UNIT - I: SETS AND FUNCTIONS
Sets and their representations. Empty set. Finite & Infinite sets. Equal sets.Subsets. Subsets of the 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 and Functions
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, codomain and range of a relation. Function as a special kind of relation from one set to another. Pictorial representation of a function, domain, co-domain & range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum and greatest integer functions, with their graphs. Sum, difference, product and quotients of functions.
3. Trigonometric Functions
Positive and negative angles. Measuring angles in radians & 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. Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x. General solution of trigonometric equations of the type sinθ = sin α, cosθ = cos α and tanθ = tan α. Proof and simple applications of sine and cosine formulae.
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 and simple applications.
Complex Numbers and Quadratic Equations
Need for complex numbers, especially , 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 in the complex number system. Square root of a complex number.
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.
Permutations and Combinations
Fundamental principle of counting. Factorial n. (n!)Permutations and combinations, derivation of formulae and their connections, simple applications.
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.
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
1. 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 axes, point-slope form, slope-intercept 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.
Sections of a cone: circles, 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 of equation of a circle.
Introduction to Three-dimensional Geometry
Coordinate axes and coordinate planes in three dimensions.
Coordinates of a point.
Distance between two points and section formula.
1. Limits and Derivatives
Limit of function introduced as rate of change of distance function and its geometric meaning.
Definition of Derivative, relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of functions.
Derivatives of polynomial and
UNIT-V: MATHEMATICAL REASONING
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
Measures of dispersion,
standard deviation of ungrouped/grouped data.
Analysis of frequency distributions with equal means but different variances
Random experiments, outcomes, sample spaces(set representation). Events, occurence of event, "not", 'and' and 'or' events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with the theories with the theories of earlier classes.
Probability of an event, Probability of 'not', 'and' and 'or' events.