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# Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

## Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

### Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

Karnataka Class 11 Commerce Maths UNIT II ALGEBRA : Karnataka Pre-University board is a government body which organizes the higher secondary examination in the state.  The board functions under the Department of Primary & Secondary Education. The board has a total of 1202 government pre-university colleges, 165 unaided Pre-University colleges and about 13 Corporation pre-university colleges.

Each year at Karnataka board of pre-university education, about 10 lakh students enroll in the 2 year Pre-University courses. The courses offered by the Pre-University board Karnataka are Humanities (Arts), Science & Commerce. There are 23 subjects, 11 languages and 50 combinations in the Pre-University curriculum. Further, for the academic year 2010-11, the Karnataka Pr-University examination board has 4,43,185 students in Humanities in I & II PUC, 2,47,421 students in Science in I & II PUC & 2,77,189 students in commerce in I & II PUC.

### Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

1 NUMBER THEORY 08 hrs
2 SETS, RELATIONS AND FUNCTIONS 15 hrs
3 THEORY OF INDICES 04 hrs
4 LOGARITHMS 05 hrs
5 PROGRESSIONS 12 hrs
6 THEORY OF EQUATIONS 12 hrs
7 LINEAR INEQUALITIES 06 hrs
TOTAL 62 Hours

### Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

UNIT II : ALGEBRA
CHAPTER 1 NUMBER THEORY (Total – 8 hrs)
1.0 Introduction
1.1 Natural Numbers
1.2 whole numbers
1.3 Integers
1.4 Odd and Even Numbers
1.5 Prime Numbers
1.6 Composite Numbers
1.7 Fundamental theorem of arithmetic
1.8 Least Common Multiple
1.9 Highest common factor
1.10 Relation between H.C.F and L.C.M
1.11 Finding H.C.F of fractions
1.12 Rational Numbers
1.13 Irrational Numbers
1.14 Real Numbers
1.15 Complex numbers

### Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

CHAPTER 2 SETS, RELATIONS AND FUNCTIONS (Total – 16 hrs)
2.0 Introduction
2.1 Sets
2.2 Methods of describing a set
2.3 Null set
2.4 Singleton set
2.5 Finite set and Infinite sets
2.6 Equal and equivalent sets
2.7 Subset
2.8 Universal Set
2.9 Operation on Sets
2.10 Complement of a set
2.11 Algebra of sets
2.12 Venn diagrams
2.13 Ordered pairs
2.14 Equality of ordered pairs
2.15 Cartesian product pairs
2.16 Worked examples
2.17 Relation
2.18 Domain and range of a relation
2.19 Inverse relation
2.20 Types of relations
2.21 Worked Examples
2.22 Functions
2.23 Domain, co-domain and range
2.24 Different types of functions
2.25 Worked examples

### Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

CHAPTER 3 THEORY OF INDICES (Total – 4Hr)
3.1 Introductions
3.2 Meaning of an
3.3 Laws of Indices

### Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

CHAPTER 4 LOGARITHMS (Total – 6Hrs)
4.1 Introduction
4.2 Definition of logarithm
4.3 Laws of logarithm
4.4 Common Logarithm

### Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

CHAPTER 5 PROGRESSSIONS (Total-12 Hrs)
5.1 Introduction
5.2 Sequences
5.3 Series
5.4 Arithmetic progressions
5.5 nth term of an A.P
5.6 Sum to ‘n’ terms of an A.P
5.7 Geometric progression
5.8 nth term of G.P
5.9 Sum to n terms of G.P
5.10 Sum to infinite G.P
5.11 Harmonic progression
5.12 nth term of H.P
5.13 Arithmetic, Geometric and harmonic means

### Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

CHAPTER 6 THEORY OF EQUATIONS (Total – 12 Hrs)
6.1 Introduction and definition of equation
6.2 Degrees of the equation and different types of equations
6.3 Linear equation in one variable
6.4 Simultaneous linear equation in two variables and different methods
6.5 Quadratic equation and its solution
6.6 Nature of the roots of quadratic equation
6.7 Cubic equation, examples and solution
6.8 Synthetic division

### Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

CHAPTER 7 LINEAR INEQUALITIES (Total – 6 Hrs)
7.1 Introduction
7.2 Inequalities
7.3 Linear inequalities in one variable
7.4 System of linear inequations in one variable
7.5 Application of Linear inequalities
7.6 Linear inequalities in two variable
7.7 System of Linear Inequations in two variables and their graphical solution

### Karnataka Class 11 Commerce Maths UNIT II ALGEBRA

NUMBER THEORY

Introduction:
In this chapter we shall study about the numbers and their properties and applications.

