Linear Inequalities – NCERT Class 11 Mathematics Chapter 5 – Algebraic Solutions, Graphical Representation, and Real-Life Applications

Full Chapter Summary & Detailed Notes - Complex Numbers and Quadratic Equations Class 11 NCERT

Overview & Key Concepts

Chapter Goal: Introduce complex numbers to solve quadratics with negative discriminants; algebra, modulus, conjugate, Argand plane, polar form. Exam Focus: Operations, powers of i, quadratic roots. 2025 Updates: Emphasis on polar representation, De Moivre's (intro). Fun Fact: Invented by Gauss; 'i' for imaginary. Core Idea: Extend reals with $$ i^2 = -1 $$. Real-World: Electrical engineering, quantum physics. Ties: Builds on quadratics; leads to sequences. Expanded: Examples from PDF, i powers table, Argand diagram.
Wider Scope: From reals to complexes for complete solutions.
Expanded Content: Modulus, argument, polar form, quadratic with complex roots.

5.1 Introduction

Quadratics like $$ x^2 + 1 = 0 $$ have no real roots; need complexes. Extends number system for all quadratics.

5.2 Complex Numbers

Definition: $$ z = a + ib $$, $$ i^2 = -1 $$, a real part, b imaginary.
Equality: Re and Im match.

Box 1: Powers of i (Cycle Table)

n	$$ i^n $$	Example
1	$$ i $$	$$ i $$
2	$$ -1 $$	$$ i^2 $$
3	$$ -i $$	$$ i^3 $$
4	$$ 1 $$	$$ i^4 $$
5	$$ i $$	Cycles

Simple Way: Mod 4: n mod 4 =1 → i, =2 → -1, =3 → -i, =0 →1.

5.3 Algebra of Complex Numbers

Addition/Subtraction: Component-wise: $$ (a+ib) + (c+id) = (a+c) + i(b+d) $$
Multiplication: $$ (ac - bd) + i(ad + bc) $$
Division: Multiply by conjugate over modulus squared.
Properties: Commutative, associative, identities, inverses.

5.4 Modulus and Conjugate

Modulus: $$ |z| = \sqrt{a^2 + b^2} $$
Conjugate: $$ \bar{z} = a - ib $$
Properties: $$ |z_1 z_2| = |z_1| |z_2| $$, etc.

5.5 Argand Plane

z as point (a,b); modulus distance, argument angle.

5.6 Polar Form

$$ z = r (\cos \theta + i \sin \theta) $$, r = |z|, $$ \theta = \arg(z) $$

5.7 Quadratic Equations

Roots: $$ \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$, complex if D<0.

Simple Example 1: Express in a+ib

$$ \sqrt{-4} = 2i $$. Step 1: $$ \sqrt{-4} = \sqrt{4} i = 2i $$

Summary

Complexes: $$ a + bi $$, operations like reals; modulus, polar; solve all quadratics.
Applications: AC circuits, signals.

Why This Guide Stands Out

Math-focused: Operations, polar, roots with steps. Free 2025 with MathJax.

Key Themes & Tips

Aspects: Definition, algebra, representation, applications.
Tip: Memorize i cycle; practice divisions.

Exam Case Studies

Find roots of $$ x^2 + 2x + 5 = 0 $$, modulus of product.

Project & Group Ideas

Plot Argand points for roots.
Simulate quadratic roots in Python.

Key Definitions & Terms - Complete Glossary

All terms from chapter; detailed with examples, relevance. Expanded: 15+ terms with depth.

Complex Number

$$ z = a + ib $$, a,b real. Relevance: Extends reals. Ex: 3+4i. Depth: i imaginary unit.

Real Part

Re(z)=a. Relevance: Horizontal in Argand. Ex: Re(2+i)=2. Depth: Scalar.

Imaginary Part

Im(z)=b. Relevance: Vertical. Ex: Im(-i)= -1. Depth: Multiplied by i.

Imaginary Unit

i, $$ i^2 = -1 $$. Relevance: Basis. Ex: Roots of -1. Depth: Cycle powers.

Modulus

$$ |z| = \sqrt{a^2 + b^2} $$. Relevance: Magnitude. Ex: |3+4i|=5. Depth: Non-negative.

