CBSE 2009 XI and XII Syllabus Mathematics

COURSE STRUCTURE
Class XI
One Paper Three Hours Max Marks. 100
Units Marks
I. SETS AND FUNCTIONS 29
II. ALGEBRA 37
III. COORDINATE GEOMETRY 13
IV. CALCULUS 06
V. MATHEMATICAL REASONING 03
VI. STATISTICS AND PROBABILITY 12
100
61
UNIT-I: SETS AND FUNCTIONS
1. Sets : (12) Periods
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.
2. Relations & Functions: (14) Periods
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 kind of relation from one set to another. Pictorial representation of a
function, domain, co-domain & range of a function. Real valued function of the real variable,
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: (18) Periods
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 and sketch of their graphs. Expressing sin (x+y) and cos (x+y) in terms of
sinx, siny, cosx & cosy. Deducing the identities like the following:
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.
UNIT-II: ALGEBRA
1. Principle of Mathematical Induction: (06) Periods
Processes 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: (10) Periods
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Need for complex numbers, especially , to be motivated by inability to solve every
quadratic equation. Brief description of 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.
3. Linear Inequalities: (10) Periods
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.
Solution of system of linear inequalities in two variables- graphically.
4. Permutations & Combinations: (12) Periods
Fundamental principle of counting. Factorial n. (n!)Permutations and combinations,
derivation of formulae and their connections, simple applications.
5. Binomial Theorem: (08) Periods
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: (10) Periods
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., geometric mean
(G.M.), relation between A.M. and G.M. Sum to n terms of the special series Σn, Σn2 and
Σn3.
UNIT-III: COORDINATE GEOMETRY
1. Straight Lines: (09) Periods
Brief recall of 2D from earlier classes. 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, twopoint
form, intercepts form and normal form. General equation of a line. Distance of a
point from a line.
2. Conic Sections: (12) Periods
Sections of a cone: circle, ellipse, parabola, hyperbola, a point, a straight line and 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 (08) Periods
Coordinate axes and coordinate planes in three dimensions. Coordinates of a point.
Distance between two points and section formula.
UNIT-IV: CALCULUS
1. Limits and Derivatives: (18) Periods
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Derivative introduced as rate of change both as that of distance function and geometrically,
intuitive idea of limit. 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
1. Mathematical Reasoning: (08) Periods
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 wordsdifference
between contradiction, converse and contrapositive.
UNIT-VI: STATISTICS & PROBABILITY
1. Statistics: (10) Periods
Measure of dispersion; mean deviation, variance and standard deviation of ungrouped/grouped
data. Analysis of frequency distributions with equal means but different variances.
2. Probability: (10) Periods
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' & 'or' events.




CLASS XII
One Paper Three Hours Marks: 100
Units Marks
I. RELATIONS AND FUNCTIONS 10
II. ALGEBRA 13
III. CALCULUS 44
IV. VECTORS AND THREE - DIMENSIONAL GEOMETRY 17
V. LINEAR PROGRAMMING 06
VI. PROBABILITY 10
Total 100
UNIT I. RELATIONS AND FUNCTIONS
1. Relations and Functions : (10) Periods
Types of relations: reflexive, symmetric, transitive and equivalence relations. One
to one and onto functions, composite functions, inverse of a function. Binary
operations.
2. Inverse Trigonometric Functions: (12) Periods
Definition, range, domain, principal value branches. Graphs of inverse trigonometric
functions. Elementary properties of inverse trigonometric functions.
UNIT-II: ALGEBRA
1. Matrices: (18) Periods
Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix,
symmetric and skew symmetric matrices. Addition, multiplication and scalar
multiplication of matrices, simple properties of addition, multiplication and scalar
multiplication. Non-commutativity of multiplication of matrices and existence of
non-zero matrices whose product is the zero matrix (restrict to square matrices of order
2). Concept of elementary row and column operations. Invertible matrices and proof of
the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
2. Determinants: (20) Periods
Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants,
minors, cofactors and applications of determinants in finding the area of a triangle.
Adjoint and inverse of a square matrix. Consistency, inconsistency and number
of solutions of system of linear equations by examples, solving system of linear
equations in two or three variables (having unique solution) using inverse of a
matrix.
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UNIT-III: CALCULUS
1. Continuity and Differentiability: (18) Periods
Continuity and differentiability, derivative of composite functions, chain rule, derivatives of
inverse trigonometric functions, derivative of implicit function.Concept of exponential and
logarithmic functions and their derivative. Logarithmic differentiation. Derivative of functions
expressed in parametric forms. Second order derivatives. Rolle's and Lagrange's Mean
Value Theorems (without proof) and their geometric interpretations.
2. Applications of Derivatives: (10) Periods
Applications of derivatives: rate of change, increasing/decreasing functions, tangents
& normals, approximation, maxima and minima (first derivative test motivated
geometrically and second derivative test given as a provable tool). Simple problems
(that illustrate basic principles and understanding of the subject as well as real-life
situations).
3. Integrals: (20) Periods
Integration as inverse process of differentiation. Integration of a variaty of functions by
substitution, by partial fractions and by parts, only simple integrals of the type
to be evaluated.
Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without
proof). Basic properties of definite integrals and evaluation of definite integrals.
4. Applications of the Integrals: (10) Periods
Applications in finding the area under simple curves, especially lines, areas of circles/
parabolas/ellipses (in standard form only), area between the two above said curves
(the region should be clearly identifiable).
5. Differential Equations: (10) Periods
Definition, order and degree, general and particular solutions of a differential
equation. Formation of differential equation whose general solution is given.
Solution of differential equations by method of separation of variables,
homogeneous differential equations of first order and first degree. Solutions of
linear differential equation of the type:
+ py = q, where p and q are functions of x.

UNIT-IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY
1. Vectors: (12) Periods
Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of
vectors. Types of vectors (equal, unit, zero, parallel and collinear vectors), position
vector of a point, negative of a vector, components of a vector, addition of vectors,
multiplication of a vector by a scalar, position vector of a point dividing a line
segment in a given ratio. Scalar (dot) product of vectors, projection of a vector on a
line. Vector (cross) product of vectors.
2. Three - dimensional Geometry: (12) Periods
Direction cosines/ratios of a line joining two points. Cartesian and vector equation
of a line, coplanar and skew lines, shortest distance between two lines. Cartesian
and vector equation of a plane. Angle between (i) two lines, (ii) two planes. (iii) a
line and a plane. Distance of a point from a plane.
UNIT-V: LINEAR PROGRAMMING
1. Linear Programming: (12) Periods
Introduction, definition of related terminology such as constraints, objective function,
optimization, different types of linear programming (L.P.) problems, mathematical
formulation of L.P. problems, graphical method of solution for problems in two
variables, feasible and infeasible regions, feasible and infeasible solutions, optimal feasible
solutions (up to three non-trivial constraints).
UNIT-VI: PROBABILITY
1. Probability: (18) Periods
Multiplication theorem on probability. Conditional probability, independent events, total
probability, Baye's theorem, Random variable and its probability distribution, mean and
variance of haphazard variable. Repeated independent (Bernoulli) trials and Binomial
distribution.