Unit-I: Sets and Functions


Sets

  • Sets and their representation
  • Types of sets: empty sets, finite and infinite sets, power set, universal set
  • Cardinality of sets
  • Subset and superset
  • Venn diagram
  • Operation on sets: union of sets
  • Operation on sets: intersection and difference of sets
  • Complement of sets
  • Practical problems on union and intersection of two sets

Relations & Functions

  • Cartesian product of sets
  • Relations
  • Function: domain and range
  • identity function, constant function and modulus function
  • Graphs of polynomial function
  • Algebra of real functionsn

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 trignometric functions and their graphs
  • Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple application
  • Deducing identities like the following:
  • Identities related to sin 2x, cos 2x, tan 2x, sin 3x, cos 3x and tan 3x
  • General solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a.

UNIT II: ALGEBRA


Principle of Mathematical Induction

  • 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
  • Simple Application of principle of mathematical induction

Complex Numbers and Quadratic Equations

  • Need for complex numbers, especially √1, to be motivated by inability to solve some of the quardratic equations
  • Algebraic properties of complex numbers
  • Modulus and conjugate 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
  • Euler's formula and De Moivre's theorem
  • Square root of a complex number.

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.

Permutations and Combinations

  • Fundamental principle of counting
  • Permutations
  • Combinations

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.

Sequence and Series

  • Sequence and Series
  • Arithmetic Progression (A.P.).
  • Geometric progression
  • Relationship between Aithmetic mean and geometric mean
  • Formula for the following special sum:

UNIT III: COORDINATE GEOMETRY


Straight Lines

  • 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, slope-intercept form, two-point form, intercept form and normal form. General equation of a line
  • AEquation of family of lines passing through the point of intersection of two lines
  • Distance of a point from a line.

Conic Sections

  • Sections of cone
  • Circle
  • Parabola
  • Ellipse
  • 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.

Introduction to Three–dimensional Geometry

  • Coordinates of point in space
  • Distance between two points
  • Section formula

Unit-IV: Calculus & Mathematical Reasoning


Limits and Derivatives

  • Derivative introduced as rate of change both as that of distance function and geometrically.
  • Intutive idea of limit
  • Limits of polynomials and rational functions, trignometric, 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 trignometric functions

Mathematical Reasoning

  • Introduction
  • Statements
  • New statements from old
  • Special word/phrase
  • Implications
  • validating statements

Unit-VI: Statistics and Probability


Statistics

  • Measures of dispersion and range
  • Mean deviation
  • mean deviation for ungrouped data
  • mean deviation for discrete frequency distribution
  • mean deviation for continuous frequency distribution
  • Variance and standard deviation
  • Shortcut method for finding variance and standard deviation
  • Analysis of frequency distribution

Probability

  • Random experiments; outcomes, sample spaces
  • Event and types of event
  • Algebra of event
  • Axiomatic approach to probability