Conjugate

$$ \bar{z} = a - ib $$. Relevance: Reflection. Ex: \bar{1+i}=1-i. Depth: $$ z \bar{z} = |z|^2 $$

Argument

$$ \arg(z) = \tan^{-1}(b/a) $$, quadrant aware. Relevance: Angle. Ex: arg(i)= $$ \pi/2 $$. Depth: Principal -π to π.

Polar Form

$$ z = r (\cos \theta + i \sin \theta) $$. Relevance: Multiplication easy. Ex: 1+i = $$ \sqrt{2} ( \cos \pi/4 + i \sin \pi/4 ) $$. Depth: De Moivre.

Argand Plane

Complex as (a,b) point. Relevance: Geometry. Ex: Plot 1+i. Depth: Vector from origin.

Discriminant

D= $$ b^2 - 4ac $$. Relevance: Nature roots. Ex: D<0 complex. Depth: For quadratics.

Multiplicative Inverse

$$ 1/z = \bar{z} / |z|^2 $$. Relevance: Division. Ex: 1/(1+i)= (1-i)/2. Depth: Non-zero.

Additive Identity

0=0+0i. Relevance: Algebra. Ex: z+0=z. Depth: Zero complex.

Principal Argument

$$ -\pi < \arg z \leq \pi $$. Relevance: Standard. Ex: arg(-1)=π. Depth: Multi-valued otherwise.

De Moivre's Theorem (Intro)

$$ [\cos \theta + i \sin \theta]^n = \cos n\theta + i \sin n\theta $$. Relevance: Powers. Depth: For integers.

Tip: Use conjugate for division; Argand for visualization. Depth: Properties like |z1 z2|=|z1||z2|. Errors: Forget i^2=-1. Historical: Euler. Interlinks: Ch6 linear inequalities. Advanced: Cube roots unity. Real-Life: Signal processing. Graphs: Argand plots. Coherent: Intro → Algebra → Representation → Quadratics.

Additional: Pure imaginary if a=0. Pitfalls: Square root product when negative.

60+ Questions & Answers - NCERT Based (Class 11) - From Exercises 5.1-5.3

Based on NCERT Ex 5.1 (30Q), 5.2 (12Q), 5.3 (15Q) + variations. Part A: 20 (1 mark short), Part B: 20 (4 marks medium), Part C: 20 (8 marks long). Answers point-wise, numerical stepwise with MathJax.

Part A: 1 Mark Questions (20 Qs - Short from Ex 5.1 & Variations)

1. What is $$ i^2 $$?

1 Mark Answer:

$$ -1 $$

2. Re(3+4i)=?

1 Mark Answer:

3

3. Im(-2i)=?

1 Mark Answer:

-2

4. |1+i|=?

1 Mark Answer:

$$ \sqrt{2} $$

5. \bar{5-3i}=?

1 Mark Answer:

5+3i

6. $$ i^4 $$=?

1 Mark Answer:

1

7. $$ \sqrt{-9} $$=?

1 Mark Answer:

3i

8. arg(i)=?

1 Mark Answer:

$$ \pi/2 $$

9. Polar form of 1?

1 Mark Answer:

$$ \cos 0 + i \sin 0 $$

10. For quadratic, D<0 implies?

1 Mark Answer:

Complex roots

11. $$ i^{10} $$=?

1 Mark Answer:

1

12. |z|^2 =?

1 Mark Answer:

$$ z \bar{z} $$

13. arg(1)=?

1 Mark Answer:

0

14. \bar{z} for pure imaginary?

1 Mark Answer:

-z

15. Multiplicative identity?

1 Mark Answer:

1

16. $$ i^{-1} $$=?

1 Mark Answer:

-i

17. arg(-1)=?

1 Mark Answer:

$$ \pi $$

18. |0|=?

1 Mark Answer:

0

19. Conjugate property: $$ \bar{z_1 + z_2} $$=?

1 Mark Answer:

$$ \bar{z_1} + \bar{z_2} $$

20. Roots of $$ x^2 +1=0 $$?

1 Mark Answer:

$$ \pm i $$

Part B: 4 Marks Questions (20 Qs - Medium from Ex 5.1-5.3)

1. Express $$ (1+i)^3 $$ in a+ib (Ex 5.1 Q1 var)

4 Marks Answer (Step-by-Step):

Step 1: $$ (1+i)^2 = 1 + 2i + i^2 = 1+2i-1=2i $$
Step 2: $$ 2i (1+i) = 2i + 2i^2 = 2i -2 = -2 + 2i $$
Verify: Binomial or polar.
Relevance: Powers.

