KENDRIYA VIDYALAYA SANGATHAN RAIPUR REGION

MODULES

for

CLASS-XI

MATHEMATICS

Session -2016-17

Class XI MLLs

INDEX

Module No.

Chapter

Marks (Weightage)

Page No.

1.

Sets

9

3

2.

Relations and Functions

5

6

3.

Principle of Mathematical induction

6

8

4.

Complex Numbers

4

9

5.

Linear Inequalities

6

10

6.

Binomial Theorem

6

11

7.

Conic Section

4

12

8.

Introduction to 3-dimentional Geometry

4

15

9.

Limits and Derivatives

6

16

10.

Mathematical Reasoning

3

18

11.

Statistics

6

20

12.

Probability

4

23

13.

Blue Print Half Yearly Examination

100

25

14.

Blue Print Session Ending Examination

100

26

15.

Things To Remember

27

Module 1

SETS

Set: Collection of well-defined and distinct objects is called as Set.

Sets are denoted by capital letters of English alphabets.

Remarks:

(i) Here well-defined means, the statement which contains no adjective.

(ii) Small letter represents the elements or members of a set.

e.g.(i) Collection of best singers in India. It is not a set as it contains adjective best.

(ii) Collection of singers in India. It is a set.

S.No.

Symbol

Meaning

Example

1.

Belongs to

2.

Does not belongs to

3.

There exists

4.

For all

5.

Union

6.

Intersection

7.

Difference

8.

Symmetric difference

9.

Complement

10.

Empty Set

11.

Subset

12.

Set of natural numbers

{1, 2, 3, }

13.

Set of whole numbers

{0, 1, 2, 3, }

14.

Set of integers

15.

Set of all positive integers

16.

Set of all negative integers

17.

Set of rational numbers

18.

Irrational Numbers

Numbers which cannot be written as

,

e.g ,

19.

Set of real numbers

All rational and irrational numbers

20.

Set of all positive real numbers

21.

Set of all negative real numbers

22.

Brackets used to enclose the elements of a set

23.

n(A)

Cardinal Number

Number of elements in a set.

Representation of Set

Roster / Tabular Form

Set Builder Form

In this representation, elements are listed in curly braces separated by commas.

For Example,

In this representation common property of elements is defined.

For example,

A = Collection of natural numbers less than 6.

or

Types of Set:

S. No.

Type of Set

Meaning

Example

1

Empty Set, Null Set, Void Set

{}

A set which has no element.

2

Singleton Set

A set containing only one element.

A= set of vowels in the word SET = {E}

3

Subset

Set A is said to be subset of set B if all the elements of A are in B.

Set of vowels is subset of set of Alphabets.

3

Super Set

Set B is said to be subset of set A if B contains all the elements of A

Set of alphabets is super set of set of vowels.

4

Finite Set

Set which contains countable number of elements.

5

Infinite Set

Set which contains uncountable number of elements.

Set of Natural Numbers.

6

Equal Sets

Two or more sets which have same elements.

A =,

B =

7

Equivalent Sets

Two sets A and B are said to be equivalent if n(A) = n(B)

A= {1, 2, 3} and B= {a, b, c}.

Clearly n(A) = n(B)=3

8

Universal Set, U

Set which is Super Set of all sets under consideration.

Let A={1,2,3}, B={2,3,5}and C={1,4,6}

Then U={1,2,3,4,5,6} is universal set for A, B and C.

9

Power Set ,P

Set of all possible subsets of a given set.

