ISC Class 12 Syllabus 2027 Applied Mathematics

ISC YEAR 2027

INDIAN SCHOOL CERTIFICATE
EXAMINATION

APPLIED
MATHEMATICS
(885)

February 2025
____________________________________________________________________________________________

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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 6 Marks

2. Algebra 10 Marks

3. Calculus 25 Marks

4. Probability 14 Marks

5. Linear Programming 5 Marks

6. Financial Mathematics 14 Marks

7. Index numbers & Moving averages 6 Marks

TOTAL 80 Marks

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1. Relations and Functions 2. Algebra
(i) Types of relations: reflexive, symmetric, Matrices and Determinants
transitive and equivalence relations. One to (i) Matrices
one and onto functions, composite function
Concept, notation, order, equality, types of
and inverse of a function.
matrices, zero and identity matrix, transpose
• Relations as: of a matrix, symmetric and skew symmetric
- Relation on a set A matrices. Operation on matrices: Addition and
multiplication and multiplication with a
- Identity relation, empty relation,
scalar. Simple properties of addition,
universal relation.
multiplication and scalar multiplication. Non-
- Types of Relations: reflexive, commutativity of multiplication of matrices
symmetric, transitive and equivalence and existence of non-zero matrices whose
relation. product is the zero matrix (restrict to square
• Functions: matrices of order upto 3). Invertible matrices
and proof of the uniqueness of inverse, if it
- As special relations, concept of
exists (here all matrices will have real entries).
writing “y is a function of x” as y =
f(x). (ii) Determinants
- Types: one to one, many to one, into, Determinant of a square matrix (up to 3 x 3
onto. matrices), minors, co-factors and applications
- Real Valued function. of determinants in finding the area of a
triangle. Adjoint and inverse of a square
- Domain and range of a function.
matrix. Consistency, inconsistency and
- Conditions of inevitability. number of solutions of system of linear
- Sketching of graph of a function and equations by examples, solving system of
its inverse. linear equations in two or three variables
- Composite functions and Invertible (having unique solution) using inverse of a
functions (algebraic functions only). matrix/Cramer’s Rule.
(ii) Inverse Trigonometric Functions - Types of matrices (m × n; m, n ≤ 3),
order; Diagonal matrix, Scalar matrix,
Definition, domain, range, principal value Identity matrix, Triangular matrix.
branch. Graphs of inverse trigonometric
functions. Elementary properties of inverse - Symmetric, Skew symmetric matrices.
trigonometric functions. Properties of Symmetric, Skew symmetric
matrices.
- Principal values.
- Operation – addition, subtraction,
- sin-1x, cos-1x, tan-1x etc multiplication of a matrix with scalar,
x multiplication of two matrices (the
- sin-1x = cos −1 1 − x 2 =
tan −1 . compatibility).
1 − x2
1 1 
−1 1 π 1 2 
- -1
sin x= cosec -1
; sin x+cos x= -1
and E.g. 0 2  = AB( say ) but BA is
x 2  2 2
similar relations for cot-1x, tan-1x, etc. 1 1 
not possible.
- Singular and non-singular matrices.
- Existence of two non-zero matrices
whose product is a zero matrix.

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- Properties of adjoint of a square matrix. Rate of change, Increasing/ decreasing
functions, maxima and minima.
AdjA
- Inverse (2×2, 3×3) A −1 = • Differentiation
A
- Derivatives of trigonometric and
- Properties of inverse inverse trigonometric functions.
- Derivatives of exponential functions.
• Martin’s Rule (i.e. using matrices) - Derivatives of logarithmic functions.
- Derivatives of implicit functions and
a1x + b1y + c1z = d1
chain rule.
a2x + b2y + c2z = d2 - Derivatives of Parametric functions.
a3x + b3y + c3z = d3 - Differentiation of a function with
respect to another function e.g.
 a 1 b 1 c1   d1   x differentiation of sinx3 with respect to
A = a 2 b2 c 2  B = d 2  X =  y 
   
