The broad objectives of teaching Mathematics at senior school stage intend to help the students:
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 gender
biases.
to develop reverence and respect towards great Mathematicians for their contributions
to the field of Mathematics.
COURSE STRUCTURE
CLASS XI (2023-24)
One Paper Total Period–240 [35 Minutes each]
Three Hours Max Marks: 80
No. Units No. of Periods Marks
I. Sets and Functions 60 23
II. Algebra 50 25
III. Coordinate Geometry 50 12
IV. Calculus 40 08
V. Statistics and Probability 40 12
Total 240 80
Internal Assessment 20
*No chapter/unit-wise weightage. Care to be taken to cover all the chapters.
Unit-I: Sets and Functions
1. Sets (20) Periods
Sets and their representations, Empty set, Finite and Infinite sets, Equal sets, Subsets, Subsets of
a set of real numbers especially intervals (with notations). Universal set. Venn diagrams. Union
and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement.
2. Relations & Functions (20) Periods
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 quotients of functions.
3. Trigonometric Functions (20) Periods
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 sin2x + cos2x = 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 sinx, siny,
cosx & cosy and their simple applications. Deducing identities like the following:
tan(x ± y) =
tan x ± tan y
1 ∓ tan x tan y
, cot(x ± y) =
cot x cot y ∓ 1
cot y ± cot x
sinα ± sinβ = 2sin
12(α ± β)cos
12(α ∓ β) cosα + cosβ = 2cos
12(α + β)cos
12(α − β) 𝑐𝑜𝑠𝛼 − 𝑐𝑜𝑠𝛽 = −2𝑠𝑖𝑛
12 (𝛼 + 𝛽)𝑠𝑖𝑛 12(𝛼 − 𝛽)
Identities related to sin2x, cos2x, tan2 x, sin3x, cos3x and tan3x.
Unit-II: Algebra
1. Complex Numbers and Quadratic Equations (10) Periods
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
2. Linear Inequalities (10) Periods
Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation
on the number line.
3. Permutations and Combinations (10) Periods
Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of
Formulae for nPr and nCr and their connections, simple applications.
4. Binomial Theorem (10) Periods
Historical perspective, statement and proof of the binomial theorem for positive integral indices.
Pascal’s triangle, simple applications.
5. Sequence and Series (10) Periods
Sequence and Series. Arithmetic Mean (A.M.) Geometric Progression (G.P.), general term of a
G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between
A.M. and G.M.
Unit-III: Coordinate Geometry
1. Straight Lines (15) Periods
Brief recall of two dimensional geometry from earlier classes. 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, Distance of a point from a line.
2. Conic Sections (25) Periods
Sections of a cone: circles, 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 (10) Periods
Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance
between two points.
Unit-IV: Calculus
1. Limits and Derivatives (40) Periods
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.
Unit-V Statistics and Probability
1. Statistics (20) Periods
Measures of Dispersion: Range, Mean deviation, variance and standard deviation of
ungrouped/grouped data.
2. Probability (20) Periods
Events; occurrence of events, ‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive
events, Axiomatic (set theoretic) probability, connections with other theories of earlier classes.
Probability of an event, probability of ‘not’, ‘and’ and ‘or’ events.
MATHEMATICS
QUESTION PAPER DESIGN
CLASS – XI
No. Typology of Questions Total
Remembering: Exhibit memory of previously learned material by
recalling facts, terms, basic concepts, and answers.
Understanding: Demonstrate understanding of facts and ideas by
organizing, comparing, translating, interpreting, giving descriptions,
and stating main ideas
44 55
Applying: Solve problems to new situations by applying acquired
knowledge, facts, techniques and rules in a different way. 20 25
Analysing :
Examine and break information into parts by identifying motives or
causes. Make inferences and find evidence to support
generalizations
Evaluating:
Present and defend opinions by making judgments about
information, validity of ideas, or quality of work based on a set of
criteria.
Creating:
Compile information together in a different way by combining
elements in a new pattern or proposing alternative solutions
16 20
Total 80 100
1. No chapter wise weightage. Care to be taken to cover all the chapters
2. Suitable internal variations may be made for generating various templates keeping the overall
weightage to different form of questions and typology of questions same.
Choice(s):
There will be no overall choice in the question paper.
However, 33% internal choices will be given in all the sections
Note: Please refer the guidelines given under XII Mathematics Syllabus:
INTERNAL ASSESSMENT 20 MARKS
Periodic Tests ( Best 2 out of 3 tests conducted) 10 Marks
Mathematics Activities 10 Marks
CLASS-XII
(2023-24)
One Paper Max Marks: 80
No. Units No. of Periods Marks
I. Relations and Functions 30 08
II. Algebra 50 10
III. Calculus 80 35
IV. Vectors and Three – Dimensional Geometry 30 14
V. Linear Programming 20 05
VI. Probability 30 08
Total 240 80
Internal Assessment 20
Unit-I: Relations and Functions
1. Relations and Functions 15 Periods
Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto
functions.
2. Inverse Trigonometric Functions 15 Periods
Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.
Unit-II: Algebra
1. Matrices 25 Periods
Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix,
symmetric and skew symmetric matrices. Operations on matrices: Addition and multiplication and
multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the
zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of
inverse, if it exists; (Here all matrices will have real entries).
