*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 = 5 ^{1} × 7^{1}*

*ii) The prime factors of 24 are 24 = 2*

^{3}× 3^{1}*iii) The prime factors of 28 are 28 = 2*

^{2}*×*7^{1}*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 = P _{1}^{α1}* ,

*P*,

_{1}^{α2}*P*,

_{1}^{α3}*P*…………………………………….

_{1}^{α4}*P*

_{n}^{αn }*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 +α*

_{1}) (1 + α_{2}) …………(1 + α_{n})*S(n) = P*– 1 ×

_{1}^{α1+1}– 1 / P_{1}*P*– 1 ×

_{2}^{α2+1}– 1 / P_{2}*P*– 1 × ………………………………..×

_{3}^{α3+1}– 1 / P_{3}*P*– 1

_{n}^{αn+1}– 1 / P_{n}**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 prod*uct 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 2 ^{n} × 5^{m} . 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 2*

^{n}× 5^{m}. 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 z _{1} = a + ib and z_{2} =c + id be two complex numbers*

*⇒ z _{1/z2 = }*

*a + ib/c + id.*

_{⇒ }*a + ib/c + id = c – id/ c – id.*

⇒ *[ ac + bd/c ^{2} + d^{2 } ] + i[bc – ac/c^{2} + d^{2}].*

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