Thursday 27 May 2021

Ch-6 Notes class -11th

 

Thursday 20 May 2021

class - 11th chapter - 5 NOTES COMPLEX NUMBER

 NOTES  OF COMPLEX NUMBER

Class 11 Maths Notes Chapter 5 Complex Numbers and Quadratic Equations

Imaginary Numbers
The square root of a negative real number is called an imaginary number, e.g. √-2, √-5 etc.
The quantity √-1 is an imaginary unit and it is denoted by ‘i’ called Iota.

Integral Power of IOTA (i)
i = √-1, i2 = -1, i3 = -i, i4 = 1
So, i4n+1 = i, i4n+2 = -1, i4n+3 = -i, i4n = 1

Note:

  • For any two real numbers a and b, the result √a × √b : √ab is true only, when atleast one of the given numbers i.e. either zero or positive.
    √-a × √-b ≠ √ab
    So, i2 = √-1 × √-1 ≠ 1
  • ‘i’ is neither positive, zero nor negative.
  • in + in+1 + in+2 + in+3 = 0

Complex Number
A number of the form x + iy, where x and y are real numbers, is called a complex number, x is called real part and y is called imaginary part of the complex number i.e. Re(Z) = x and Im(Z) = y.

Purely Real and Purely Imaginary Complex Number
A complex number Z = x + iy is a purely real if its imaginary part is 0, i.e. Im(z) = 0 and purely imaginary if its real part is 0 i.e. Re (z) = 0.

Equality of Complex Number
Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal, iff x1 = x2 and y1 = y2 i.e. Re(z1) = Re(z2) and Im(z1) = Im(z2)
Note: Order relation “greater than’’ and “less than” are not defined for complex number.

Algebra of Complex Numbers
Addition of complex numbers
Let z1 = x1 + iy1 and z2 = x2 + iy2 be any two complex numbers, then their sum defined as
z1 + z= (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i (y1 + y2)

Properties of Addition

  • Commutative: z1 + z2 = z2 + z1
  • Associative: z1 + (z2 + z3) = (z1 + z2) + z3
  • Additive identity z + 0 = z = 0 + z
    Here, 0 is additive identity.

Subtraction of complex numbers
Let z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex numbers, then their difference is defined as
z1 – z2 = (x1 + iy1) – (x2 + iy2) = (x1 – x2) + i(y1 – y2)

Multiplication of complex numbers
Let z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex numbers, then their multiplication is defined as
z1z2 = (x1 + iy1) (x2 + iy2) = (x1x2 – y1y2) + i (x1y2 + x2y1)

Properties of Multiplication

  • Commutative: z1z2 = z2z1
  • Associative: z1(z2z3) = (z1z2)z3
  • Multiplicative identity: z . 1 = z = 1 . z
    Here, 1 is multiplicative identity of an element z.
  • Multiplicative inverse: For every non-zero complex number z, there exists a complex number z1 such that z . z1 = 1 = z1 . z
  • Distributive law: z1(z2 + z3) = z1z2 + z1z3

Division of Complex Numbers
Let z1 = x1 + iy1 and z2 = x2 + iy2 be any two complex numbers, then their division is defined as
Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 5

Conjugate of Complex Number
Let z = x + iy, if ‘i’ is replaced by (-i), then said to be conjugate of the complex number z and it is denoted by z¯, i.e. z¯ = x – iy

Properties of Conjugate
Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 5

Modulus of a Complex Number
Let z = x + iy be a complex number. Then, the positive square root of the sum of square of real part and square of imaginary part is called modulus (absolute values) of z and it is denoted by |z| i.e. |z| = x2+y2
It represents a distance of z from origin in the set of complex number c, the order relation is not defined
i.e. z1 > z2 or z1 < z2 has no meaning but |z1| > |z2| or |z1|<|z2| has got its meaning, since |z1| and |z2| are real numbers.

Properties of Modulus of a Complex number
Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 5

SOLUTIONS OF CH -5

https://www.learncbse.in/ncert-solutions-for-class-11th-maths-chapter-5-complex-numbers-and-quadratic-equations/


Sunday 9 May 2021

class 11th math introduction of sets (NOTES AND PPT)

 

SETS-  COLLECTION OF WELL DEFINED OBJECTS.

 SELECT THE BELOW LINK AND RIGHT THEN go to http  

https://www.slideshare.net/ASHadventurelover/introduction-to-sets

Notes Ch - 1 

Set
A set is a well-defined collection of objects.

Representation of Sets
There are two methods of representing a set

  • Roster or Tabular form In the roster form, we list all the members of the set within braces { } and separate by commas.
  • Set-builder form In the set-builder form, we list the property or properties satisfied by all the elements of the sets.

