Unit-I: Sets and Functions

Chapter 1: Sets

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

Practical Problems based on sets

Chapter 2: Relations & Functions

Ordered pairs

Cartesian product of sets

Number of elements in the cartesian product of two finite sets

Cartesian product of the sets of real (up to R Ã— R)

Definition of âˆ’

Relation

Pictorial diagrams

Domain

Co-domain

Range of a relation

Function as a special kind of relation from one set to another

Pictorial representation of a function, domain, co-domain and range of a function

Real valued functions, domain and range of these functions âˆ’

Constant

Identity

Polynomial

Rational

Modulus

Signum

Exponential

Logarithmic

Greatest integer functions (with their graphs)

Sum, difference, product and quotients of functions.

Chapter 3: Trigonometric Functions

Positive and negative angles

Measuring angles in radians and in degrees and conversion of one into other

Definition of trigonometric functions with the help of unit circle

Truth of the sin2x + cos2x = 1, for all x

Signs of trigonometric functions

Domain and range of trigonometric functions and their graphs

Expressing sin (xÂ±y) and cos (xÂ±y) in terms of sinx, siny, cosx & cosy and their simple application

Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x

General solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a.

Unit-II: Algebra

Chapter 1: Principle of Mathematical Induction

Process of the proof by induction âˆ’

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

Chapter 2: Complex Numbers and Quadratic Equations

Need for complex numbers, especially âˆš1, to be motivated by inability to solve some of the quadratic equations

Algebraic properties of complex numbers

Argand plane and polar representation of complex numbers

Statement of Fundamental Theorem of Algebra

Solution of quadratic equations in the complex number system

Square root of a complex number

Chapter 3: Linear Inequalities

Linear inequalities

Algebraic solutions of linear inequalities in one variable and their representation on the number line

Graphical solution of linear inequalities in two variables

Graphical solution of system of linear inequalities in two variables

Chapter 4: Permutations and Combinations

Fundamental principle of counting

Factorial n

(n!) Permutations and combinations

Derivation of formulae and their connections

Simple applications.

Chapter 5: Binomial Theorem

History

Statement and proof of the binomial theorem for positive integral indices

Pascal’s triangle

General and middle term in binomial expansion

Simple applications

Chapter 6: Sequence and Series

Sequence and Series

Arithmetic Progression (A.P.)

Arithmetic Mean (A.M.)

Geometric Progression (G.P.)

General term of a G.P.

Sum of n terms of a G.P.

Arithmetic and Geometric series infinite G.P. and its sum

Geometric mean (G.M.)

Relation between A.M. and G.M.

Unit-III: Coordinate Geometry

Chapter 1: Straight Lines

Brief recall of two dimensional geometries from earlier classes

Shifting of origin

Slope of a line and angle between two lines

Various forms of equations of a line âˆ’

Parallel to axis

Point-slope form

Slope-intercept form

Two-point form

Intercept form

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

Chapter 2: Conic Sections

Sections of a cone âˆ’

Circles

Ellipse

Parabola

Hyperbola âˆ’ 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 âˆ’

Parabola

Ellipse

Hyperbola

Standard equation of a circle

Chapter 3. Introduction to Threeâ€“dimensional Geometry

Coordinate axes and coordinate planes in three dimensions

Coordinates of a point

Distance between two points and section formula

Unit-IV: Calculus

Chapter 1: Limits and Derivatives

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

Intuitive idea of limit

Limits of âˆ’

Polynomials and rational functions

Trigonometric, exponential and logarithmic functions

Definition of derivative, relate it to slope of tangent of a curve, derivative of sum, difference, product and quotient of functions

The derivative of polynomial and trigonometric functions

Unit-V: Mathematical Reasoning

Chapter 1: Mathematical Reasoning

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

Chapter 1: Statistics

Measures of dispersion âˆ’

Range

Mean deviation

Variance

Standard deviation of ungrouped/grouped data

Analysis of frequency distributions with equal means but different variances.

Chapter 2: Probability

Random experiments âˆ’

Outcomes

Sample spaces (set representation)

Events âˆ’

Occurrence of events, ‘not’, ‘and’ and ‘or’ events

Exhaustive events

Mutually exclusive events

Axiomatic (set theoretic) probability

Connections with the theories of earlier classes

Probability of âˆ’

An event

probability of ‘not’, ‘and’ and ‘or’ events

The course is six months long with weekday classes. The fee for the course is Rs7000, to be paid monthly.