R.D. Sharma Objective Mathematics

3.1 Function

3.2 Domain, Co-Domain and range of a function

3.3 Description of a function

3.4 Equal function

3.5 Number of functions

3.6 Function as a relation

3.7 Kinds of functions

3.8 Composition of functions

3.9 Properties of composition of functions

3.10 Inverse of an element

3.11 Inverse of a function

3.12 Properties of inverse of a function

Study Plan

5 days main study - 5 days revision

Main study period 1 hour each day

Day 1

3.1 Function

3.2 Domain, Co-Domain and range of a function

3.3 Description of a function

3.4 Equal function

Day 2

3.5 Number of functions

3.6 Function as a relation

3.7 Kinds of functions

3.8 Composition of functions

Day 3

3.9 Properties of composition of functions

3.10 Inverse of an element

3.11 Inverse of a function

3.12 Properties of inverse of a function

Day 4

Objective Type Exercises 1 to 20

Revision Period 30 minutes each day

Day 5

O.T.E. 21 to 40

Day 6

O.T.E. 41 to 53

Day 7

Practice Exercise 1 to 20

## Wednesday, May 13, 2015

## Sunday, May 10, 2015

### Ch. 2. Cartesian Product of Sets and Relations - Concepts Review

Contents

2.1 Cartesian product of sets

2.2 Relations

2.3 Types of relations

2.4 Some results on relations

2.5 Composition of relations

2.1 Cartesian product of sets

Cartesian product is an operation on sets.

Ordered pair: An Ordered pair consists of two objects or elements in a given

Cartesian product: Let A and B be any two non empty sets. The set of all ordered pairs (a,b) such that a ЄA and b ЄB is called the Cartesian product of the sets A and B and is denoted by A×B

Theorems

Theorem 1; For any three sets

(i) A×(B U C) = (A×B) U (A×C)

(ii) A×(B∩C) = (A×B) ∩(A×C)

Theorem 2: For any three sets

A×(B – C) = (A×B) – (A×C)

Theorem 3: If and A and B are any two non-empty sets, then

A×B = B×A => A = B

Theorem 4: If A is a subset of B, A×A is a sub set of (A×B) ∩(B×A)

2.2 Relation

Let A and B be two sets. Then a relation R from A to B is a subset of A×B.

R is a relation from A to B => R is a subset of A×B.

Total number of relations: If A and B are two non empty sets with m and n elements respectively, A×B consists of mn ordered pairs.

Since each subset defines a relation from A to B, so total number of relations from A to B is 2

2.3 Types of relations

Void relation

Universal relation

Identity relation

Reflexive relation

Symmetric relation

Transitive relation

Antisymmetric relation

Equivalence relation

Cartesian Products of Sets and Relations - Part 2

2.4 Some more properties and results on relations

1. If R and S are two equivalence relations on a set A, then R∩S is also an equivalence relation on A.

2. The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.

3. If R is an equivalence relation on a set A, the R

2.5 Composition of Relations

When r and S are two relations from set A to B and B to C respectively, we can define a relation SoR from A to C such that

(a.c) Є SoR imples for all b Є B subject to the relations (a,b) ЄR and (b.c) ЄS.

SoR is called the composition of R and S.

Properties of SoR

In general RoS is not equal to SoR.

(SoR)

2.1 Cartesian product of sets

2.2 Relations

2.3 Types of relations

2.4 Some results on relations

2.5 Composition of relations

2.1 Cartesian product of sets

Cartesian product is an operation on sets.

Ordered pair: An Ordered pair consists of two objects or elements in a given

**fixed order.**Cartesian product: Let A and B be any two non empty sets. The set of all ordered pairs (a,b) such that a ЄA and b ЄB is called the Cartesian product of the sets A and B and is denoted by A×B

Theorems

Theorem 1; For any three sets

(i) A×(B U C) = (A×B) U (A×C)

(ii) A×(B∩C) = (A×B) ∩(A×C)

Theorem 2: For any three sets

A×(B – C) = (A×B) – (A×C)

Theorem 3: If and A and B are any two non-empty sets, then

A×B = B×A => A = B

Theorem 4: If A is a subset of B, A×A is a sub set of (A×B) ∩(B×A)

2.2 Relation

Let A and B be two sets. Then a relation R from A to B is a subset of A×B.