x3.
a 3 b3 c3   d 3   z  - Logarithmic Differentiation - Finding
x

AX = B ⇒ X = A −1 B dy/dx when y = x x .
Successive differentiation up to 2nd
-
Problems based on above. order.
NOTE: The conditions for consistency of NOTE: Derivatives of composite functions
equations in two and three variables, using using chain rule.
matrices/Determinants, are to be covered.
• Rate measure.
• Determinants
• Increasing and decreasing functions.
- Order.
• Maxima and minima.
- Minors.
- Stationary /turning points,
- Cofactors. - First derivatives test and second
- Expansion. derivatives test
- Applications of determinants in finding (ii) Integrals
the area of triangle and collinearity.
Integration as inverse process of
- Cramer’s rule: Solving system of differentiation. Integration of a variety of
equations of two/three variables functions by substitution, by partial
fractions and by parts, Evaluation of simple
3. Calculus integrals of the following types and
(i) Differentiation, derivative of composite problems based on them.
functions, chain rule, derivatives of inverse Fundamental Theorem of Calculus (without
trigonometric functions, derivative of implicit proof). Basic properties of definite integrals
functions. Concept of exponential and and evaluation of definite integrals.
logarithmic functions.
• Indefinite integral
Derivatives of logarithmic and exponential - Integration as the inverse of
functions. Logarithmic differentiation, differentiation. (anti-derivative).
derivative of functions expressed in
parametric forms. Second order derivatives. - Anti-derivatives of polynomials and
functions (ax +b)n , sinx, cosx, sec2x,
cosec2x etc .

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- Integrals of the type sin2x, sin3x, sin4x, b c b

cos2x, cos3x, cos4x.
x
∫ f ( x)dx = ∫ f ( x)dx + ∫ f ( x)dx
- Integration of 1/x, e . a a c
where a < c < b
- Integration by substitution.
b b
- Integrals of the type f ' (x)[f (x)]n,
f ′( x)
.
∫ f ( x)dx = ∫ f (a + b − x)dx
a a
f ( x)
a a
-
-
Integration of tanx, cotx, secx, cosecx.
Integration by parts.
∫
0
f (=
x)dx ∫ f (a − x)dx
0

- Integration using partial fractions.  a
f ( x) 2a 2 ∫ f ( x)dx, if f (2a − x) = f ( x)
Expressions of the form when ∫ f ( x)dx =  0
g ( x) 0  0, f (2a − x) =− f ( x)

degree of f(x) < degree of g(x)
 a
x+2 A B
∫
a
E.g. = + 2 f ( x)dx,if f is an even function
( x − 3)( x + 1) x − 3 x + 1 ∫
−a
f ( x)dx =  0

x+2 A B C  0,if f is an odd function
= + +
( x − 2)( x − 1) 2
x − 1 ( x − 1) 2
x−2
x +1 Ax + B C (iii) Application of Calculus in Commerce and
= 2 +
( x + 3)( x − 1) x + 3 x − 1
2
Economics in the following:
When degree of f (x) ≥ degree of g(x), - Cost function,
x2 +1  3x + 1  - average cost,
e.g. 2
= 1−  2  - marginal cost and its interpretation
x + 3x + 2  x + 3x + 2 
- demand function,
• Integrals of the type:
- revenue function,
dx dx px + q px + q - marginal revenue function and its
∫ 2 2 ,∫ 2 2 ,∫ 2 dx, ∫ dx
x ±a x ± a ax + bx + c ax 2 + bx + c interpretation,
- Profit function and breakeven point.
• Definite Integral
- Rough sketching of the following curves:
- Fundamental theorem of calculus AR, MR, R, C, AC, MC and their
(without proof) mathematical interpretation using the
concept of maxima & minima and
- Properties of definite integrals.
increasing- decreasing functions.
- Problems based on the following - Identify the region representing C.S. and
properties of definite integrals are to P.S. graphically. Apply the definite
be covered. integral to find consumer surplus-
b b producer surplus etc.
∫ f ( x)dx = ∫ f (t )dt - Problems based on finding -Total cost
a a when Marginal Cost is given - Total
b a Revenue when Marginal Revenue is given
-Equilibrium price and equilibrium
∫ f ( x)dx = −∫ f ( x)dx
a b
quantity and hence consumer and
producer surplus etc.

