11M05��Complex Numbers and Quadratic Equations
No real roots exist!!
11M05.1
Introduction to Complex Numbers
11M05.1 – Introduction to Complex Numbers
Learning Objectives
Introduction to iota
Powers of iota
Understanding Complex Number
Square Roots of Negative Real Number
Equality of Complex Numbers
5
11M05.1
CV 0
Revision
Natural Numbers
Whole Numbers
Rational Numbers
Irrational Numbers
Integers
Real Numbers
Recall Test
Q. Which of the following is an Irrational number ?
Sol. C
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(Time Duration : 1 Minute)
11M05.1
CV1�Introduction to iota
11M05.1
CV2�Powers of iota
11M05.1
PSV 1
Sol.
Conclusion: Sum of 4 consecutive powers of iota will be always zero.
ConcepTest 1
Ready for a challenge
Sol.
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(Time Duration : 3 Minutes)
ConcepTest 2
Ready for a Challenge
Q. Solve
Sol.
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(Time Duration : 4 Minutes)
11M05.1
CV 3�Understanding Complex Number
Real part
Imaginary Part
Complex Number
Purely Imaginary Number
Purely Real Number
11M05.1
PSV 2
Q. Which of the following is not a Complex number ?
Sol. D
11M05.1
CV 4�Square Roots of Negative Real Number
Conclusion: Do not separate the two negative numbers in
square root while they are in multiplication.
11M05.1
CV 5�Equality of Complex Numbers
Conclusion:Two complex number are equal when their corresponding real and imaginary parts are equal
For Real numbers
For complex numbers
11M05.1
PSV 3
Sol.
ConcepTest 3
Ready for a Challenge
Sol.
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(Time Duration : 2 Minutes)
Summary
11M05.1 – Introduction to Complex Numbers
Reference Questions
NCERT Exercise 5.1 : 1, 2, 3
Work Book : 1, 12
11M05.2
Algebra of Complex Numbers
11M05.2 – Algebra of Complex Numbers
Learning Objectives
Addition and Subtraction of Complex Numbers
Multiplication of Complex Numbers
Division of Complex Numbers
Multiplicative Inverse of Complex Numbers
Identities of Complex Numbers
37
11M05.2
CV 1�Addition and Subtraction of Complex Numbers
Real part added/Subtracted
with real one
Imaginary part added/Subtracted
with imaginary one
Sol.
1. Closure Property
Will be also a Complex Number
Properties of Addition of Complex Numbers
Complex Number
Order doesn’t affect
2. Commutative Property
Properties of Addition of Complex Numbers
Properties of Addition of Complex Numbers
Order doesn’t affect answer
3. Associative Property
4. Existence of Additive Identity
5. Existence of Additive Inverse
Properties of Addition of Complex Numbers
11M05.2
CV 2�Multiplication of Complex Numbers
Real Part
Imaginary Part
Sol.
Properties of Multiplication of Complex Numbers
Complex Number
Will be also a Complex Number
Properties of Multiplication of Complex Numbers
Order doesn’t affect
Properties of Multiplication of Complex Numbers
Order Doesn’t Affect Answer
3. Associative Property
4. Existence of Multiplicative Identity
5. Existence of Multiplicative Inverse
Properties of Multiplication of Complex Numbers
11M05.2
CV 3�Division of Complex Numbers
Sol.
Rationalise it & Separate the Real part and Imaginary part
Rationalising the denominator
11M05.2
PSV 1
Sol.
(NCERT- Ex. 5.1- Q3, Q4,Q5)
Rationalising the denominator
ConcepTest 1
Ready for a Challenge
Q. Solve the following expression and find real and imaginary parts
of the Complex Number
Sol.
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(Time duration : 3 Minutes)
11M05.2
CV 4�Multiplicative Inverse of Complex Numbers
Multiplicative Inverse
Reciprocal
11M05.2
PSV 2
Sol.
Rationalizing the denominator
ConcepTest 2
Ready for a challenge
Sol.
Rationalizing the denominator
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(Time duration : 3 Minutes)
11M05.2
CV 5�Identities of Complex Numbers
Did you know?
For Real Numbers
For Complex Numbers
11M05.2
PSV 3
Sol.
ConcepTest 3
Ready for a Challenge
Q.
Sol.
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(Time duration : 3 Minutes)
Summary
11M05.2 – Algebra of Complex Numbers
Reference Questions
NCERT Exercise 5.1 : 4,5,6,7,8,9,10,11,12,13,14
Work Book : 2,3,7,9,11,14,16,17,18,19
11M05.3
Polar Representation of Complex Numbers
11M05.3 – Polar Representation of Complex Numbers
Learning Objectives
Modulus of Complex Number
Argument of Complex Number
Conjugate of Complex Number
Polar Representation of a Complex Number
11M05.3
CV 1
Modulus of Complex Number
For Complex Number,
Sol.
Properties of Modulus
Triangular Inequality
11M05.3
CV 2
Argument of Complex Number
Abbreviated as Argument of z
or (Amplitude of z)
NO!!!!
Principle Argument
| | | | |
| | | | |
Abbreviated as Argument of z
or (Amplitude of z)
Principle Argument
Sol.
ConcepTest 1
Ready for a Challenge
Sol.
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(Time duration : 4 Minutes)
11M05.3
CV 3
Conjugate of Complex Number
Conjugate of Complex number
Change the sign. of Imaginary part
Sol.
Properties of Conjugate
11M05.3
PSV 1
Sol.
ConcepTest 2
Ready for a Challenge
Sol. a
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(Time duration : 1 Minute)
ConcepTest 3
Ready for a Challenge
Sol.
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(Time duration : 4 Minutes)
11M05.3
CV 4
Argand Plane
Argand Plane
O
Real Axis
Imaginary Axis
Jean Robert Argand
O
0
O
0
O
0
O
0
Quadrant | I | II | III | IV |
| | | | |
arg z | | | | |
O
P
Q
Sol.
11M05.3
CV 5
Polar Form of Complex Number
O
By squaring and adding (1) and (2), we get
Sol.
11M05.3
CV 6
Another Method to find the Polar Form
Sol.
11M05.3
PSV 2
By squaring and adding (1) and (2), we get
Sol.
ConcepTest 4
Ready for a Challenge
Sol.
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(Time duration : 4 Minutes)
11M05.4
Quadratic Equations
11M05.4 – Quadratic Equations
Learning Objectives
Quadratic Equation with Imaginary roots
106
11M05.4�CV 0�Revision
Factorisation
Completing the square method
Quadratic Formula
Recall Test
Using Quadratic formula,
Sol.
11M05.4�CV1�Quadratic Equation with Imaginary roots
Resolved by the inception of complex numbers
Sol.
11M05.4
PSV 1
Rearranging it ,
Using Quadratic formula,
Sol.
11M05.4
PSV 2
Using Quadratic Formula,
Sol.
ConcepTest
Ready for a Challenge
Rearranging it ,
Using Quadratic formula,
Sol.
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(Time duration : 4 Minutes)