*Karnataka Class 11 Commerce Maths Sets*

*Karnataka Class 11 Commerce Maths Sets*

*Karnataka Class 11 Commerce Maths Sets : 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.*

*The theory of sets was developed by German Mathematician George Cantor. He is regarded as the father of set theory. It is proved to be of great importance in the foundation of relations and functions, sequences, Geometry, Probability theory etc. Also it has wide application in logic and philosophy.*

*Karnataka Class 11 Commerce Maths Sets*

**Design of the Question Paper**

**MATHEMATICS CLASS : I PUC**

**Time: 3 hours 15 minute; Max. Mark:100**

*The weightage of the distribution of marks over different dimensions of the question paper shall be as follows:*

*Karnataka Class 11 Commerce Maths Sets*

**I. Weightage to Objectives:**

Objective | Weightage | Marks |

Knowledge | 40% | 60/150 |

Understanding | 30% | 45/150 |

Application | 20% | 30/150 |

Skill | 10% | 15/150 |

*Karnataka Class 11 Commerce Maths Sets*

**II. Weightage to level of difficulty:**

Level | Weightage | Marks |

Easy | 35% | 53/150 |

Average | 55% | 82/150 |

Difficult | 10% | 15/150 |

*Karnataka Class 11 Commerce Maths Sets*

**III. Weightage to content:**

Chapter No. | Chapter | No. of teaching Hours | Marks |

1. | SETS | 8 | 8 |

2. | RELATIONS AND FUNCTIONS | 10 | 11 |

3. | TRIGONOMETRIC FUNCTIONS | 18 | 19 |

4. | PRINCIPLE OF MATHEMATICAL INDUCTION | 4 | 5 |

5. | COMPLEX NUMBERS AND QUADRATICEQUATIONS | 8 | 9 |

6. | LINEAR INEQUALITIES | 8 | 7 |

7. | PERMUTATION AND COMBINATION | 9 | 9 |

8. | BINOMIAL THEOREM | 7 | 8 |

9. | SEQUENCE AND SERIES | 9 | 11 |

10. | STRAIGHT LINES | 10 | 10 |

11. | CONIC SECTIONS | 9 | 9 |

12. | INTRODUCTION TO 3D GEOMETRY | 5 | 7 |

13. | LIMITS AND DERIVATIVES | 14 | 15 |

14. | MATHEMATICAL REASONING | 6 | 6 |

15. | STATISTICS | 7 | 7 |

16. | PROBABILITY | 8 | 9 |

| Total | 150 | 150 |

*Karnataka Class 11 Commerce Maths Sets*

**IV. Pattern of the question paper:**

PART | Type of questions | Number of questions to be set | Number of questions to be answered | Remarks |

A | 1 mark questions | 10 | 10 | Compulsory part |

B | 2 mark questions | 14 | 10 | —————— |

C | 3 mark questions | 14 | 10 | —————— |

D | 5 mark questions | 10 | 6 | Questions must be asked from the specific set of topics as mentioned below, under section V |

E | 10 mark questions
| 2 | 1 |

*Karnataka Class 11 Commerce Maths Sets*

**GUIDELINES TO THE QUESTION PAPER SETTER**

*1. The question paper must be prepared based on the individual blue print without changing the weightage of marks fixed for each chapter.*

*2. The question paper pattern provided should be adhered to.*

*Part A : 10 compulsory questions each carrying 1 mark;**Part B : 10 questions to be answered out of 14 questions each carrying 2 mark ;**Part C : 10 questions to be answered out of 14 questions each carrying 3 mark;**Part D : 6 questions to be answered out of 10 questions each carrying 5 mark**Part E : 1 question to be answered out of 2 questions each carrying 10 mark with subdivisions (a) and (b) of 6 mark and 4 mark respectively. (The questions for PART D and PART E should be taken from the content areas as explained under section V in the design of the question paper)*