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Self-explanatory • Theoretical probability distribution,
NOTE: Application involving differentiation, probability distribution function; mean and
increasing and decreasing function and variance of random variable.
maxima and minima to be covered. • Binomial distribution
Application involving integration, definite - Bernoulli’s trials.
integration to be covered.
- Binomial distribution
(iv) Differential Equations
- Mean and variance
Definition, order and degree, general and
particular solutions of a differential equation. • Poisson Distribution
Solution of differential equations by method - Definition of Poisson distribution
of separation of variables.
- Differential equations, order and degree. - Characteristics
- Formation of differential equation by - Mean and variance
eliminating arbitrary constant(s). • Normal distribution
- Solution of differential equations.
- Concept of continuous of distribution
- Variable separable.
- Homogeneous equations. - Understanding the normal distribution is
a Continuous distribution.
dy
- Linear form + Py = Q where P and Q - Standard normal variate
dx
are functions of x/constant. Similarly, for - Mean and Standard deviation.
dx/dy. - Total area under the curve
NOTE 1: Equations reducible to variable
separable type are included. - Area relationship between Mean and
Standard deviation.
NOTE 2: The second order differential equations
are excluded. 5. Linear Programming
4. Probability Introduction, related terminology such as
constraints, objective function, optimization,
Conditional probability, multiplication theorem different types of linear programming (L.P.)
on probability, independent events, total problems, mathematical formulation of L.P.
probability, Bayes’ theorem, Random variable and problems, graphical method of solution for
its probability distribution, mean and variance of problems in two variables, feasible and infeasible
r a n d o m variable. Binomial, Poisson and Normal regions (bounded and unbounded), feasible and
distributions and its application in real life situation. infeasible solutions, optimal feasible solutions (up
Advantages & disadvantages of each type of to three non-trivial constraints).
distributions.
• Introduction, definition of related
• Independent and dependent events terminology such as constraints, objective
conditional events. function, optimization, advantages of linear
• Laws of Probability, addition theorem, programming; limitations of linear
multiplication theorem, conditional programming; application areas of linear
probability. programming; different types of linear
programming (L.P.) problems, mathematical
• Theorem of Total Probability. formulation of L.P problems, graphical
• Baye’s theorem. method of solution for problems in two
variables, feasible (bounded/ unbounded)
and infeasible regions, feasible and

