 ## CBSE Class 10 , Class 11 , Class 12 Subjects

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#### Sets and Functions

1. Sets
Sets and their representations.Empty set.Finite and Infinite sets.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 Complement Sets.
2. Relations & 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, co-domain 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 functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotient 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 sin2 x+cos2 x=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, cos x & cos y and their simple applications. Deducing the identities like the following:
$$tan\quad (x\pm y)=\frac { tanx\pm tany }{ 1\mp tanx\quad tany } ,cot(x\pm y)=\frac { cotx\quad coty\quad \mp 1 }{ coty\pm cotx }$$
$$sin\alpha \pm sin\beta =2sin\frac { 1 }{ 2 } (\alpha \pm \beta )cos\frac { 1 }{ 2 } (\alpha \mp \beta )$$
$$cos\alpha +cos\beta =2cos\frac { 1 }{ 2 } (\alpha +\beta )cos\frac { 1 }{ 2 } (\alpha -\beta )$$
$$cos\alpha -cos\beta =-2sin\frac { 1 }{ 2 } (\alpha +\beta )sin\frac { 1 }{ 2 } (\alpha -\beta )$$

Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x. General solution of trigonometric equations of the type sin y = sin a, cos y = cos a and tan y = tan a.

#### Algebra

1. 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.
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 (with real coefficients) 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 solutions of linear inequalities in two variables.Graphical method of finding a 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
Sequence and Series. Arithmetic Progression (A. P.). Arithmetic Mean (A.M.) Geometric Progression (G.P.), general term of a G.P., sum of first 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.
$$\sum _{ k=1 }^{ m }{ k } ,\sum _{ k=1 }^{ n }{ { k }^{ 2 } } and\quad \sum _{ k=1 }^{ n }{ { k }^{ 3 } }$$

#### 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 axis, 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: circle, 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.

#### Calculus

1. 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 logarithmic functions. Definition of derivative relate it to scope of tangent of the curve, Derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.

#### 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.

#### Statistics and Probability

1. Statistics
Measures of dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions 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 other theories studied in earlier classes. Probability of an event, probability of 'not', 'and' and 'or' events.

#### Sets

Introduction-Sets and their Representations-The Empty Set-Finite and Infinite Sets-Equal Sets-Subsets-Power Set-Universal Set-Venn Diagrams-Operations on Sets-Complement of a Set-Practical Problems on Union and Intersection of Two Sets

#### Relations and Functions

Introduction-Cartesian Product of Sets-Relations-Functions

#### Trigonometric Functions

Introduction-Angles-Trigonometric Functions-Trigonometric Functions of Sum and Difference of Two Angles-Trigonometric Equations

#### Principle of Mathematical Induction

Introduction-Motivation-The Principle of Mathematical Induction

#### Complex Numbers and Quadratic Equations

Introduction-Complex Numbers-Algebra of Complex Numbers-The Modulus and the Conjugate of a Complex Number-Argand Plane and Polar Representation-Quadratic Equations

#### Linear Inequalities

Introduction-Inequalities-Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation-Graphical Solution of Linear Inequalities in Two Variables-Solution of System of Linear Inequalities in Two Variables

#### Permutations and Combinations

Introduction-Fundamental Principle of Counting-Permutations-Combinations

#### Binomial Theorem

Introduction-Binomial Theorem for Positive Integral Indices-General and Middle Terms

#### Sequences and Series

Introduction-Sequences-Series-Arithmetic Progression-Geometric Progression-Relationship Between A.M. and G.M.-Sum to n terms of Special Series

#### Straight Lines

Introduction-Slope of a Line-Various Forms of the Equation of a Line-General Equation of a Line-Distance of a Point From a Line

#### Conic Sections

Introduction-Sections of a Cone-Circle-Parabola-Ellipse-Hyperbola

#### Introduction to Three Dimensional Geometry

Introduction-Coordinate Axes and Coordinate Planes in Three Dimensional Space-Coordinates of a Point in Space-Distance between Two Points-Section Formula

#### Limits and Derivatives

Introduction-Intuitive Idea of Derivatives-Limits-Limits of Trigonometric Functions-Derivatives

#### Mathematical Reasoning

Introduction-Statements-New Statements from Old-Special Words/Phrases-Implications-Validating Statements

#### Statistics

Introduction-Measures of Dispersion-Range-Mean Deviation-Variance and Standard Deviation-Analysis of Frequency Distributions 