ISC Class 12 Syllabus 2028 Mathematics

ISC
INDIAN SCHOOL CERTIFICATE
EXAMINATION

YEAR 2028

MATHEMATICS
(860)

Developed by:
Research, Development and Curriculum Division (RDCD)
CISCE

January 2026
____________________________________________________________________________________________

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Council for the Indian School Certificate Examinations (CISCE)

MISSION STATEMENT

The Council for the Indian School Certificate
Examinations is committed to serving the nation's
children, through high quality educational
endeavours, empowering them to contribute towards
a humane, just and pluralistic society, promoting
introspective living, by creating exciting learning
opportunities, with a commitment to excellence.

ETHOS OF CISCE

Trust and fair play.
Minimum monitoring.
Allowing schools to evolve their own niche.
Catering to the needs of the children.
Giving freedom to experiment with new ideas
and practices.
Diversity and plurality - the basic strength for
evolution of ideas.
Schools to motivate pupils towards the
cultivation of:
Excellence - The Indian and Global
experience.
Values - Spiritual and cultural - to be the bedrock
of the educational experience.
Schools to have an 'Indian Ethos', strong roots in
the national psyche and be sensitive to national
aspirations.

CLASS XII
There will be two papers in the subject:
Paper I : Theory (3 hours) ……80 marks
Paper II: Project Work ……….20 marks

PAPER I (THEORY) : 80 Marks

DISTRIBUTION OF MARKS FOR THE THEORY PAPER

S.No. UNIT TOTAL WEIGHTAGE

1. Relations and Functions 10 Marks

2. Algebra 10 Marks

3. Calculus 35 Marks

4 Vector Algebra 5 Marks

5 Three - Dimensional Geometry 6 Marks

6. Linear Programming 5 Marks

7. Probability 9 Marks

TOTAL 80 Marks

1. Relations and Functions
(i) Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto
functions, composite function and inverse of a function.
• Relations as:
- Relation on a set A.
- Identity relation, empty relation, universal relation.
- Types of Relations: reflexive, symmetric, transitive and equivalence relation.
• Functions:
- As special relations, concept of writing ‘y is a function of x’ as y = f(x).
- Types: one to one, many to one, into, onto.
- Real Valued function.
- Domain and range of a function.
- Conditions of invertibility.
- Sketching of graph of a function and its inverse.
- Composite functions and Invertible functions (algebraic functions only).
(ii) Inverse Trigonometric Functions
Definition, domain, range, principal value branch. Graphs of inverse trigonometric functions.
Elementary properties of inverse trigonometric functions.
- Principal values.
- sin-1x, cos-1x, tan-1x, etc.

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x
- sin-1x = cos −1 1 − x 2 =
tan −1 .
1 − x2
1 π
- sin-1x= cosec −1 ; sin-1x + cos-1x=
x 2
and similar relations for cot-1x, tan-1x, etc.

𝑠𝑠𝑠𝑠𝑛𝑛−1 𝑥𝑥 ± 𝑠𝑠𝑠𝑠𝑛𝑛−1 𝑦𝑦 = 𝑠𝑠𝑠𝑠𝑠𝑠 −1 �𝑥𝑥�1 − 𝑦𝑦 2 ± 𝑦𝑦�1 − 𝑥𝑥 2 �
𝑐𝑐𝑐𝑐𝑠𝑠 −1 𝑥𝑥 ± 𝑐𝑐𝑐𝑐𝑠𝑠 −1 𝑦𝑦 = 𝑐𝑐𝑐𝑐𝑠𝑠 −1 �𝑥𝑥𝑥𝑥 ∓ �1 − 𝑦𝑦 2 �1 − 𝑥𝑥 2 �
𝑥𝑥 + 𝑦𝑦
similarly, t𝑎𝑎𝑛𝑛−1 𝑥𝑥 + 𝑡𝑡𝑡𝑡𝑛𝑛−1 𝑦𝑦 = 𝑡𝑡𝑡𝑡𝑛𝑛−1 , 𝑥𝑥𝑥𝑥 < 1
1 − 𝑥𝑥𝑥𝑥
𝑥𝑥 − 𝑦𝑦
t𝑎𝑎𝑛𝑛−1 𝑥𝑥 − 𝑡𝑡𝑡𝑡𝑛𝑛−1 𝑦𝑦 = 𝑡𝑡𝑡𝑡𝑛𝑛−1 , 𝑥𝑥𝑥𝑥 > −1
1 + 𝑥𝑥𝑥𝑥
- Formulae for 2sin-1x, 2cos-1x, 2tan-1x, 3tan-1x etc. and application of these formulae.