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infeasible solutions, optimum feasible PAPER II – PROJECT WORK – 20 Marks
solution(may/may not exists).
Candidates will be expected to have completed two
6. Financial Mathematics projects. The project work will be assessed by the
- Perpetuity, Sinking Funds (Meaning). Real life subject teacher and a Visiting Examiner appointed
examples of sinking fund. Advantages of locally and approved by the Council.
sinking fund. Sinking fund vs. Savings Mark allocation for each Project [10 marks]:
account
- EMI (Methods to calculate EMI (Flat rate Overall format 1 mark
method, Reducing balance method). Real life Content 4 marks
examples to calculate EMI of various types of
loans, purchase of assets etc. Findings 2 marks
- Rate of Return, Nominal rate of return Viva-voce based on the Project 3 marks
(meaning and use) and their formula.
Total 10 marks
- Compound Annual Growth Rate (Meaning
and use) and their formula List of suggested assignments for Project Work:
- Linear Method of Depreciation (Meaning) 1. Using a graph, demonstrate a function which is
and its formula. Advantages and one-one but not onto.
disadvantages of Linear Method
2. Using a graph demonstrate a function which is
7. Index Numbers and Moving Averages invertible.
3. Draw the graph of y = sin-1 x (or any other inverse
(i) Index Numbers trigonometric function), using the graph of y =
- Price index or price relative. sin x (or any other relevant trigonometric
- Simple average of price relatives. function). Demonstrate the concept of mirror line
- Weighted average of price relatives (about y = x) and find its domain and range.
(cost of living index, consumer price 4. Explore the principal value of the function
index). sin-1 x (or any other inverse trigonometric
- Simple aggregate method. function) using a unit circle.
- All types of weighted aggregate index 5. Find the derivatives of a determinant of the order
number methods and their of 3 x 3 and verify the same by other methods.
advantages/disadvantages with reference 6. Verify the consistency of the system of three linear
to real life situation equations of two variables and verify the same
graphically. Give its geometrical interpretation.
(ii) Moving Averages
7. For a dependent system (non-homogeneous) of
- Meaning and purpose of the moving
three linear equations of three variables, identify
averages.
infinite number of solutions.
- Calculation of moving averages with the 8. Explain the concepts of increasing and decreasing
given periodicity and plotting them on a functions, using geometrical significance of dy/dx.
graph. Illustrate with proper examples.
If the period is even, then the centered moving 9. Explain the geometrical significance of point of
average is to be found out and plotted. inflexion with examples and illustrate it using
graphs.
10. Explain and illustrate (with suitable examples) the
concept of local maxima and local minima using
graph.

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11. Explain and illustrate (with suitable examples) the 19. For a given data, find regression equations by the
concept of absolute maxima and absolute minima method of least squares. Also find angles between
using graph. regression lines.
12. Explain the conditional probability, the theorem of 20. Draw the scatter diagram for a given data. Use it
total probability and the concept of Bayes’ to draw the lines of best fit and estimate the value
theorem with suitable examples. of Y when X is given and vice-versa.
13. Explain the types of probability distributions and 21. Different methods to calculate depreciation. Main
derive mean and variance of binomial probability inputs to calculate depreciation. Linear or straight
distribution for a given function. line method to find depreciation with a suitable
14. Using any suitable data, find the minimum cost by real life example.
applying the concept of Transportation problem. 22. Stock price movement.
15. Using any suitable data, find the minimum cost 23. Predicting stock market crash.
and maximum nutritional value by applying the 24. Risk assessment of insurance farm from data.
concept of Diet problem.
25. Identify the purchasing power using the concept of
16. Using any suitable data, find the Optimum cost in cost of living index number.
the manufacturing problem by formulating a linear
26. Identify the purchasing power using the concept of
programming problem (LPP).
weighted aggregate price index number.
17. Demonstrate application of differential equations
27. Calculate moving averages with the given even
to solve a given problem (example, population
increase or decrease, bacteria count in a culture, Periodicity. Plot them and as well as the original
etc.). data on the same graph.
18. Draw a rough sketch of Cost (C), Average Cost 28. Real life application of Binomial distribution,
(AC) and Marginal Cost (MC) Poisson and Normal distribution in the field of
Or medical, games, banking, election result etc.
Revenue (R), Average Revenue (AR) and 29. Applications of Sequence and series in Banking
Marginal Revenue (MR). and Finance.
Give their mathematical interpretation using the
concept of increasing - decreasing functions and NOTE: No question paper for Project Work will be
maxima-minima. set by the CISCE.

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SAMPLE TABLE FOR PROJECT WORK

S. Unique PROJECT 1 PROJECT 2 TOTAL
No. Identification MARKS
Number A B C D E F G H I J
(Unique ID) Teacher Visiting Average Viva- Total Teacher Visiting Average Viva- Total (E + J)
of the Examiner Marks Voce by Marks Examiner Marks Voce by Marks
candidate (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
the Teacher and the Visiting Examiner is as Name of Teacher:
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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