Ch-6 Notes class -11th

 

Linear Inequalities class 11 Notes Mathematics

1. Inequalities – Algebraic Solutions and Graphical Representation
2. Graphical Solution of Linear Inequalities
3. Solution of System of Linear Inequalities

• Inequalities: Two real numbers or two algebraic expressions related by the symbols <, >, ≤ or ≥ form an inequality. For example: 3x<20, 4x+y<12.
• Equal numbers may be  added to (or subtracted from) both sides of an inequality.
• Both sides of an inequality can be  multiplied (or divided) by the same  positive number. But when both sides are multiplied (or divided) by a negative number, then the inequality is reversed.
• Linear Inequality: An inequality is said to be linear, if each variable occurs in first degree only and there is no term involving the product of the variables.   .
• Closed Interval: If  and  are real numbers, such that  then the set of all real numbers  such that  is called a closed interval and is denoted by R. Therefore, .
• Open Interval: If  and  are real numbers, such that  then the set of all real numbers  such that  is called a open interval and is denoted by  or . Therefore, .
• Solution of an Inequality: The values of x, which make an inequality a true statement, are called solutions of the inequality.
• To represent x <  a (or x >  a) on a number line, put a circle on the number and dark line to the left (or right) of the number a.
• To represent x ≤ a (or  x ≥ a) on a number line, put a dark circle on the number and dark the line to the left (or right) of the number x.
• If an inequality is having ≤ or ≥ symbol, then the points on the line are also included in the solutions of the inequality and the graph of the inequality lies left (below) or right (above) of the graph of the equality represented by dark line that satisfies an arbitrary point in that part.
• If an inequality is having < or > symbol, then the points on the line are not included in the solutions of the inequality and the graph of the inequality lies tothe left (below) or right (above) of the graph of the corresponding equality represented by dotted line that satisfies an arbitrary point in that part.
• To remove the denominator when the sign of the value of the denominator is unknown, may be +ve or -ve: We multiply by the square of the denominator. Square of the denominator is always positive. On multiplication of an inequation by a positive number the sign of inequality does not change.
• Solution of a system of Linear Inequality: The solution set of a system of linear inequality in one variable is defined as the intersection of the solution sets of the linear inequality of the system. For example: If we have two solution sets  and  then solution of the system is the intersection of  and  i.e., 
• Properties of absolute values:
  (i)  i.e., 
  (ii)  i.e., 
  (iii)  or  i.e., 
  (iv)  or  i.e., 
• Triangle Inequality: , 

To find plane containing the origin:
– If the origin satisfies the inequality of the half of the half plane, then the half plane contains the origin. This inequality represents the half plane containing origin.
– If the origin does not satisfy the inequality of the half of the half plane, then the half plane does not contain the origin. This inequality represent the half plane, which does not contain the origin.
– Shading: A line divides the plane in two half planes. We check whether the given half plane contains the origin. If so, then the inequality represents the half plane containing the origin and we shade the half plane.
– If the origin does not satisfy the inequality, then the inequality represents the other half plane not containing origin and we shade this half plane.
– Graphical Solution of Linear Inquality in two variables: After identification of the half plane, we shade the half planes of the given inequality. The points of the enclosed area of the shaded portion are the solution of the given system of inequality.

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 + z2 = (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

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

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

SOLUTIONS OF CH -5

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

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)