2. Algebra
Matrices and Determinants
(i) Matrices
Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix,
symmetric and skew symmetric matrices. Operation 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 upto 3). Invertible matrices and proof of the
uniqueness of inverse, if it exists (here all matrices will have real entries).
(ii) Determinants
Determinant of a square matrix (up to 3×3 matrices), properties of determinants, 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.
- Types of matrices (m × n; m, n ≤ 3), order; Diagonal matrix, Scalar matrix, Identity matrix,
Triangular matrix.
- Symmetric, Skew symmetric matrices. Properties of Symmetric, Skew symmetric matrices.
- Operation – addition, subtraction, multiplication of a matrix with scalar, multiplication of two
matrices (the compatibility).
1 1 
  1 2 = AB( say )
E.g. 0 2  but BA is not possible.
 2 2
1 1 

- Singular and non-singular matrices.
- Existence of two non-zero matrices whose product is a zero matrix.
- Properties of adjoint of a square matrix.
AdjA
- Inverse (2×2, 3×3) A −1 =
A
- Properties of inverse
• Martin’s Rule (i.e. using matrices)
a1x + b1y + c1z = d1

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a2x + b2y + c2z = d2
a3x + b3y + c3z = d3

 a 1 b 1 c1   d1   x
A = a 2 b2 c 2  B = d 2  X =  y 
a 3 b3 c3   d 3   z 

AX = B ⇒ X = A −1 B
Problems based on above.
NOTE: The conditions for consistency of equations in two and three variables, using matrices, are to
be covered.
• Determinants
- Order.
- Minors.
- Cofactors.
- Expansion.
- Applications of determinants in finding the area of triangle and collinearity.
- Properties of determinants. Problems based on properties of determinants.

3. Calculus
(i) Continuity, Differentiability and Differentiation. Continuity and differentiability, derivative of
composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit
functions.
Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of
functions expressed in parametric forms. Second order derivatives.
• Continuity
- Continuity of a function at a point x = a.
- Continuity of a function in an interval.
- Algebra of continuous function.
- Removable discontinuity. Types of removable discontinuity.
• Differentiation
- Concept of continuity and differentiability of x , [x], etc.
- Derivatives of trigonometric functions.
- Derivatives of exponential functions.
- Derivatives of logarithmic functions.
- Derivatives of inverse trigonometric functions - differentiation by means of substitution.
- Derivatives of implicit functions and chain rule.
- Derivatives of Parametric functions.
- Differentiation of a function with respect to another function e.g. differentiation of sin x3 with
respect to x3.
x
- Logarithmic Differentiation - Finding dy/dx when y = x x .
- Successive differentiation up to 2nd order.
NOTE: Derivatives of composite functions using chain rule.