R is a relation from A to B => R is a subset of A×B.

Total number of relations: If A and B are two non empty sets with m and n elements respectively, A×B consists of mn ordered pairs.

Since each subset defines a relation from A to B, so total number of relations from A to B is 2

^{mn}.2.3 Types of relations

Void relation

Universal relation

Identity relation

Reflexive relation

Symmetric relation

Transitive relation

Antisymmetric relation

Equivalence relation

Cartesian Products of Sets and Relations - Part 2

2.4 Some more properties and results on relations

1. If R and S are two equivalence relations on a set A, then R∩S is also an equivalence relation on A.

2. The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.

3. If R is an equivalence relation on a set A, the R

^{-1}is also an equivalence relation on A.2.5 Composition of Relations

When r and S are two relations from set A to B and B to C respectively, we can define a relation SoR from A to C such that

(a.c) Є SoR imples for all b Є B subject to the relations (a,b) ЄR and (b.c) ЄS.

SoR is called the composition of R and S.

Properties of SoR

In general RoS is not equal to SoR.

(SoR)

^{-}= R^{-}oS^{-}## Friday, May 8, 2015

### CBSE - XI Class Syllabus for 2015 - 2016

MATHEMATICS (Code No. 041)

Senior Secondary stage is a launching stage from where the students go either for higher academic education in Mathematics or for professional courses like Engineering, Physical and Bioscience, Commerce or Computer Applications. The present revised syllabus has been designed in accordance with National Curriculum Framework 2005 and as per guidelines given in Focus Group on Teaching of Mathematics 2005 which is to meet the emerging needs of all categories of students.

Motivating the topics from real life situations and other subject areas, greater emphasis has been laid on application of various concepts.

Objectives

The broad objectives of teaching Mathematics at senior school stage intend to help the students:

to acquire knowledge and critical understanding, particularly by way of motivation and visualization, of basic concepts, terms, principles, symbols and mastery of underlying processes and skills.

to feel the flow of reasons while proving a result or solving a problem.

to apply the knowledge and skills acquired to solve problems and wherever possible, by more than one method.

to develop positive attitude to think, analyze and articulate logically.

to develop interest in the subject by participating in related competitions.

to acquaint students with different aspects of Mathematics used in daily life.

to develop an interest in students to study Mathematics as a discipline.

to develop awareness of the need for national integration, protection of environment, observance of

small family norms, removal of social barriers, elimination of gender biases.

to develop reverence and respect towards great Mathematicians for their contributions to the field of Mathematics.

COURSE STRUCTURE CLASS XI (2015-16)

One Paper Total Hours-

Three Hours Max Marks. 100

Topic Periods Marks

I. Sets and Functions 60 29

II. Algebra 70 37

III. Coordinate Geometry 40 13

IV. Calculus 30 06

V. Mathematical Reasoning 10 03

VI. Statistics and Probability 30 12

Total 240 100

Unit-I: Sets and Functions

1. Sets (20) Periods

Sets and their representations.Empty set.Finite and Infinite sets.Equal sets.Subsets.Subsets of a set of

real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and

Intersection of sets.Difference of sets. Complement of a set. Properties of Complement Sets.

2. Relations & Functions: (20) Periods

Ordered pairs, Cartesian product of sets.Number of elements in the cartesian product of two finite sets.

Cartesian product of the set of reals with itself (upto R x R x R). Definition of relation, pictorial

diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorial

representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotient of functions.

3. Trigonometric Functions: (20) Periods

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one

measure to another.Definition of trigonometric functions with the help of unit circle. Truth of the

identity sin2x+cos2x=1, for all x. Signs of trigonometric functions. Domain and range of trignometric

functions and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applications. Deducing the identities like the following:

tanx ± tany cotxcoty 1

tan(x ± y) = , cot(x ± y) =

1 tanxtany coty ± cotx

1 1

sinα ± sinβ = 2sin (α ± β)cos (α β)

2 2

1 1

cosα + cosβ = 2cos (α + β)cos (α - β)

2 2

1 1

cosα - cosβ = -2sin (α + β)sin (α - β)

2 2

Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x. General solution of trigonometric

equations of the type siny = sina, cosy = cosa and tany = tana.

