grasp the basic concepts effectively. NCERT solutions for class 12 maths and NCERT solutions for other subjects and classes can be downloaded from the NCERT Solutions.

NCERT Class 11 Maths Chapter 4 Notes

Complex Number

A complex number is defined as the combination of real and imaginary numbers.

Example: $3+4i$

Here, $3$ is real and $4i$ is the imaginary part.

Here is $i =\sqrt{(-1)}$, then we can say that $i^2 = -1$

Algebra Of Complex Numbers

Addition Of Two Complex Numbers

Let $x=a+ib$ and $y=c+id$

Therefore, $x+y= (a+c) +i(b+d)$

Properties

• The closure law: The sum of two complex numbers is also a complex number

• The commutative law $x+y=y+x$

• The associative law $(x+y)+z=x+(y+z)$

Difference Of Two Complex Numbers
Let $x=a+ib$ and $y=c+id$

Therefore $x-y= x+(-y)$

Multiplication of Two Complex Numbers
Let $x=a+ib$ and $y=c+id$

Therefore, $xy=ac-bd+i(ad+bc)$

Properties

• The closure law. The multiplication of two complex numbers is also a complex number

• The commutative law $xy=yx$

• The associative law $(xy)z=x(yz)$

• The distributive law $x(y+z) = xy+xz$ and $(x+y)z=xz+yz$

Division Of Two Complex Numbers

Let $x=a+ib$ and $y=c+id$

Therefore $\frac{x}{y}$ = $x.(\frac{1}{y})$

Power of i

In general, any integer k

$i^{4 k}=1, i^{4 k+1}=i, i^{4 k+2}=-1, i^{4 k+3}=-i$

The square roots of a negative real number

$i^2=-1, \text { and }(-i)^2=i^2=-1$

The Modulus And The Conjugate Of A Complex Number

If $x=a+i b$, then $|x|=\sqrt{a^2+b^2}$

Properties

1. $|x y|=|x||y|$

2. $|\frac{x}{y}|=\frac{|x|}{|y|}$

3. $(xy)'=x'y'$

4. $(x±y)’=x’±y’$

5. $(\frac{x}{y})’=\frac{x’}{y’}$

Argand Plane And Polar Representation

The complex number $a+ib$ corresponding to ordered pairs $(a, b)$ can also be represented geometrically in the XY plane.

In the Argand plane, the modulus of a complex number $a+ib =\sqrt{(a^2+b^2)}$ is the distance from the origin to the point.

Polar Representation Of A Complex Number

We note that the point P is uniquely determined by the ordered pair of real numbers $(r, θ),$ called the polar coordinates of the point P.

We get $a=r\cosθ, b=r\sinθ,$ thus $x=r(\cosθ +i \sin θ)$

Example:

Express the following in the form $\mathrm{a}+\mathrm{ib}: i^{-35}$
Solution : $i^{-35}=\frac{1}{i^{35}}=\frac{1}{\left(i^2\right)^{17} i}=\frac{1}{-i} × \frac{i}{i}=i$

Quadratic Equation

The equation that consists of the highest power in the polynomial is 2.

Let $ax^2+bx+c=0$ be a quadric equation then its discriminant is

$D=b^2-4ac$

Let us assume that $D<0$

Then $x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}$

Example:

Solve $x^2+x+1=0$

Solution:

Here, $b^2-4ac=1^2-4×1×1=-3<0$

$\begin{aligned} & \text { Therefore } x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} \\ &= \frac{-1 \pm \sqrt{1^2-4 × 1 × 1}}{2 × 1}=\frac{-1 \pm \sqrt{3} i}{2}\end{aligned}$

This concludes the chapter.

Importance of NCERT Class 11 Maths Chapter 3 Notes:

NCERT Class 11 Maths Chapter 3 Notes play a vital role in helping students grasp the core concepts of the chapter easily and effectively, so that they can remember these concepts for a long time. Some important points of these notes are:

• Effective Revision: These notes provide a detailed overview of all the important theorems and formulas, so that students can revise the chapter quickly and effectively.
• Clear Concepts: With these well-prepared notes, students can understand the basic concepts effectively. Also, these notes will help the students remember the key concepts by breaking down complex topics into simpler and easier-to-understand points.
• Time Saving: Students can look to save time by going through these notes instead of reading the whole lengthy chapter.
• Exam Ready Preparation: These notes also highlight the relevant contents for various exams, so that students can get the last-minute minute very useful guidance for exams.

NCERT Class 11 Notes Chapter Wise

Subject-Wise NCERT Exemplar Solutions

After finishing the textbook exercises, students can use the following links to check the NCERT exemplar solutions for a better understanding of the concepts.

• NCERT Exemplar Class 11 Solutions
• NCERT Exemplar Class 11 Maths
• NCERT Exemplar Class 11 Physics
• NCERT Exemplar Class 11 Chemistry
• NCERT Exemplar Class 11 Biology

Subject-Wise NCERT Solutions

Students can also check these well-structured, subject-wise solutions.

• NCERT Solutions for Class 11 Mathematics
• NCERT Solutions for Class 11 Chemistry
• NCERT Solutions for Class 11 Physics

• NCERT Solutions for Class 11 Biology

NCERT Books and Syllabus

Students should always analyze the latest CBSE syllabus before making a study routine. The following links will help them check the syllabus. Also, here is access to more reference books.

• NCERT Book for Class 11
• NCERT Syllabus for Class 11

Happy learning !!!