A ={1,2} then P(A)={ ,{1},{1, 2},{2} }

Remark:

1. Empty set is subset of itself.

2. Every set is subset of itself.

3. If A and , then

4. If A and , then

5. If A and then

6. If A and Then A is called as proper subset of B.

7. If then total number of subsets of A = and number of proper subsets is -1.

8. If XP(A) ,then

9. n[P(A)]=2 n(A)

Properties of Sets:

S. No.

Property

Meaning

1

Commutative Law

2

Associative Law

3

Law of

4

Law of U

5

Idempotent Law

6

Distributive Law

7

Complement Laws

8

De Morgans Law

9

Law of Double Complementation

10

Law of Empty Set and Universal Set

Questions for Practice:

1. . Fill in the blanks by putting appropriate symbols.

i) ii) iii) iv)

2.Write in roster form

i)ii) The set of letters of the word PANIC

3.Write the followings in set builder form.

i) ii)

4. Represent the following in Roster form

(ii) (iii)

(iv) (v)

5. Represent in roster form

(i) Set of odd numbers. (ii) {xN : x2 < 25} (iii)Set of integers from -5 to 5

6. Classify the following as finite or infinite set.

(i) Set of integers greater than 100.

(ii) Set of concentric circles

(iii) {x}

(iv) The set of prime numbers less than 100

(v) {x}

7 .State True or False

Let A= {1,2,{3,4},5} state as True or False for the following statements

(i) {3,4} A (ii) {1,2,5} A (iii) {} A (iv) 1A (v) {3,4} A

Intervals

Q1. write the below intervals in set builder form.

(i) [ a,b][ii] [a,b)(iii)[a,b](iv) (a,b](v) (-1,3)

(vi) (-1,3)(vii) [3,10](viii) (-4,0)(ix) [2,9)(x) (-1,3]

Q2. write the below sets in interval form.

(i) {x: 0< x 10}(ii) ) {x: - 4 < x < 2}(iii){ x: 2x7}

Q.3.Find the intersection of given two intervals.

(i) [-1,3] and [0,5](ii) (2,6) and(-1,2] (iii) [1,4) and (2,5)

Level I

1. Find the cardinal number of the set

2.

Let A=and B=. Find and .

3. In a school there are 20 teachers who teach Mathematics or Physics. Out of these, 12 teach Mathematics and 4 teach Physics and Mathematics. How many teach Physics?

4.

Let A, B and C be three sets such that and Show that B=C

5. If then verify that

i) ii)

6. Draw appropriate Venn diagram for

i) ii)

Level-II

1. If A= B=, C=, D=, find:

(i) (ii) (iii) (iv)

2. Draw appropriate Venn diagram for each of the following:

1.

(ii)

Level III

1. In a survey of 25 students, it was found that 15 had taken Mathematics, 12 had taken Physics and 11 had taken Chemistry, 5 had taken Mathematics and Chemistry, 9 had taken Mathematics and Physics , 4 had taken Physics and Chemistry and 3 had taken all the three subjects. Find the number of students that had:

i) Only Chemistry ii) at least one of the three subjects

iii) None of the subjects

Module-2 : Relations and Functions

S. No.

Concept

Meaning

Example

1

Cartesian Product

AB

A={a,b,c} and B={e,f},

then AB ={(a,e),(a,f),(b,e),(b,f),(c,e),(c,f)}

2

Relation

A subset of cartesian product AB

If A={1,2,3,4,5,6}, B={2,3,5,7,9}

and R is a relation from A to B such that R={(a,b): aA, bB and a is the multiple of b}, then R={(2,2),(3,3),(4,2),(5,5),(6,2),(6,3)}.

3

Domain

Set of all the first elements of the ordered pairs of R

Domain of R = {

In the above example,

domain of R={2,3,4,5,6}

4

Range of a Relation

Set of all the second elements of the ordered pairs of R

Range of R = {

In the above example, range of R={2,3,5}

5

Function

A relation from a set A to a set B is called a function from A to B (denoted by ) if every element of A has one and only one image in B

6

Range of a function

Set of all the elements of the codomain of the function which have pre-image in domain under

7

Real Valued Function

A function whose range is the subset of R

8

Real Function

A function whose domain and range are the subsets of R

9

Identity Function

, for each

10

Constant Function

, for each , where c is a constant

11

Modulus Function

, for each , i.e.

12

Signum Function

13

Greatest Integer Function

, which assumes the value of the greatest integer, less than or equal to x, for each

Note:

(i) If any of the sets A and B is empty then AB is empty.

(ii) If any of the sets A and B is infinite then AB is infinite.

(iii) If both the sets A and B are finite then AB is finite.

(iv) S