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(ii) Applications of Derivatives
Applications of derivatives: rate of change of quantities, increasing/decreasing functions, tangent
and normal, maxima and minima (first derivative test motivated geometrically and second
derivative test given as a provable tool). Problems that illustrate basic principles and understanding of
the subject as well as real-life situations.
• Equation of Tangent and Normal, Angle between two curves
• Rate measure.
• Increasing and decreasing functions.
• Maxima and minima.
- Critical points, Stationary/turning points, Extreme points.
- Absolute maxima/minima
- Local maxima/minima
- First derivatives test and second derivatives test
- Application problems based on maxima and minima.
(iii) Integrals
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.
Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and
evaluation of definite integrals.
• Indefinite integral
- Integration as the inverse of differentiation.(anti-derivative).
- Anti-derivatives of polynomials and functions (ax +b)n, sin x, cos x, sec2x, cosec2x etc.
- Integrals of the type sin2x, sin3x, sin4x, cos2x, cos3x, cos4x.
- Integration of 1/x, ex.
- Integration by substitution.
f ′( x)
- Integrals of the type f ' (x)[f (x)]n, .
f ( x)
- Integration of tan x, cot x, sec x, cosec x.
- Integration by parts.
- Integration using partial fractions.
f ( x)
Expressions of the form when degree of f(x) < degree of g(x)
g ( x)
x+2 A B
E.g. = +
( x − 3)( x + 1) x − 3 x + 1
x+2 A B C
= + +
( x − 2)( x − 1) 2
x − 1 ( x − 1) 2
x−2

x +1 Ax + B C
= 2 +
( x + 3)( x − 1) x + 3 x − 1
2

When degree of f (x) ≥ degree of g(x),
x2 +1  3x + 1 
e.g. 2
= 1−  2 
x + 3x + 2  x + 3x + 2 

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• Integrals of the type:
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑝𝑝𝑝𝑝+𝑞𝑞 𝑝𝑝𝑝𝑝+𝑞𝑞
∫ 𝑥𝑥 2 ±𝑎𝑎2 , ∫ ,∫ 𝑑𝑑𝑑𝑑, ∫ 𝑑𝑑𝑑𝑑 and ∫ �𝑎𝑎2 ± 𝑥𝑥 2 𝑑𝑑𝑑𝑑, ∫ √𝑥𝑥 2 − 𝑎𝑎2 𝑑𝑑𝑑𝑑,
�𝑥𝑥 2 ±𝑎𝑎2 𝑎𝑎𝑥𝑥 2 +𝑏𝑏𝑏𝑏+𝑐𝑐 √𝑎𝑎𝑥𝑥 2 +𝑏𝑏𝑏𝑏+𝑐𝑐

∫ √𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 𝑑𝑑𝑑𝑑, ∫(𝑝𝑝𝑝𝑝 + 𝑞𝑞)√𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 𝑑𝑑𝑑𝑑, integrations reducible to the above forms.
dx 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 (a cos x + b sin x)dx
∫ a cos x + b sin x , ∫ 𝑎𝑎+𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐 𝑥𝑥 , ∫ 𝑎𝑎+𝑏𝑏 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥 ∫ 𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐 𝑥𝑥+𝑏𝑏 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥+𝑐𝑐 , ∫ c cos x + d sin x ,
2
dx 1± x dx
∫ a cos2 x + b sin 2 x + c , 1 + x 4 dx , ∫ ∫ 1 + x , ∫ tan x dx, ∫ cot x dx etc.
4

• Definite Integral
- Fundamental theorem of calculus (without proof)
- Properties of definite integrals.
- Problems based on the following properties of definite integrals are to be covered.
b b

∫ f ( x)dx = ∫ f (t )dt
a a

b a

∫ f ( x)dx = − ∫ f ( x)dx
a b

b c b

∫ f ( x)dx = ∫ f ( x)dx + ∫ f ( x)dx , where a < c < b
a a c

b b

∫ f ( x)dx = ∫ f (a + b − x)dx
a a

a a

∫0
f (=
x)dx ∫ f (a − x)dx
0

𝑎𝑎
2𝑎𝑎 2 ∫0 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑 , 𝑖𝑖𝑖𝑖 𝑓𝑓(2𝑎𝑎 − 𝑥𝑥) = 𝑓𝑓(𝑥𝑥)
∫0 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑 = �
0, 𝑓𝑓(2𝑎𝑎 − 𝑥𝑥) = −𝑓𝑓(𝑥𝑥)
𝑎𝑎
𝑎𝑎
2 � 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑,if 𝑓𝑓 is an even function
� 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑 = � 0
−𝑎𝑎
0, if 𝑓𝑓 is an odd function
(iv) Application of Integrals (Area under the curve) - Application in finding the area bounded b y simple
curves and coordinate axes. Area enclosed between two curves.
- Application of definite integrals - area bounded by curves, lines and coordinate axes is required
to be covered.
- Simple curves: lines, circles/ parabolas/ ellipses, polynomial functions, modulus function,
exponential function, logarithmic function.
(v) Differential Equations
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
+ p𝑦𝑦= q, where p and q are functions of x or constants.
dx
dx
+ p𝑥𝑥 = q, where p and q are functions of 𝑦𝑦 or constants.
dy