Unit-II: Algebra

1. Principle of Mathematical Induction: (10) Periods

Process of the proof by induction, motivating the application of the method by looking at natural

numbers as the least inductive subset of real numbers. The principle of mathematical induction and

simple applications.

2. Complex Numbers and Quadratic Equations (15) Periods

Need for complex numbers, especially √−1, to be motivated by inability to solve some of the quardratic equations. Algebraic properties of complex numbers.Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations (with real coefficients) in the complex number system. Square root of a complex number.

3. Linear Inequalities (15) Periods

Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line.Graphical representation of linear inequalities in two variables.Graphical method of finding a solution of system of linear inequalities in two variables.

4. Permutations and Combinations (10) Periods

Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of

formulae for 𝑛𝑃𝑟

and 𝑛𝐶𝑟

and their connections, simple applications.

5. Binomial Theorem (10) Periods

History, statement and proof of the binomial theorem for positive integral indices.Pascal's triangle,

General and middle term in binomial expansion, simple applications.

6. Sequence and Series (10) Periods

Sequence and Series. Arithmetic Progression (A. P.). Arithmetic Mean (A.M.) Geometric Progression

(G.P.), general term of a G.P., sum of first n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M. Formulae for the following special sums

2 3

1 1 1

,

n n n

k k k

k k and k

Unit-III:Coordinate Geometry

1. Straight Lines (10) Periods

Brief recall of two dimensional geometry from earlier classes. Shifting of origin. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point-slope form, slopeintercept form, two-point form, intercept form and normal form. General equation of a line.Equation of family of lines passing through the point of intersection of two lines.Distance of a point from a line.

2. Conic Sections (20) Periods

Sections of a cone: circle, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola.Standard equation of a circle.

3. Introduction to Three-dimensional Geometry (10) Periods

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between

two points and section formula.

Unit-IV: Calculus

1. Limits and Derivatives (30) Periods

Derivative introduced as rate of change both as that of distance function and geometrically.

Intutive idea of limit.Limits of polynomials and rational functions trigonometric, exponential and

logarithmic functions. Definition of derivative relate it to scope of tangent of the curve, Derivative of

sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric

functions.

Unit-V: Mathematical Reasoning

1. Mathematical Reasoning (10) Periods

Mathematically acceptable statements. Connecting words/ phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words, Difference between contradiction, converse and contrapositive.

Unit-VI: Statistics and Probability

1. Statistics (15) Periods

Measures of dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances.

2. Probability (15) Periods

Random experiments; outcomes, sample spaces (set representation). Events; occurrence of events, 'not', 'and' and 'or' events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories studied in earlier classes. Probability of an event, probability of 'not', 'and' and 'or' events.

Senior Secondary stage is a launching stage from where the students go either for higher academic education in Mathematics or for professional courses like Engineering, Physical and Bioscience, Commerce or Computer Applications. The present revised syllabus has been designed in accordance with National Curriculum Framework 2005 and as per guidelines given in Focus Group on Teaching of Mathematics 2005 which is to meet the emerging needs of all categories of students.

Motivating the topics from real life situations and other subject areas, greater emphasis has been laid on application of various concepts.

Objectives

The broad objectives of teaching Mathematics at senior school stage intend to help the students:

to acquire knowledge and critical understanding, particularly by way of motivation and visualization, of basic concepts, terms, principles, symbols and mastery of underlying processes and skills.

to feel the flow of reasons while proving a result or solving a problem.

to apply the knowledge and skills acquired to solve problems and wherever possible, by more than one method.

to develop positive attitude to think, analyze and articulate logically.

to develop interest in the subject by participating in related competitions.

to acquaint students with different aspects of Mathematics used in daily life.

to develop an interest in students to study Mathematics as a discipline.

to develop awareness of the need for national integration, protection of environment, observance of

small family norms, removal of social barriers, elimination of gender biases.

to develop reverence and respect towards great Mathematicians for their contributions to the field of Mathematics.

COURSE STRUCTURE CLASS XI (2015-16)

One Paper Total Hours-

**Periods of 35 Minutes each**Three Hours Max Marks. 100

Topic Periods Marks

I. Sets and Functions 60 29

II. Algebra 70 37

III. Coordinate Geometry 40 13

IV. Calculus 30 06

V. Mathematical Reasoning 10 03

VI. Statistics and Probability 30 12

Total 240 100

Unit-I: Sets and Functions

1. Sets (20) Periods

Sets and their representations.Empty set.Finite and Infinite sets.Equal sets.Subsets.Subsets of a set of

real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and

Intersection of sets.Difference of sets. Complement of a set. Properties of Complement Sets.

