MATHS AND STATISTICS

MATHEMATICS CLASS 11

UNIT I: SETS AND FUNCTIONS

1. Sets (Periods 12)
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 (Periods 14)
Ordered pairs, Cartesian product of sets. The 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. The 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 (Periods 18)
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:
tan (x + y)
tan tan cot cot 1 ,cot ( ) 1 tan tan cot cot
x y xy x y
x y yx
±
= ±=
× ±
∓
∓
sin x + sin y = 2sin .cos ,cos cos 2cos cos 2 2 22
x y xy xy xy x y
+ − +−
+ =
sin x – sin y = 2cos .sin ,cos cos 2sin sin 2 2 22
x y xy xy xy x y
+ − +− − =−
Identities related to sin2x, cos2x, tan2x, sin3x, cos3x and tan3x. General solution of trigonometric
equations of the type sinθ = sin α, cosθ = cosα and tanθ = tan α. Proofs and simple applications
of sine and cosine formulae.
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UNIT II : ALGEBRA

1. Principle of Mathematical Induction (Periods 06)
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 (Periods 10)
Need for complex numbers, especially −1, 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, Square-root of a Complex number.

3. Linear Inequalities (Periods 10)
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 and Combinations (Periods 12)
The fundamental principle of counting. Factorial n. Permutations and combinations derivation of
formulae and their connections, simple applications.

5. Binomial Theorem (Periods 08)
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 (Periods 10)
Sequence and Series. Arithmetic Progression (A.P.), Arithmetic Mean (A.M.), Geometric
Progression (G.P.), general term of a G.P., the 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: 2 3 ∑∑ ∑ nn n , and

UNIT III: COORDINATE GEOMETRY

1. Straight Lines (Periods 09)
Brief recall of 2-D from earlier classes, shifting of origin. The slope of a line and the 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 (Periods 12)
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 (Periods 08)
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 (Periods 18)
Derivative introduced as rate of change both as that of distance function and geometrically,
intuitive idea of limit. → →
+
0 0
log (1 ) – 1 lim , lim
x
e
x x
x e
x x
. Definition of derivative, relate it to slope
of the tangent of the curve, a derivative of the sum, difference, product, and quotient of functions. Derivatives
of polynomial and trigonometric functions.

UNIT V: MATHEMATICAL REASONING (Periods 08)
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 a variety of examples related to real life and
Mathematics. Validating the statements involving the connecting words – the difference between
contradiction, converse, and contrapositive.

UNIT VI: STATISTICS AND PROBABILITY
1. Statistics (Periods 10)
The 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 (Periods 15)
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.

MATHEMATICS  CLASS XII (Total Periods 180)

UNIT I: RELATIONS AND FUNCTIONS

1. Relations and Functions (Periods 10)
Types of relations: Reflexive, symmetric, transitive, and equivalence relations. One to one and onto
functions, composite functions, the inverse of a function. Binary operations.

2. Inverse Trigonometric Functions (Periods 12)
Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions.
Elementary properties of inverse trigonometric functions.

UNIT II: ALGEBRA

1. Matrices (Periods 18)
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 matrices (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 (Periods 20)
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 (Periods 18)  Continuity and differentiability, a derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. Concepts of exponential, logarithmic functions.
Derivatives of loge
x
and ex. 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 (Periods 10)
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).
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3. Integrals (Periods 20)
Integration as the inverse process of differentiation. Integration of a variety of functions by substitution,
by partial fractions and by parts, only simple integrals of the type –
± ++ ± − ++ ∫∫ ∫ ∫ ∫ 22 2 22 22 2 ,,, , , dx dx dx dx dx
x a ax bx c x a a x ax bx c
( ) + + ( ) ± −
+ + + + ∫ ∫ ∫∫ 22 22
2 2 , , and , px q px q dx dx a x dx x a dx
ax bx c ax bx c
++ + ++ ( ) ∫ ∫ 2 2 ax bx c dx px q ax bx c dx and
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 (Periods 10)
Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ellipses (in standard form only), and area between the two above said curves (the region should be clearly
identifiable).

5. Differential Equations (Periods 10)
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 the method of
separation of variables, homogeneous differential equations of the first order and first degree. Solutions
of linear differential equation of the type –
, dy Py Q dx
+ = where P and Q are functions of x or constant
+ = , dx Px Q dy where P and Q are functions of y or constant

UNIT IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY

1. Vectors (Periods 10)

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, the 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 (Periods 12)
Direction cosines/ratios of a line joining two points. Cartesian and vector equation of a line, coplanar
and skew lines, the shortest distance between two lines. Cartesian and vector equation of a plane. The angle
between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane.
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Unit V: Linear Programming (Periods 12)

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 non-trivial constraints).

Unit VI: Probability (Periods 18)

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.