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- Differential equations, order and degree.
- Formation of differential equation by eliminating arbitrary constant(s).
- Solution of differential equations.
- Variable separable.
- Homogeneous equations.
dy
- Linear form + Py = Q ,where P and Q are functions of x/constant. Similarly, for dx/dy.
dx
NOTE 1: Equations reducible to variable separable type are included.
NOTE 2: The second order differential equations are excluded.

4. Vector Algebra
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.
- As directed line segments.
- Magnitude and direction of a vector.
- Types: equal vectors, unit vectors, zero vector.
- Position vector.
- Components of a vector.
- Vectors in two and three dimensions.
- iˆ, ˆj , kˆ as unit vectors along the x, y and the z axes; expressing a vector in terms of the unit vectors.
- Operations: Sum and Difference of vectors; scalar multiplication of a vector.
- Section formula.
- Scalar (dot) product of vectors and its geometrical significance.
- Scalar and vector projection.
- Cross product and its geometrical significance. Its properties - area of a triangle, area of
parallelogram, collinear vectors.
NOTE: Proofs of geometrical theorems by using Vector algebra are excluded.

5. Three - dimensional Geometry
Direction cosines and direction ratios of a line joining two points. Cartesian equation 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.
- Equation of x-axis, y-axis, z-axis and lines parallel to them.
- Equation of xy-plane, yz–plane, zx–plane.
- Direction cosines, direction ratios.
- Angle between two lines in terms of direction cosines /direction ratios.
- Condition for lines to be perpendicular/parallel.
• Lines
- Cartesian and vector equations of a line through one and two points.
- Coplanar and skew lines.
- Conditions for intersection of two lines.
- Distance of a point from a line.
- Shortest distance between two lines.

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• Planes
- Cartesian and vector equation of a plane.
- Direction ratios of the normal to the plane.
- One point form.
- Normal form.
- Intercept form.
- Distance of a point from a plane.
- Intersection of the line and plane.
- Angle between two planes, a line and a plane.

6. 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 (bounded and unbounded) and infeasible regions, feasible
and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).
Introduction, definition of related terminology such as constraints, objective function, optimization,
advantages of linear programming; limitations of linear programming; application areas of linear
programming; different types of linear programming (L.P.) problems, mathematical formulation of L.P
problems, graphical method of solution for problems in two variables, feasible (bounded/ unbounded)
and infeasible regions, feasible and infeasible solutions, optimum feasible solution(may/may not exists).

7. Probability
Conditional probability, multiplication theorem on probability, independent events, total probability,
Bayes’ theorem, Random variable and its probability distribution, mean of r a n d o m variable.
- Independent and dependent events conditional events.
- Laws of Probability, addition theorem, multiplication theorem, conditional probability.
- Theorem of Total Probability.
- Bayes’ theorem.
- Theoretical probability distribution, probability distribution function; mean of random variable.

