|I.||Sets and Functions||29|
|VI.||Statistics and Probability||12|
Unit-1 : Sets and Functions
Sets and their representations. Empty set. Finite and 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. Practice Problems based on 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 (upto R × R × 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 and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum exponential, logarithmic functions 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 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 trigonometric functions and their graphs. Expressing sin (x ± y) and cos (x ± y) in terms of sin x, sin y, cosx and cos y. Deducing the identities 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.
1. Principle of Mathematical Induction:
Process of the proof by induction, motivating the applications 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:
Need for complex numbers, especially √1, 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.
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.
4. Permutations and Combinations:
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:
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:
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. Sum to n terms of the special series
Unit-3: 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.
2. Conic Sections:
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 equation of a circle.
3. 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:
Derivative introduced as rate of change of distance function and geometrically. Intuitive idea of limit. Limits of Polynomials and rational function, 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. Derivatives of polynomial and trigonometric functions.
Unit-5: 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-6: Statistics and Probability
Measure of dispersion, range, mean deviation, variance and standard deviation of ungrouped/grouped data.
Analysis of frequency distributions with equal mean but different variances.
Random experiments, outcomes; Sample spaces (set representation); Event; 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 an event, probability of “not”, “and” and “or” events.