OFFICIAL SYLLABUS - 2022
Unit-1:Sets and Functions
1. Sets: 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 .
2. Relations & Functions: Ordered pairs, Cartesian product of sets. Number of elements in the Cartesian product of two finite sets. Cartesian product of the 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 type of relation. Pictorial representation of a function, domain, co-domain and range of a function. Real valued function, 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 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 function and their graphs. Expressing sin (x ± y) and cos(x ± y) in terms of sinx, siny, cosx & cosy and their simple applications.
Unit-II Algebra
4. 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.
5. Complex Numbers and Quadratic Equations: Need for complex numbers, especially , to be motivated by inability to solve some of the quadratic equation. Algebraic properties of complex number. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations (with real coefficients) in the complex number system. Square root of a complex number.
6. 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 method of finding a solution of system of linear inequalities in two variables .
7. Permutations & Combinations: Fundamental principle of counting, Factorial n(n!) Permutations and combinations, derivation of formulae for ?npr and ?ncr and their connections, simple applications
8. 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.
9. 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. Infinite G.P. and its sum, geometric mean (G.M), relation between A.M. and G.M. Formulae for the following special sums.
Unit-III: Coordinate Geometry
10. 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, slop–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.
11. 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 equations of a circle.
12. Introduction to Three-dimensional Geometry:Coordinate axes and coordinate planes in three dimensions.Coordinates of a point. Distance between two points and section formula.
Unit-IV: Calculus
13. Limits and Derivatives: Derivative introduced as rate of change both as that of distance function and geometrically. intuitive idea of limit. Limits of polynomials and rational functions trigonometric, exponential and logarithimic functions. 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 trigonometric functions.
Unit-V: Mathematical Reasoning
14. 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 among contradiction, converse and contrapositive.
Unit-VI: Statistics and Probability
15. Statistics: Measure of Dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances.
16. Probability: Random experiments: outcomes, sample spaces (set representation). Events: Occurrence of events, ‘not’, ‘and’ & ‘or’ events, exhaustive events, mutually exclusive events. Axiomatic (set theoretic) probability, connections with others theories of earlier classes. Probability of an event , probability of ‘not’, ‘and’ & ‘or’ events.
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