PAPER II (PROJECT WORK) : 20 Marks
Candidates will be expected to have completed two projects.
The project work will be assessed by the subject teacher and a Visiting Examiner appointed locally and
approved by the Council.
Mark allocation for each Project [10 marks]:

Overall format 1 mark
Content 4 marks
Findings 2 marks
Viva-voce based on the Project 3 marks
Total 10 marks

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List of suggested assignments for Project Work:
1. Using a graph, demonstrate a function which is one-one but not onto.
2. Using a graph demonstrate a function which is invertible.
3. Draw the graph of y = sin-1 x (or any other inverse trigonometric function), using the graph of y = sin x
(or any other relevant trigonometric function). Demonstrate the concept of mirror line (about y = x) and
find its domain and range.
4. Explore the principal value of the function sin-1 x (or any other inverse trigonometric function) using a unit
circle.
5. Find the derivatives of a determinant of the order of 3 x 3 and verify the same by other methods.
6. Verify the consistency of the system of three linear equations of two variables and verify the same
graphically. Give its geometrical interpretation.
7. For a dependent system (non-homogeneous) of three linear equations of three variables, identify infinite
number of solutions.
8. Explain the concepts of increasing and decreasing functions, using geometrical significance of dy/dx.
Illustrate with proper examples.
9. Explain and illustrate (with suitable examples) the concept of local maxima and local minima using graph.
10. Explain and illustrate (with suitable examples) the concept of absolute maxima and absolute minima using
graph.
11. Explain the conditional probability, the theorem of total probability and the concept of Bayes’ theorem
with suitable examples.
12. Using any suitable data, find the minimum cost by applying the concept of Transportation problem.
13. Using any suitable data, find the minimum cost and maximum nutritional value by applying the concept
of Diet problem.
14. Using any suitable data, find the Optimum cost in the manufacturing problem by formulating a linear
programming problem (LPP).
15. Using vector algebra, find the area of a parallelogram/triangle. Also, derive the area analytically and verify
the same.
16. Using Vector algebra, prove the formulae of properties of triangles (sine/cosine rule, etc.)
17. Using Vector algebra, prove the formulae of compound angles, e.g. sin (A + B) = Sin A Cos B + Sin B
Cos A, etc.
18. Find the image of a line with respect to a given plane.
19. Find the distance of a point from a given plane measured parallel to a given line.
20. Find the distance of a point from a line measured parallel to a given plane.
21. Find the area bounded by a parabola and an oblique line.
22. Find the area bounded by a circle and an oblique line.
23. Find the area bounded by an ellipse and an oblique line.
24. Find the area bounded by a circle and a circle.
25. Find the area bounded by a parabola and a parabola.
26. Find the area bounded by a circle and a parabola.
(Any other pair of curves which are specified in the syllabus may also be taken.)
27. Analyse - Three methods (proofs) to find the area under the curve.
28. Tessellation: Types of tessellations, Geometrical shape of tessellation, Tessellation in nature, Man made
tessellation, application in real life situation.
29. Sacred Geometry and Euclid geometry comparison and different approach (Mathematically)
30. Derivation of 𝑒𝑒 ∫ 𝑝𝑝𝑝𝑝𝑝𝑝 and its applications.
31. Differentiation – instantaneous rate of change of displacement, why not rate of change of one variable with
reference to other variable.
32. Derivation of determinant.
NOTE: No question paper for Project Work will be set by CISCE.

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SAMPLE TABLE FOR PROJECT WORK

S. No. Unique PROJECT 1 PROJECT 2 TOTAL
Identification MARKS
Number A B C D E F G H I J
(Unique ID) of Teacher Visiting Average Viva- Total Teacher Visiting Average Viva- Total (E + J)
the candidate Examiner Marks Voce by Marks Examiner Marks Voce by Marks
(A + B ÷ Visiting (C + D) (F + G ÷ Visiting (H + I)
2) Examiner 2) Examiner
7 Marks* 7 Marks* 7 Marks 3 Marks 10 Marks 7 Marks* 7 Marks* 7 Marks 3 Marks 10 Marks 20
Marks
1
2
3
4
5
6
7
8
9
10

*Breakup of 7 Marks to be awarded separately by
Name of Teacher:
the Teacher and the Visiting Examiner is as follows:
Signature: Date
Overall Format 1 Mark
Content 4 Marks Name of Visiting Examiner
Findings 2 Marks
Signature: Date
NOTE: VIVA-VOCE (3 Marks) for each Project is to be conducted only by the Visiting Examiner, and should be based on the Project only.

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