(Download) NCERT Revised Syllabus of Mathematics (Class 11 &
MATHEMATICS (CLASSES XI –XII)
(i) All concepts/identities must be illustrated by
(ii) The language of ‘word problems’ must be clear, simple and unambiguous.
(iii) Problems given should be testing the understanding of the subject.
(iv) All proofs to be produced in a manner that allow the learner to see flow of
reasons. Wherever possible, give more than one proof.
(v) Motivate results, wherever possible. Prove explicitly those results where a
short and clear argument reinforces mathematical thinking and reasoning. There
must be emphasis on correct way of expressing the arguments.
(Total Periods 180)
UNIT I: 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.
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 reals with itself (upto R ×
R × R).
Definition of relation, pictorial diagrams, 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 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
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 and
sketch of their graphs. Expressing sin (x+ y) and cos (x + y) in terms of sin x,
sin y, cos x and cos y. Deducing the identities like following:
Identities related to sin2x, cos2x, tan2x, sin3x, cos3x and tan3x. General
solution of trigonometric equations of the type sinθ = sinα, cosθ = cosα and
tanθ = tanα. Proofs and simple applications of sine and cosine formulae.
UNIT II : 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
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. Solution of system of linear inequalities
in two variables - graphially.
4. Permutations and Combinations
Fundamental principle of counting. Factorial n. Permutations and combinations
derivation of formulae 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,
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 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 : ∑ n, ∑ n2 and ∑ n3
UNIT III : COORDINATE GEOMETRY
1. Straight Lines
Brief recall of 2-D 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, intercepts 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 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.
UNIT IV : CALCULUS
Limits and Derivatives
UNIT V: 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 VI : STATISTICS AND PROBABILITY
Measure of dispersion; mean deviation, variance and standard deviation of
ungrouped/grouped data. Analysis of frequency distributions with equal means but
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 the
theories of earlier classes. Probability of an event, probability of ‘not’,
‘and’, & ‘or’ events.
(Total Periods 180)
UNIT I: RELATIONS AND FUNCTIONS
1. Relations and Functions
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
Definition, range, domain, principal value branches. Graphs of inverse
Elementary properties of inverse trigonometric functions.
UNIT II: ALGEBRA
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
Determinant of a square matrix (up to 3 × 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.
UNIT III: CALCULUS
1. Continuity and Differentiability
Continuity and differentiability, derivative of composite functions, chain
rule, derivatives of inverse trigonometric functions, derivative of implicit
function. Concepts of exponential, logarithmic functions. Derivatives of loge
. Logarithmic differentiation. Derivative of functions expressed in parametric
forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems
and their geometric interpretations.
2. Applications of Derivatives
Applications of derivatives: Rate of change, increasing/decreasing functions,
tangents and 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).
Integration as inverse process of differentiation. Integration of a variety
of functions by substitution,
by partial fractions and by parts, only simple integrals of the type –
Definite integrals as a limit of a sum. Fundamental Theorem of Calculus (without
properties of definite integrals and evaluation of definite integrals.
4. Applications of the Integrals
Applications in finding the area under simple curves, especially lines, arcs
of circles/parabolas/ellipses (in standard form only), area between the two
above said curves (the region should be cleraly identifiable).
5. Differential Equations
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 –
UNIT IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY
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, scalar triple product.
2. Three-dimensional Geometry
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
Introduction, 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
Unit VI: Probability
Multiplications 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.