*3. There is nothing like a single blue print for all the question papers to be set. The paper setter should prepare a blue print of his own and set the paper accordingly without changing the weightage of marks given for each chapter.*

*4. Position of the questions from a particular topic is immaterial.*

*5. In case of the problems, only the problems based on the concepts and exercises discussed in the text book (prescribed by the Department of Pre-university education) can be asked. Concepts and exercises different from text book given in Exemplar text book should not be taken. Question paper must be within the frame work of prescribed text book and should be adhered to weightage to different topics and guidelines.*

*6. No question should be asked from the historical notes and appendices given in the text book.*

*7. Supplementary material given in the text book is also a part of the syllabus.*

*8. Questions should not be split into subdivisions. No provision for internal choice question in any part of the question paper.*

*9. Questions should be clear, unambiguous and free from grammatical errors. All unwanted data in the questions should be avoided.*

*10. Instruction to use the graph sheet for the question on LINEAR PROGRAMMING in PART E should be given in the question paper.*

*11. Repetition of the same concept, law, fact etc., which generate the same answer in different parts of the question paper should be avoided.*

*Karnataka Class 11 Commerce Maths Sets*

**UNIT I: SETS AND FUNCTIONS**

**1. Sets**

*Sets and their representations:**Definitions, examples, Methods of Representation in roster and rule form, examples**Types of sets: Empty set. Finite and Infinite sets. Equal sets. Subsets.**Subsets of the set of real numbers especially intervals (with notations).**Power set.**Universal set. examples**Operation on sets: Union and intersection of sets. Difference of sets. Complement of a set,**Properties of Complement sets. Simple practical problems on union and intersection of two sets.**Venn diagrams: simple problems on Venn diagram representation of operation on sets.*

*Karnataka Class 11 Commerce Maths Sets*

**Sets:**

*A set is a well defined collection of distinct objects. Each member is called the element of the set.*

**Note:**

*1. A set is always represented by capital letters*

*2. If a is an element of set A then we write a ∈ A.*

*3. If b is not an element of set A then we write b ∉ A*

**Examples:**

*1. The set of boys in class V ^{th}A.*

*2. The set of even natural numbers*

*3. The set of days of a week*

*4. The set of vowels in the English alphabet.*

*Karnataka Class 11 Commerce Maths Sets*

**Methods of describing a set:**

*A set can be represented in two forms*

*1. Roster form or Tabular form*

*2. Set builder form or rule form*

**Roster Form:**

*In the roster Form, all the elements are listed and separated by commas and are enclosed within brackets.*

*A = The set of all even numbers between 0 and 10*

*Roster Method given by A = {2, 4, 6, 8}*

**Set builder Form:**

*In this method all the elements of a set possess a single common property, which is not possessed by any element outside the set.*

*If A = {1, 2, 3, 4, 5} then the set builder form is represented by A = { x : x N and x < 6}.*

*Karnataka Class 11 Commerce Maths Sets*

**Null set or Empty set:**

*A set containing no elements is called an empty set.*

*It is denoted by Φ or { }*

*For eg.: A = {The set of all even prime numbers other than 2 }*

*A = Φ or { }*

*2. A = set of all n*atural numbers < 0

A = *Φ * or{ }

**Singleton set:**

*A set containing only one element is called a singleton set.*

*Eg. 1. A = { x : x – 1 = 0 , x ∈ N}*

*A = { 1 }*

*2. B = {x : x is an even prime number}*

*B = { 2 }*

*Karnataka Class 11 Commerce Maths Sets*

**Finite set and infinite set:**

*A set is called a finite set if it contains finite numbers of elements.*

*Example*

*1) A = {1, 2, 3}*

*n(A) = 3*

*2) B = {set of prime numbers < 9}*

*B = {2, 3, 5,7}*

*n(B) = 4*

*A set which is not finite is called an infinite set.*

**Examples**

*1. The set of natural numbers.*

*2. The set of real numbers.*

*Karnataka Class 11 Commerce Maths Sets*

*Equal and Equivalent sets:*

*Equal Sets: Two sets A and B are said to be equal if they have exactly the **same elements.*

