Page 1
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. 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). Invertible matrices and proof of the uniqueness of
inverse, if it exists; (Here all matrices will have real entries).
2. Determinants 25 Periods
Page 2
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. 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). 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,
???????? sin
-1
?? , cos
-1
?? and tan
-1
?? , 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 real-
life 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
x
2
± a
2,
?
dx
vx
2
± a
2
, ?
dx
va
2
- x
2
, ?
dx
ax
2
+ bx + c
, ?
dx
vax
2
+ ???? + ??
?
px + q
ax
2
+ bx + c
dx, ?
px + q
vax
2+
bx + c
dx, ? va
2
± x
2
dx, ?
v
x
2
- a
2
dx
?v????
2
+ ???? + ?? ???? ,
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:
Page 3
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. 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). 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,
???????? sin
-1
?? , cos
-1
?? and tan
-1
?? , 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 real-
life 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
x
2
± a
2,
?
dx
vx
2
± a
2
, ?
dx
va
2
- x
2
, ?
dx
ax
2
+ bx + c
, ?
dx
vax
2
+ ???? + ??
?
px + q
ax
2
+ bx + c
dx, ?
px + q
vax
2+
bx + c
dx, ? va
2
± x
2
dx, ?
v
x
2
- a
2
dx
?v????
2
+ ???? + ?? ???? ,
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:
dy
dx
+ py = q, where p and q are functions of x or constants.
d?? d?? + 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.
Page 4
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. 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). 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,
???????? sin
-1
?? , cos
-1
?? and tan
-1
?? , 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 real-
life 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
x
2
± a
2,
?
dx
vx
2
± a
2
, ?
dx
va
2
- x
2
, ?
dx
ax
2
+ bx + c
, ?
dx
vax
2
+ ???? + ??
?
px + q
ax
2
+ bx + c
dx, ?
px + q
vax
2+
bx + c
dx, ? va
2
± x
2
dx, ?
v
x
2
- a
2
dx
?v????
2
+ ???? + ?? ???? ,
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:
dy
dx
+ py = q, where p and q are functions of x or constants.
d?? d?? + 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
S.
No.
Typology of Questions
Total
Marks
%
Weightage
1
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 25
3
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: 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
Page 5
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. 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). 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,
???????? sin
-1
?? , cos
-1
?? and tan
-1
?? , 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 real-
life 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
x
2
± a
2,
?
dx
vx
2
± a
2
, ?
dx
va
2
- x
2
, ?
dx
ax
2
+ bx + c
, ?
dx
vax
2
+ ???? + ??
?
px + q
ax
2
+ bx + c
dx, ?
px + q
vax
2+
bx + c
dx, ? va
2
± x
2
dx, ?
v
x
2
- a
2
dx
?v????
2
+ ???? + ?? ???? ,
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:
dy
dx
+ py = q, where p and q are functions of x or constants.
d?? d?? + 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
S.
No.
Typology of Questions
Total
Marks
%
Weightage
1
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 25
3
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: 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.
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