2. Determinants 25 Periods
Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors 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 20 Periods
Continuity and differentiability, chain rule, derivative of inverse trigonometric functions,
𝑙𝑖𝑘𝑒 sin−1 , cos−1 𝑥 and tan−1 𝑥, derivative of implicit functions. Concept of exponential and logarithmic
functions.
Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions
expressed in parametric forms. Second order derivatives.
2. Applications of Derivatives 10 Periods
Applications of derivatives: rate of change of quantities, increasing/decreasing functions, 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 reallife situations).
3. Integrals 20 Periods
Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by
partial fractions and by parts, Evaluation of simple integrals of the following types and problems based
on them.
, ∫ dx √ax2 + 𝑏𝑥 + 𝑐 ∫ px + q ax2 + bx + c dx, ∫ px + q √ax2+bx + c dx, ∫ √a
2 ± x2 dx, ∫ √x 2 − a
2 dx∫√𝑎𝑥2 + 𝑏𝑥 + 𝑐 𝑑𝑥,
Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation
of definite integrals.
4. Applications of the Integrals 15 Periods
Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in
standard form only)
5. Differential Equations 15 Periods
Definition, order and degree, general and particular solutions of a differential equation. Solution of
differential equations by method of separation of variables, solutions of homogeneous differential
equations of first order and first degree. Solutions of linear differential equation of the type:
dydx + py = q, where p and q are functions of x or constants.
d𝑥d𝑦+ px = q, where p and q are functions of y or constants.
Unit-IV: Vectors and Three-Dimensional Geometry
1. Vectors 15 Periods
Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a
vector. 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. Definition, Geometrical Interpretation,
properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.
2. Three – dimensional Geometry 15 Periods
Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation
of a line, skew lines, shortest distance between two lines. Angle between two lines.
Unit-V: Linear Programming
1. Linear Programming 20 Periods
Introduction, related terminology such as constraints, objective function, optimization, graphical method
of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded),
feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).
Unit-VI: Probability
1. Probability 30 Periods
Conditional probability, multiplication theorem on probability, independent events, total probability,
Bayes’ theorem, Random variable and its probability distribution, mean of random variable.
MATHEMATICS (Code No. – 041)
QUESTION PAPER DESIGN CLASS – XII
(2023-24)
Time: 3 hours Max. Marks: 80
No. Typology of Questions Total
Remembering: Exhibit memory of previously learned material
by recalling facts, terms, basic concepts, and answers.
Understanding: Demonstrate understanding of facts and
ideas by organizing, comparing, translating, interpreting, giving
descriptions, and stating main ideas
44 55
2
Applying: Solve problems to new situations by applying
acquired knowledge, facts, techniques and rules in a different
way.
20 2
Analysing :
Examine and break information into parts by identifying
motives or causes. Make inferences and find evidence to
support generalizations
Evaluating:
Present and defend opinions by making judgments about
information, validity of ideas, or quality of work based on a set
of criteria.
Creating:
Compile information together in a different way by combining
elements in a new pattern or proposing alternative solutions
1. No chapter wise weightage. Care to be taken to cover all the chapters
2. Suitable internal variations may be made for generating various templates keeping the overall
weightage to different form of questions and typology of questions same.
Choice(s):
There will be no overall choice in the question paper.
However, 33% internal choices will be given in all the sections
Note: For activities NCERT Lab Manual may be referred.
INTERNAL ASSESSMENT 20 MARKS
Periodic Tests ( Best 2 out of 3 tests conducted) 10 Marks
Mathematics Activities 10 Marks
Conduct of Periodic Tests:
Periodic Test is a Pen and Paper assessment which is to be conducted by the respective
subject teacher. The format of periodic test must have questions items with a balance mix,
such as, very short answer (VSA), short answer (SA) and long answer (LA) to effectively
assess the knowledge, understanding, application, skills, analysis, evaluation and synthesis.
Depending on the nature of subject, the subject teacher will have the liberty of incorporating
any other types of questions too. The modalities of the PT are as follows:
a) Mode: The periodic test is to be taken in the form of pen-paper test.
b) Schedule: In the entire Academic Year, three Periodic Tests in each subject may be
conducted as follows:
Test Pre Mid-term (PT-I) Mid-Term (PT-II) Post Mid-Term (PT-III)
Tentative Month July-August November December-January
This is only a suggestive schedule and schools may conduct periodic tests as per their
convenience. The winter bound schools would develop their own schedule with similar time
gaps between two consecutive tests.
c) Average of Marks: Once schools complete the conduct of all the three periodic tests,
they will convert the weightage of each of the three tests into ten marks each for identifying
best two tests. The best two will be taken into consideration and the average of the two
shall be taken as the final marks for PT.
d) The school will ensure simple documentation to keep a record of performance as
suggested in detail circular no.Acad-05/2017.
e) Sharing of Feedback/Performance: The students’ achievement in each test must be
shared with the students and their parents to give them an overview of the level of learning
that has taken place during different periods. Feedback will help parents formulate
interventions (conducive ambience, support materials, motivation and morale-boosting)
to further enhance learning. A teacher, while sharing the feedback with student or parent,
should be empathetic, non- judgmental and motivating. It is recommended that the
teacher share best examples/performances of IA with the class to motivate all learners.