*Ex.*

*1). A = {1, 3, 8}*

*B = {8, 3, 1}*

*Then A = B as A and B have the same elements.*

*2). A = { x : x is a letter in the word ‘flow’}*

*B = { x : x is a letter in the word ‘wolf’)*

*Then A = B as A and B have the same elements.*

*Equivalent sets:*

*Two finite sets A and B are said to be equivalent if they **have the same cardinal number i.e. if the same number of elements. i.e. if*

*n (A) = n (B).*

*Let A = {a, e, i, o, u}*

*B = {1, 2, 3, 4, 5}*

*Then n (A) = 5 and n (B) = 5*

*⇒ the sets A and B are equivalent.*

**Subset:**

*If each and every element of A is an element of B, then A is called a subset of B or A is contained in B. We write A B.*

**Example** ** 1**

*A = {1, 2}*

*B = {1, 2, 3, 4}*

*A ⊂ B*

**Note:**

*1. If atleast one element of A does not belong to set B then A is not a subset of B. It is symbolically represented by A⊆B*

*2. Every set A is a subset of itself i.e. A⊆A.*

*3. Φ is a subset of every set.*

*4. If A⊆B and B⊆ A then A = B*

**Example 2**

*Set of Natural Numbers ⊆ set of whole numbers*

*Karnataka Class 11 Commerce Maths Sets*

**Super Set:**

*Set A and B are two non empty sets such that A is contained in B and A ≠ B then B is called the super set of A*

*It is symbolically represented by B ⊃A*

**EXAMPLE :** Set of complex numbers is a super set of set of real numbers.

*Karnataka Class 11 Commerce Maths Sets*

*Operation on Sets:*

*a) Union of Sets:*

*Let A and B be any two sets. Then the union of A and **B denoted by A∪ B is defined to be the set of all those elements. which **are in A or in B or in both.*

*Examples:*

*1. Let A = {a, b, c} B = {c, d, e, f}*

*∴ A∪ B = {a, b, c, d, e, f}*

*2. A = {1, 2, 3, 4, 5}*

*B = { 1, 2, …………9}*

*A∪ B = {1, 2, 3 …… ..9}*

**Note:**

*1) A ∪ A = A*

*2) A∪ Ø = A*

*3) Ø∪ Ø = Ø*

*4) If A ⊆ B then A∪ B = B*

*b) Intersection of sets:*

*Let A and B be any two sets. Then the intersection **of A and B denoted by A ∩ B is defined to be the set of all common **elements between A and B*

*Example:*

*1. Let A = {a, b, c, d}*

*B = {c, d, e, f, g, h}*

*A ∩ B = {c,d}*

*2. Let A = {1, 2, 3, 4, 5}*

*B = {1, 2, 3, ….. 9}*

*A ∩ B = {1, 2, 3, 4, 5} = A itself*

**Note**:

*1. A ∩ A = A*

*2. A ∩ Ø = Ø*

*3 Ø ∩ Ø = Ø*

*4. It A ⊆ B then A ∩ B = A itself.*

**c) Difference between any two sets :**

*Let A and B be any two sets. Then **the difference A-B is defined to be the set of all those elements of A **which are not in B.*

*It is also called the complement of B w.r.t. A. Similarly B – A is defined **to be the set of all those elements of B which are not in A. It is also **called the complement of A w.r.t. B.*

*Example:*

*1. Let A = {a, b, c, d}*

*B = {d, e, f, g, h, i}*

*A – B = {a, b, c}*

*B – A = {e, f, g, h, i}*

*Function is a special type of relation. Each function is a relation but each relation is not a function. In this lesson we shall discuss some basic definitions and operations involving sets, Cartesian product of two sets, relation between two sets, the conditions under when a relation becomes a function, different types of function and their properties.*

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