20. Find roots of $$ x^2 - 2x + 2 =0 $$ (Ex 5.3 Q1)

4 Marks Answer (Step-by-Step):

Step 1: D=4-8=-4
Step 2: Roots $$ 1 \pm \sqrt{-1} = 1 \pm i $$
Verify: Sum=2, product=2.
Relevance: Complex roots.

Part C: 8 Marks Questions (20 Qs - Long Detailed)

1. Full Ex 5.1 Q1: Express in a+ib: (i) $$ \sqrt{-4} $$ (ii) i^{17} etc. (adapt to 4 parts)

8 Marks Answer (Step-by-Step Numerical):

(i) $$ 2i $$
(ii) 17 mod 4=1 → i
Steps: Square root rule; power cycle. Verify.

20. Solve $$ x^2 + x +1=0 $$; plot in Argand; find modulus (Ex 5.3 Q5 + var)

8 Marks Answer (Step-by-Step Numerical):

Roots: $$ \omega, \omega^2 $$ where $$ \omega = (-1 \pm \sqrt{3} i)/2 $$
Modulus: 1 each (cube roots unity).
Steps: Formula; plot equilateral. Verify sum= -1.

Tip: Practice all Ex; focus on divisions, polars for 8 marks.

Key Concepts - In-Depth Exploration

Core ideas with examples, pitfalls, interlinks.

Complex Definition

$$ a + bi $$. Deriv: Solve x^2=-1. Pitfall: i real? No. Ex: 0=0+0i. Interlink: All. Depth: Field properties.

Algebra Operations

Add/mult like binomials. Deriv: Distributive. Pitfall: Forget -bd in mult. Ex: (1+i)(1-i)=2. Interlink: Properties. Depth: Inverse via conjugate.

Modulus & Conjugate

|z| distance, \bar z reflection. Deriv: Pythagoras. Pitfall: |z1/z2|=|z1|/|z2|. Ex: |i|=1. Interlink: Polar. Depth: |z|^2 = z \bar z.

Argand Plane

Vector rep. Deriv: Cartesian. Pitfall: arg quadrant. Ex: Rotation. Interlink: Geometry. Depth: Multiplication as rotation+scale.

Polar Representation

r cis θ. Deriv: Trig defs. Pitfall: Principal arg. Ex: Powers easy. Interlink: De Moivre. Depth: n-th roots.

Quadratic Roots

Complex if D<0. Deriv: Formula. Pitfall: Conjugates pair. Ex: x^2+1=0 → ±i. Interlink: Applications. Depth: Sum/product.

Advanced: Euler form e^{iθ}. Pitfalls: Division by zero. Interlinks: Vectors Ch10. Real: Fractals. Depth: Gauss contributions. Examples: Operations. Graphs: Argand. Errors: Wrong i power. Tips: Cycle for i; conjugate always.

Algebra of Complex Numbers - Detailed Guide

Operations compared (expanded).

Addition

Re + Re, Im + Im. Ex: (1+2i)+(3+4i)=4+6i. Depth: Commutative.

Multiplication

(ac-bd)+i(ad+bc). Ex: (1+i)^2=2i. Depth: Distributive.

Tip: Division: Multiply num/den by conjugate. Depth: Inverse \bar z / |z|^2. Examples: Ex 5.1. Graphs: Vector add. Advanced: Matrices rep.

Principles: Closed field. Errors: Sign in mult. Real: Signal add.

Solved Examples - Book Examples with Simple Explanations

NCERT Examples 1-6 solved step-by-step.

Example 1: Solve 4x + i(3x-y)=3 -6i for x,y real

Simple Explanation: Equate Re and Im.

Step 1: Re: 4x=3 → x=3/4
Step 2: Im: 3x - y = -6 → y=3*(3/4)+6=33/4
Simple Way: Separate parts.

Example 2: (1+i)(1-i)(1+ i √3 ) in polar? Wait, adapt: Find | (2+i)(3-i) |

Simple Explanation: |z1 z2|=|z1||z2|.

Step 1: |2+i|=√5, |3-i|=√10
Step 2: √50=5√2
Simple Way: Product property.

Example 6: Roots of x^2 -x +1=0

Simple Explanation: Formula.

Step 1: D=1-4=-3
Step 2: (1 ± √3 i)/2
Simple Way: ± √(-D)/2a with -b/2a.