Natural numbers:

Counting numbers 1, 2, 3, 4, 5,…… are called natural numbers. They are also called positive integers N = { 1, 2, 3, 4, …… }

Whole numbers:

All the natural numbers together with zero (0) form the set of whole numbers.
If 0 is added to any natural number ‘n’ then it gives the same number n
0 + n = n + 0 = n ∀ n ∈ N
W = { 0, 1, 2, 3 4, 5 …… }

Integers:

The set of all positive integers, negative integers together with 0 is called the set of integers and is denoted by Z
Z = {. -3, -2, -1, 0, 1, 2, 3 …….}

Odd and even integers:

The integers 1, 3, 5, 7 ….. are called odd integers.
The integers 0, 2, 4, 6 ….. -2, -4, -6 ….. are called even integers

Note:

i) 1, 2, 3, 4 … are called positive integers
ii) -1, -2, -3, -4 …. are called negative integers
iii) Zero (0) is neither positive nor negative.
We also assume that it is neither even nor odd.

Prime numbers:

An integer p > 1 is said to be a prime number if it has no other divisors except one and itself. Eg : 2, 3, 5, 7, 11, 13………

Note:

i) 2 is the only even prime number
ii) Prime numbers are infinite

Composite numbers:

An integer which is not a prime number is called a composite number
Eg : 4, 6, 8, 9, 10, 12, 14 ……
A composite number n has a divisor other than + 1 and + n

Note:

0 and 1 are neither prime nor composite

Fundamental theorem of Arithmetic.

Every composite number can be expressed as a product of primes and this decomposition is unique. Apart from
the order in which the prime factors occurs.
For Eg. i) The factors of 35 are 5 and 7
∴ 35 = 51 × 71
ii) The prime factors of 24 are 24 = 23 × 31
iii) The prime factors of 28 are 28 = 22 × 71

This is called PRIME FACTORIZATION or CANONICAL REPRESENTATION.

The above theorem can be applied to find out
i) Number of positive divisors of a number
ii) The sum of all the positive divisors of a number.

Let n be a composite number. We know that any composite number can be expressed as a product of Primes
Let n = P1α1 ,  P1α2 ,  P1α3 , P1α4 …………………………………….  Pnαn where P1, P2, P3, P4,….. Pn are distinct primes.
Let T(n) denote the number of positive divisors of n and S(n) denote the sum of all positive divisors of n. Then we have
T(n) = (1 +α1) (1 + α2 ) …………(1 + αn)
S(n) =  P1α1+1 – 1 / P1 – 1  ×   P2α2+1 – 1 / P2 – 1   ×   P3α3+1 – 1 / P3 – 1   × ………………………………..×   Pnαn+1 – 1 / Pn – 1

Least Common Multiple

The least number which is exactly divisible by each one of the given numbers is called their LCM.
Rules to find LCM

Rule I : To find the LCM by Prime factorization method.
Express each number as the product of primes then LCM = Product of highest powers of all the factors.

Highest Common Factor :

(HCF) or (Greatest Common Divisor GCD) or(Greatest common measure)

Definition: An integer d is called as the GCD or HCF of 2 integers ‘a’ and ‘b’ (both of them are not zero) if
i) d|a and d|b
ii) Every common divisor of ‘a’ and ‘b’ divides ‘d’
i.e. x|a, x|b ⇒ x|d usually GCD of ‘a’ and ‘b’ is denoted by d = (a, b)
For e.g. HCF of 8 and 16 is 8 and is denoted by (8, 16) = 8

Note:

i) HCF of 2 numbers is a unique positive integer.

ii) If (a, b) = d then d = (-a, b) = (a, -b) = (-a, -b) = a positive integer.

Relation between LCM and HCF

If A and B are two numbers then the product of their LCM and HCF is equal to the product of two numbers.

LCM × HCF = A × B

⇒ HCF = A × B / LCM

⇒ LCM = A × B / HCF

To find the HCF of fractions

If a/b, c/d, e/f are the proper fractions, the HCF is given by

HCF = HCF of numerators a c e / LCM of denominators b d f

Rational numbers:

A number in the simplest form p / q where p and q are integers and q ≠ 0 is called a reational number. The set of all rational numbers is denoted by Q.

Eg : 4/7, 3/5, -2/3 etc

Note:
1. Every integer is a rational number
2. Zero (0) is a rational number since 0 = 0/1 is of the form p/q, where p and q are integer and q ≠ 0.

