UNIT-1 : SETS & FUNCTIONS
1. Sets : Sets and their representations. Type of sets : Empty set, Finite & Infinite set and, 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 complements of sets, Applications based problems.
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 nos. with itself (up to R x R x R). Definitions of relation, Pictorial diagram for relation, Domain, Co-domain 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. Type of functions : Constant, Identity, polynomial, rational, modulus, signum and greatest integer functions with their Graphs. Sum, difference, product and quotients of functions. Concept of exponential and logarithmic function.
3. Trigonometric Functions : Positive and negative angles, Measuring angles in Radians & in Degrees and Conversions from one measure to another. Definition of trigonometric functions with the help of unit circle. Proofs of various trigonometric identities for all real values of x. Sign conventions of trigonometric functions. Domain and Range of trigonometric functions and their Graphs. Various relations viz. sin (x + y), sin (x – y), cos (x + y), cos (x – y), tan (x + y) etc. Deduction of various trigonometric relations. Identities related to sin 2x, sin 3x, cos 2x, cos 3x, tan 2x, tan 3x etc. General solutions of trigonometric equations, Principal solutions of trigonometric equations. Properties of triangles. Proof and applications of sine rules and cosine rules.
UNIT-2 : ALGEBRA
1. Principle Of Mathematical Induction : Process of the proofs 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 & Quadratic Equations : Need for complex numbers, especially square root of ‘-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. 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 and their representations on the number line, Graphical solution of linear inequalities in two variables, Graphical solution of system of linear inequalities in two variables, Applications of linear inequalities.
4. Permutations & Combinations : Fundamental principle of counting, Factorial n (i.e., n!), Permutations and combinations, derivation of formulae and their connections, simple applications.
5. Binomial Theorem : History, Statement and proof of Binomial Theorem for positive indices, Pascal’s triangle, General and middle terms in binomial expansion, simple application.
6. Sequence & Series : Sequence and Series, Arithmetic Progression (A.P.), General term of a A.P., Sum of n terms of a A.P., Arithmetic Mean (A.M.), Geometric Progression (G.P.), General term of a G.P., Sum of n terms of a G.P., Geometric Mean (G.M.), Infinite Geometric Progression, Sum of infinite G.P., Relation between A.M. and G.M., Sum to n terms of special sequences.
UNIT-3 : COORDINATE GEOMETRY
1. Straight Lines : Brief recap 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 : Point-Slope form, Two-Point form, Slope-Intercept 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. Simple applications.
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, hyperbola and circle. Simple applications.
3. Introduction To 3 Dimensional Geometry : Coordinate Axes and coordinate planes in Three Dimensions, Coordinates of a point, Distance between two points and section formulae.
UNIT-4 : CALCULUS
1. Limits & Derivatives : Derivative introduced as rate of change of both as that of distance function and geometrically. Intutive idea of limit. Limits of polynomials and rational functions, trigonometric functions, exponential functions, logarithmic functions. Definitions of derivative, relate it to slope of tangent of the curves, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.
UNIT-5 : MATHEMATICAL REASONING
1. Mathematical Reasoning : Mathematical acceptable statements, connecting words/phrase – consolidating and 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 & PROBABILITY
1. Statistics : Measures of dispersion, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distribution with equal means but different variances.
2. Probability : 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 the theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’ and ‘or’ events.
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