Interactive Quiz - Master Complex Numbers

10 MCQs with MathJax; 80%+ goal. Definitions, operations, roots.

Quick Revision Notes & Mnemonics

Concise notes, mnemonics.

Basics

z=a+ib, i^2=-1
Re=a, Im=b
Mnemonic: "Imaginary i Introduces Infinite" (i^3)

i Powers

Cycle: i, -1, -i, 1
n mod 4
Mnemonic: "Icky Negative Ice One" (i,-1,-i,1)

Operations

Add: Parts separate
Mult: FOIL with i^2=-1
Div: Conjugate / |z|^2
Mnemonic: "Add Real Im, Mult Real-Im + i(Cross)"

Modulus/Conj

|z|=√(a^2+b^2)
\bar z = a-ib
Mnemonic: "Modulus Magnitude, Conj Flip Im"

Polar/Argand

θ= tan^{-1}(b/a)
r(cosθ + i sinθ)
Mnemonic: "Arg Angle from Real to Im"

Quadratics

D<0: Complex conjugate roots
Formula applies
Mnemonic: "Disc Negative? Imaginary Pair"

Overall Mnemonic: "Complex i Algebra Mod Argand Quadratic" (CiAM AQ). Flashcards for i cycle.

Formulas & Notations - All Key

Formula/Notation	Description	Example	Usage
$$ i^2 = -1 $$	Imaginary unit	√(-1)=i	Definition
$$ z = a + ib $$	Complex form	3+4i	Standard
$$ \|z\| = \sqrt{a^2 + b^2} $$	Modulus	\|3+4i\|=5	Magnitude
$$ \bar{z} = a - ib $$	Conjugate	\bar{1+i}=1-i	Reflection
$$ z_1 + z_2 = (a+c) + i(b+d) $$	Addition	(1+i)+(1-i)=2	Sum
$$ z_1 z_2 = (ac-bd) + i(ad+bc) $$	Multiplication	(1+i)^2=2i	Product
$$ \frac{z_1}{z_2} = z_1 \frac{\bar{z_2}}{\|z_2\|^2} $$	Division	(1+i)/(1-i)=(1+i)^2 / 2 = i	Quotient
$$ i^{4k+1} = i $$ etc.	Powers of i	i^5=i	Cycle
$$ \arg z = \tan^{-1} \frac{b}{a} $$	Argument	arg(i)=π/2	Angle
$$ z = r (\cos \theta + i \sin \theta) $$	Polar	1+i = √2 cis π/4	Representation
Quadratic roots: $$ \frac{-b \pm \sqrt{D}}{2a} $$	With D=b²-4ac	x²+1=0 → ±i	Solutions

Tip: Memorize modulus, division; use for proofs.

Derivations & Proofs - Solved Step-by-Step

Proof 1: |z1 z2| = |z1| |z2|

Step-by-Step:

Step 1: |z1 z2|^2 = (z1 z2) \bar{(z1 z2)} = z1 z2 \bar z2 \bar z1 = z1 \bar z1 z2 \bar z2 = |z1|^2 |z2|^2
Step 2: Take sqrt: |z1 z2| = |z1| |z2|
Conclusion: Multiplicative. Proof: Conjugate property.

Proof 2: (z1 + z2)^2 = z1^2 + z2^2 + 2 z1 z2

Step-by-Step:

Step 1: Expand left: z1^2 + 2 z1 z2 + z2^2
Step 2: Matches right
Conclusion: Binomial. Proof: Distributive.

Proof 3: \bar{z1 + z2} = \bar z1 + \bar z2

Step-by-Step:

Step 1: \bar{(a+c + i(b+d))} = (a+c) - i(b+d) = (a-ib) + (c-id)
Conclusion: Additive. Proof: Definition.

Proof 4: Square roots of -a = ± √a i

Step-by-Step:

Step 1: (√a i)^2 = a i^2 = -a
Step 2: Similarly for -
Conclusion: Principal positive. Proof: Solve x^2 +a=0.

Proof 5: 1/z = \bar z / |z|^2

Step-by-Step:

Step 1: z ( \bar z / |z|^2 ) = (z \bar z) / |z|^2 = |z|^2 / |z|^2 =1
Conclusion: Inverse. Proof: Multiplication.

Tip: Use conjugates for modulus proofs. Practice: |z1/z2|.