2. Relations & Functions: (20) Periods

Ordered pairs, Cartesian product of sets.Number of elements in the cartesian product of two finite sets.

Cartesian product of the set of reals with itself (upto R x R x R). Definition of relation, pictorial

diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorial

representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotient of functions.

3. Trigonometric Functions: (20) Periods

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one

measure to another.Definition of trigonometric functions with the help of unit circle. Truth of the

identity sin2x+cos2x=1, for all x. Signs of trigonometric functions. Domain and range of trignometric

functions and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applications. Deducing the identities like the following:

tanx ± tany cotxcoty 1

tan(x ± y) = , cot(x ± y) =

1 tanxtany coty ± cotx

1 1

sinα ± sinβ = 2sin (α ± β)cos (α β)

2 2

1 1

cosα + cosβ = 2cos (α + β)cos (α - β)

2 2

1 1

cosα - cosβ = -2sin (α + β)sin (α - β)

2 2

Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x. General solution of trigonometric

equations of the type siny = sina, cosy = cosa and tany = tana.

Unit-II: Algebra

1. Principle of Mathematical Induction: (10) Periods

Process of the proof by induction, motivating the application of the method by looking at natural

numbers as the least inductive subset of real numbers. The principle of mathematical induction and

simple applications.

2. Complex Numbers and Quadratic Equations (15) Periods

Need for complex numbers, especially √−1, to be motivated by inability to solve some of the quardratic equations. Algebraic properties of complex numbers.Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations (with real coefficients) in the complex number system. Square root of a complex number.

3. Linear Inequalities (15) Periods

Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line.Graphical representation of linear inequalities in two variables.Graphical method of finding a solution of system of linear inequalities in two variables.

4. Permutations and Combinations (10) Periods

Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of

formulae for 𝑛𝑃𝑟

and 𝑛𝐶𝑟

and their connections, simple applications.

5. Binomial Theorem (10) Periods

History, statement and proof of the binomial theorem for positive integral indices.Pascal's triangle,

General and middle term in binomial expansion, simple applications.

6. Sequence and Series (10) Periods

Sequence and Series. Arithmetic Progression (A. P.). Arithmetic Mean (A.M.) Geometric Progression

(G.P.), general term of a G.P., sum of first n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M. Formulae for the following special sums

2 3

1 1 1

,

n n n

k k k

k k and k

Unit-III:Coordinate Geometry

1. Straight Lines (10) Periods

Brief recall of two dimensional geometry from earlier classes. Shifting of origin. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point-slope form, slopeintercept form, two-point form, intercept form and normal form. General equation of a line.Equation of family of lines passing through the point of intersection of two lines.Distance of a point from a line.

2. Conic Sections (20) Periods

Sections of a cone: circle, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola.Standard equation of a circle.

3. Introduction to Three-dimensional Geometry (10) Periods

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between

two points and section formula.

Unit-IV: Calculus

1. Limits and Derivatives (30) Periods

Derivative introduced as rate of change both as that of distance function and geometrically.

Intutive idea of limit.Limits of polynomials and rational functions trigonometric, exponential and

logarithmic functions. Definition of derivative relate it to scope of tangent of the curve, Derivative of

sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric

functions.

Unit-V: Mathematical Reasoning

1. Mathematical Reasoning (10) Periods

Mathematically acceptable statements. Connecting words/ phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words, Difference between contradiction, converse and contrapositive.

Unit-VI: Statistics and Probability

1. Statistics (15) Periods

Measures of dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances.

2. Probability (15) Periods

Random experiments; outcomes, sample spaces (set representation). Events; occurrence of events, 'not', 'and' and 'or' events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories studied in earlier classes. Probability of an event, probability of 'not', 'and' and 'or' events.

## Thursday, May 7, 2015

### Ch. 1. Sets - Concept Review

1,1 Set Explanation

Mathematicians encountered server difficulties in defining set. They realized that there is a need for some undefined (primitive) terms. Thus set and elements are undefined terms in mathematics.