3. Square root of a positive integer, which is a perfect square is rational number:
Eg: √16 = 4, √25 = 5

Rational number in lowest terms

A rational number of the form p/q where q ≠ 0 is said to be in the lowest form if the integers p and q have no other common factors other than 1 i.e., if p and q are co-prime or HCF of p and q is 1

Eg : 7/14 is a rational number but not in the lowest form. But 7/14 = 1/2 is a rational number in lowest form.

All rational numbers when expressed in decimal form are either terminating decimals or recurring decimals.

Recurring decimal:

A decimal representation in which all the digits after a certain stage are repeated is called a recurring decimal.
For Eg : 1/3 = 0.33333 …….. = 0.3‾

Mixed Recurring decimal :

A decimal in which some digits after the decimal point is not repeated and then some digit or digits are repeated is called a ‘Mixed recurring decimal’
For example : 6.12555 = 6.125¯

Note:
1. Let x = p/q be a rational number such that prime factorization of q is of the form 2n × 5m . Then decimal expansion of p/q terminates.
2. If p/q is a rational number such that prime factorization of q is not of the form 2n × 5m. Then decimal expansion of p/q is non-terminating.

Irrational numbers:

A number which cannot be put in the form of p/q where p and q are integers and q  0 is called as irrational number.
i.e. a number which is not rational is irrational.
Eg: 2, 3, 5, 7 etc. are irrational.

Note:
1. Negative of an irrational number is an irrational number eg.: – 2
2. Sum of rational and irrational number is always an irrational number.
Eg. 2 + √3
3. Product of a non-zero rational number and an irrational number is irrational.

Eg. 2/3 × (√3 + 1).

DEFINITION OF A SURD:

An irrational root of a rational number is called a surd.
For eg.: √2, √3, √5 + 3, 3 +√ 7….

TRANSCENDENTAL NUMBER

An irrational number which is not a surd is called a transcendental number. π is a transcendental number.

Real Numbers:

Rational numbers and irrational numbers taken together are said to form the set of Real numbers. The set of real numbers is denoted by R.
i.e. A real number is either rational or irrational R = Q U Q’ where Q is set of rational and Q’ is the set of irrationals.

Every real number (whether rational or irrational) can be represented by a unique point on the number line called real number line and conversely.

Properties of Real numbers:
1. For any real number a, only one of the following is true.
i) a > 0 or ii) a < 0 or iii) a = 0
2. For any two real numbers a and b only one of the following is true.
i) a < b or ii) a > b or iii) a = b
3. If a, b, c are real numbers such that a > b, b > c then a > c
4. If a and b are any two real numbers such that ab = 0 then a = 0 or b = 0
5. If a, b, c, are any three real numbers then
1) a + b = b + a Commutative laws
a. b = b. a
2) a + (b + c) = (a + b) + c
a. (b.c) = (a. b). c Associative laws
3) a .(b + c) = a.b + a.c (Distributive law)
6. Square of a real number is always positive.
For eg. (-9)2 = 81
(5)2 = 25

Complex numbers:

A number of the form a + ib where a, b ∈ R, the set of real numbers and i = √-1 is called as a complex number. Where ‘a’ is called as real part and ‘b’ is called the imaginary part.
Example: 1 + 6i, 3 – 4i, 2 + √3i etc. are complex numbers.

Algebra of complex numbers:

Addition and subtraction of two complex numbers:

Definition:

If z1 = a + ib and z2 = c + id are two complex numbers, then z1 + z2 = (a + ib) + (c +id) is the complex number = (a + c) + i (b + d);
z1 – z2 = (a + ib) – (c + id) = (a – c) + i (b – d).

Eg. z1 = 4 + 5i , z2 = 1 + 7i.

i) z1 + z2 = 5 + 12i.

ii) z1 – z2 =3 – 2i.

MULTIPLICATION OF TWO COMPLEX NUMBERS.

If z1 = a + ib and z2 = c + id are two complex numbers then z1 z2 = (a + ib) (c + id) = (ac  bd) + i (ad + bc)

Example: (2 + 3i) (4 + 3i) = (8 – 9) + i (18) = -1 + 18i.

CONJUGATE OF A COMPLEX NUMBER

If z = a + ib is a complex number then a – ib is called as its conjugate and is denoted by z .

Example if z = 2 + 3i then z¯ = 2 – 3i

If z = 5 – 4i then z¯ = 5 + 4i

DIVISION OF A COMPLEX NUMBER BY A NON-ZERO COMPLEX NUMBER.

Let z1 = a + ib and z2 =c + id be two complex numbers

⇒ z1/z2 = a + ib/c + id.

⇒  a + ib/c + id = c – id/ c – id.

[ ac + bd/c2 + d2  ] + i[bc – ac/c2 + d2].

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