** CBSE Sample Papers**

# (Syllabus) CBSE Class 11th & 12th : Mathematics Syllabus 2011 (Code No: 041)

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Courses of Studies 2011

Class : 11th & 12th

MATHEMATICS (Code No 041)

Courses of Studies 2011

Class : 11th & 12th

MATHEMATICS (Code No 041)

**The Syllabus in the subject of Mathematics has undergone changes from time to
time in accordance with growth of the subject and emerging needs of the society.
Senior Secondary stage is a launching stage from where the students go either
for higher academic education in Mathematics or for professional courses like
engineering, physical and Bioscience, commerce or computer applications. The
present revised syllabus has been designed in accordance with National
Curriculum Frame work 2005 and as per guidelines given in Focus Group on
Teaching of Mathematics 2005 which is to meet the emerging needs of all
categories of students. Motivating the topics from real life situations and
other subject areas, greater emphasis has been laid on application of various
concepts.**

**Objectives **

The broad objectives of teaching Mathematics at senior school stage intend to
help the pupil:

to acquire knowledge and critical understanding, particularly by way of
motivation and visualization, of basic concepts, terms, principles, symbols and
mastery of underlying processes and skills.

to feel the flow of reasons while proving a result or solving a problem.

to apply the knowledge and skills acquired to solve problems and wherever
possible, by more than one method.

to develop positive attitude to think, analyze and articulate logically.

to develop interest in the subject by participating in related competitions.

to acquaint students with different aspects of mathematics used in daily life.

to develop an interest in students to study mathematics as a discipline.

to develop awareness of the need for national integration, protection of
environment, observance of small family norms, removal of social barriers,
elimination of sex biases.

to develop reverence and respect towards great Mathematicians for their
contributions to the field of Mathematics.

**COURSE STRUCTURE **

Class XI

One Paper Three Hours Max Marks. 100

Class XI

One Paper Three Hours Max Marks. 100

I. SETS AND FUNCTIONS 29 Marks

II. ALGEBRA 37 Marks

III. COORDINATE GEOMETRY 13 Marks

IV. CALCULUS 06 Marks

V. MATHEMATICAL REASONING 03 Marks

VI. STATISTICS AND PROBABILITY 12 Marks

**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:

tan (x ± y) = (tan x ± tany)/ (1 ± tan x . tan y) , cot (x ± y) = (cot x . cot y ± 1)/ (cot y ± cot x)

sin x + sin y = 2 sin (x + y)/2 . cos (x – y)/2 , cos x + cos y = 2 cos (x + y)/2 . cos (x – y)/2

sin x – sin y = 2 cos (x + y)/2 . sin (x – y)/2 , cos x – cos y = – 2 sin (x + y)/2 . sin (x – y)/2

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**

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.

**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**

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 words difference between contradiction,
converse and contra positive.

**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.

**Recommended Textbooks.**

1) Mathematics Part I – Textbook for Class XI, NCERT Publication

2) Mathematics Part II – Textbook for Class XI, NCERT Publication

**CLASS XII **

One Paper Three Hours Marks: 100

One Paper Three Hours Marks: 100

I. RELATIONS AND FUNCTIONS
10 Marks

II.
ALGEBRA
13 Marks

III.
CALCULUS
44 Marks

IV. VECTORS AND THREE – DIMENSIONAL GEOMETRY 17 Marks

V. LINEAR PROGRAMMING
06 Marks

VI.
PROBABILITY
10 Marks

**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.

**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.

∫dx/(x^{2} ±a^{2}), ∫dx/√(x^{2} ±a^{2}),
∫dx/√(a^{2} – x^{2}), ∫dx/(ax^{2} + bx^{2} + c),
∫dx/√(ax^{2} + bx + c)

∫(px+q)/(ax^{2} + bx + c)dx, ∫(px+q)/√(ax^{2} + bx + c)dx,
∫√(a^{2} ± x^{2}) dx and ∫√(x^{2} – a^{2})dx

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:

dy/dx + 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 random variable. Repeated independent
(Bernoulli) trials and Binomial distribution.

**Recommended Textbooks.**

1) Mathematics Part I – Textbook for Class XII, NCERT Publication

2) Mathematics Part II – Textbook for Class XII, NCERT Publication