Set is synonymous with the words, ‘collection’, aggregate’, ‘class’, and is comprised of elements.

The words ‘element’, ‘object’, and ‘member’ are synonymous.

Sets designated by specific letters.

N: natural numbers

Z : integers

Z

^{+}: positive integers

Q: rational numbers

Q

^{+}: positive rational numbers

R: real numbers

R

^{+}: positive real numbers

C: complex numbers

1.2 Description of a set

Sets can be described by roster method or set-builder method.

1.3 Types of sets

Empty set

Singleton set

Finite set

Infinite set

Equivalent set

Equal set

Subset

Universal set

Power set

1.4 Theorems on subsets

1. Every set is a subset of itself.

2. The empty set is a subset of every set.

3. The total number of subsets of a finite set containing n elements is 2ⁿ

1.5 Universal Set

A set that contains all sets in a given context (in simple terms in the given problem) is called the universal set.

1.6 Power Set

Given a set A, the collection of all subsets of A is called the power set of A and is denoted by P(A).

1.7 Venn Diagrams

In Venn diagrams the universal set U is represented by points within a rectangle and its subsets are represented by points in closed curves (usually circles) within the the rectangle. If two sets A and B have some common elements, they will be shown two intersecting circles. If the two sets are disjointed sets, then they are represented by two non-intersecting circles. If set B is a subset of A, then B is shown as a circle inside the circle representing set A.

1.8 Operations on Sets

Union of Sets

Intersection of Sets

Difference of Sets

Symmetric Difference of Two Sets

Disjoint Sets

Complement of a Set

1.9 Laws of algebra of sets

1. Idempotent laws

(i) A U A = A

(ii) A ∩ A = A

2. Identity laws

(i) A U ф = A

(ii) A ∩ U = A

3. commutative law

(i) A U B = B U A

(ii) A ∩ B = B ∩ A

4. Associative laws

(i) (A U B) U C = A U (B U C)

(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)

5. Distributive laws

(i) A U (B ∩ C) = (A U B) ∩ (A U C)

(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

6. De-morgan’s laws

(i) (A U B)’ = A’ ∩ B’

(ii) (A ∩ B)’ = A’ U B’

1.10 Some more theorems

Theorem 1

If A and B are two sets

(i) A – B = A ∩ B’

(ii) B – A = B ∩ A’

(iii) A – B = A A ∩ B = ф

(iv) (A – B) U B = A U B

(v) (A-B) ∩ B = ф

(vi) A is a sub set of B B’ is a subset of A’

(vii) (A-B) U (B-A) = (A U B) – (A ∩ B)

Theorem 2

If A, B and C are three sets, then

(i) A – (B ∩ C) = (A-B) U (A-C)

(ii) A – (B U C) = (A-B) ∩ (A-C)

(iii) A ∩ (B-C) = (A ∩ B) - (A ∩ C)

(iv) A ∩ (B Δ C) = (A∩B) Δ (A∩C)

1.11 Number of Elements in Sets

Note union operation and universal set have the same symbol in these pages. Hence identify appropriately.

n(A) denotes the number of elements in the set A. Similarly n(B) and n(C).

If A,B and C are finite sets. U is the finite universal set, then

(i) n(A U B) = n(A) +n(B) – n(A∩B)

(ii) n(A U B) = n(A) +n(B) implies A, B are disjoint non-void sets.

(iii) n(A-B) = n(A) –n(A∩B)

(iv) n(A ΔB) = Number of elements which belong to exactly one of A or B

= n((A-B) U (B-A))

(v) n(A U B U C) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(A∩C)+n(A∩B∩C)

(vi) No. Of elements in exactly two of the sets A,B,C

= n(A∩B) + n(B∩C)+n(C∩A)-3n(A∩B∩C)

(vii) No. Of elements in exactly one of the sets A,B,C

= n(A) +n(B)+n(C)-2n(A∩B)-2n(B∩C)-2n(A∩C)+3n(A∩B∩C)

(viii) n(A’ U B’) = n((A∩B)’) = n(U) – n(A∩B)

(ix) n(A’∩B’) = n((AUB)’) = n(U)-n(A∩B)

Reference: Objective Mathematics by R.D. Sharma, Dhanpat Rai & Sons

## Friday, February 27, 2015

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