Complex Numbers and Quadratic Equations Class 11 Notes

Revision notes

Complex Numbers: revision notes

1. The number $i$

$x^{2} + 1 = 0$ has no real solution, so we introduce $i$ with $i^{2} = - 1$ . Powers repeat in a cycle of four: $i, - 1, - i, 1$ ; so $i^{4 k} = 1$ , $i^{4 k + 1} = i$ , $i^{4 k + 2} = - 1$ , $i^{4 k + 3} = - i$ . For $a > 0$ , $\sqrt{- a} = i \sqrt{a}$ .

2. Complex numbers

$z = a + i b$ with real $a = Re z$ and $b = Im z$ . Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal.

3. Algebra of complex numbers

Add and subtract parts: $(a + i b) \pm (c + i d) = (a \pm c) + i (b \pm d)$ .
Multiply like binomials and use $i^{2} = - 1$ : $(a + i b) (c + i d) = (a c - b d) + i (a d + b c)$ .
Divide by multiplying top and bottom by the conjugate of the denominator. The inverse of $z \neq 0$ is $\frac{\bar{z}}{| z |^{2}}$ .
Caution: $\sqrt{a} \sqrt{b} = \sqrt{a b}$ is not true when $a$ and $b$ are both negative: $\sqrt{- 4} \sqrt{- 9} = (2 i) (3 i) = - 6$ .

4. Modulus and conjugate

$\bar{z} = a - i b$ , $| z | = \sqrt{a^{2} + b^{2}}$ . Properties: $z \bar{z} = | z |^{2}$ , $| z_{1} z_{2} | = | z_{1} | | z_{2} |$ , $| \frac{z_{1}}{z_{2}} | = \frac{| z_{1} |}{| z_{2} |}$ , $\overset{―}{z_{1} \pm z_{2}} = {\bar{z}}_{1} \pm {\bar{z}}_{2}$ , $\overset{―}{z_{1} z_{2}} = {\bar{z}}_{1} {\bar{z}}_{2}$ .

5. The Argand plane

$z = a + i b$ is the point $(a, b)$ : real axis horizontal, imaginary axis vertical. $| z |$ is the distance from the origin; $| z_{1} - z_{2} |$ is the distance between the points; $\bar{z}$ is the reflection in the real axis. Multiplying by $i$ turns a point through $90^{\circ}$ anticlockwise about the origin.

Worked example 1

Find $i^{- 39}$ .

$i^{- 39} = i^{- 40} i = i$ .

Worked example 2

Write $\frac{5}{2 + i}$ in the form $a + i b$ .

$\frac{5 (2 - i)}{4 + 1} = 2 - i$ .

Worked example 3

Find real $x, y$ with $(x - y) + i (x + y) = 1 + 5 i$ .

$x - y = 1$ , $x + y = 5$ : $x = 3$ , $y = 2$ .

Common errors

$i^{2} = 1$ , or forgetting to replace $i^{2}$ by $- 1$ .
Rationalising with $a + i b$ instead of the conjugate $a - i b$ .
$| a + i b | = a + b$ , or including $i$ in the modulus: $| 3 - 4 i | = \sqrt{9 + 16}$ .
Using $\sqrt{a} \sqrt{b} = \sqrt{a b}$ with negative $a, b$ .

Exam tips

Give every final answer in the form $a + i b$ .
To show a complex number satisfies an equation, substitute and collect real and imaginary parts separately.

Topics in this chapter: The number i · Modulus and conjugate · Algebra of complex numbers · Complex numbers · The Argand plane.

Formulas for this chapter: Chapter 4 on the Class 11 formula sheet (free PDF and A3 poster).

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can simplify powers of i, add, multiply and take conjugates and moduli.

Read first: 1. The number i; 2. Complex numbers; 4. Modulus and conjugate 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%

Practise step 1

Step 2

Exam standard

You can divide complex numbers, compare real and imaginary parts and plot numbers on the Argand plane.

Read first: 3. Algebra of complex numbers; 5. The Argand plane; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can combine the properties of modulus and conjugate in longer calculations and in circuit or rotation contexts.

Read first: 4. Properties of modulus and conjugate; 5. Argand plane 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can solve equations in z = x + iy and interpret them on the Argand plane.

Read first: 5. Argand plane; Common errors 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 2 of the 40 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceThe number i

The value of $i^{35}$ is

(a) $1$
(b) $- i$
(c) $i$
(d) $- 1$

Q2·1 mark·Multiple choiceModulus and conjugate

The modulus of $3 - 4 i$ is

(a) $7$
(b) $25$
(c) $- 1$
(d) $5$

Q3·1 mark·Multiple choiceAlgebra of complex numbers

$(2 + 3 i) (1 - i)$ equals

(a) $5 + i$
(b) $- 1 + i$
(c) $5 - i$
(d) $2 - 3 i$

Worked solution·2 marks·Very short answerThe Argand plane

Plot $z = 1 + i$ and $i z$ in the Argand plane, and describe how the point $i z$ is obtained from $z$ .

Full worked solution Free, no sign-in

$i z = i + i^{2} = - 1 + i$ : the points $(1, 1)$ and $(- 1, 1)$ A1
$i z$ is $z$ turned through $90^{\circ}$ anticlockwise about the origin A1

Every question here has a full worked solution. See them all →

Stretch yourself: original number theory problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S47 · Problem S06 · Problem S46 · Problem S05. All 22 →

Knowledge organiser: Complex Numbers and Quadratic Equations on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Complex Numbers and Quadratic Equations

i² left as i² or taken as +1. Fix: replace i² by −1 at once.
Division without the conjugate. Fix: multiply numerator and denominator by the conjugate of the denominator.
Modulus with i inside or without squaring. Fix: |a + ib| = √(a² + b²).
√(−4)·√(−9) written as √36 = 6. Fix: write each as i√a first: (2i)(3i) = −6.
Answer not in the form a + ib. Fix: separate real and imaginary parts in the last line.

Test Complex Numbers and Quadratic Equations against the clock

Want it against the clock? Take a timed 30-mark chapter test on Complex Numbers and Quadratic Equations, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE? Try the JEE-style questions on complex numbers and quadratic equations.

Complex Numbers and Quadratic Equations: common questions

What is in CBSE Class 11 Maths Chapter 4, Complex Numbers and Quadratic Equations?

The topics are: The number i, Modulus and conjugate, Algebra of complex numbers, Complex numbers, The Argand plane. This page has revision notes for each, and 40 board-style practice questions with step-marked solutions.

How many marks is Complex Numbers and Quadratic Equations worth?

It is in the Algebra unit, which carries 25 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Complex Numbers and Quadratic Equations?

i² left as i² or taken as +1. Fix: replace i² by −1 at once. Division without the conjugate. Fix: multiply numerator and denominator by the conjugate of the denominator.

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Complex Numbers and Quadratic Equations Class 11: notes and important questions

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