Types of Sets 

  • Empty Sets: A set which does not contain any element is called an empty set or the void set or null set and it is denoted by {} or Φ.
  • Singleton Set: A set consists of a single element, is called a singleton set.
  • Finite and infinite Set: A set which consists of a finite number of elements, is called a finite set, otherwise the set is called an infinite set.
  • Equal Sets: Two sets A and 6 are said to be equal, if every element of A is also an element of B or vice-versa, i.e. two equal sets will have exactly the same element.
  • Equivalent Sets: Two finite sets A and 6 are said to be equal if the number of elements are equal, i.e. n(A) = n(B)

Subset 

A set A is said to be a subset of set B if every element of set A belongs to set B. In symbols, we write
A ⊆ B, if x ∈ A ⇒ x ∈ B

Note:

  • Every set is o subset of itself.
  • The empty set is a subset of every set.
  • The total number of subsets of a finite set containing n elements is 2n.

Intervals as Subsets of R
Let a and b be two given real numbers such that a < b, then

  • an open interval denoted by (a, b) is the set of real numbers {x : a < x < b}.
  • a closed interval denoted by [a, b] is the set of real numbers {x : a ≤ x ≤ b}.
  • intervals closed at one end and open at the others are known as semi-open or semi-closed interval and denoted by (a, b] is the set of real numbers {x : a < x ≤ b} or [a, b) is the set of real numbers {x : a ≤ x < b}.

Power Set
The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). If the number of elements in A i.e. n(A) = n, then the number of elements in P(A) = 2n.

Universal Set
A set that contains all sets in a given context is called the universal set.

Venn-Diagrams
Venn diagrams are the diagrams, which represent the relationship between sets. In Venn-diagrams the universal set U is represented by point within a rectangle and its subsets are represented by points in closed curves (usually circles) within the rectangle.

Operations of Sets
Union of sets: The union of two sets A and B, denoted by A ∪ B is the set of all those elements which are either in A or in B or in both A and B. Thus, A ∪ B = {x : x ∈ A or x ∈ B}.

Intersection of sets: The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements which are common to both A and B.
Thus, A ∩ B = {x : x ∈ A and x ∈ B}

Disjoint sets: Two sets Aand Bare said to be disjoint, if A ∩ B = Φ.

Intersecting or Overlapping sets: Two sets A and B are said to be intersecting or overlapping if A ∩ B ≠ Φ

Difference of sets: For any sets A and B, their difference (A – B) is defined as a set of elements, which belong to A but not to B.
Thus, A – B = {x : x ∈ A and x ∉ B}
also, B – A = {x : x ∈ B and x ∉ A}

Complement of a set: Let U be the universal set and A is a subset of U. Then, the complement of A is the set of all elements of U which are not the element of A.
Thus, A’ = U – A = {x : x ∈ U and x ∉ A}

Some Properties of Complement of Sets

  • A ∪ A’ = ∪
  • A ∩ A’ = Φ
  • ∪’ = Φ
  • Φ’ = ∪
  • (A’)’ = A

Symmetric difference of two sets: For any set A and B, their symmetric difference (A – B) ∪ (B – A)
(A – B) ∪ (B – A) defined as set of elements which do not belong to both A and B.
It is denoted by A ∆ B.
Thus, A ∆ B = (A – B) ∪ (B – A) = {x : x ∉ A ∩ B}.

Laws of Algebra of Sets

Idempotent Laws: For any set A, we have

  • A ∪ A = A
  • A ∩ A = A

Identity Laws: For any set A, we have

  • A ∪ Φ = A
  • A ∩ U = A

Commutative Laws: For any two sets A and B, we have

  • A ∪ B = B ∪ A
  • A ∩ B = B ∩ A

Associative Laws: For any three sets A, B and C, we have

  • A ∪ (B ∪ C) = (A ∪ B) ∪ C
  • A ∩ (B ∩ C) = (A ∩ B) ∩ C

Distributive Laws: If A, B and Care three sets, then

  • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
  • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

De-Morgan’s Laws: If A and B are two sets, then

  • (A ∪ B)’ = A’ ∩ B’
  • (A ∩ B)’ = A’ ∪ B’

Formulae to Solve Practical Problems on Union and Intersection of Two Sets
Let A, B and C be any three finite sets, then

  • n(A ∪ B) = n(A) + n (B) – n(A ∩ B)
  • If (A ∩ B) = Φ, then n (A ∪ B) = n(A) + n(B)
  • n(A – B) = n(A) – n(A ∩ B)
  • n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)