ISC Class 12 Applied Mathematics Syllabus 2026-27

The ISC (Indian School Certificate) Class 12 Applied Mathematics syllabus for the academic session 2026-27 is prescribed by the Council for the Indian School Certificate Examinations (CISCE) under subject code 885. Applied Mathematics is a new ISC subject, introduced for the Year 2027 examination as an alternative to standard Mathematics (860) for candidates heading towards Humanities, Commerce, Economics, Biosciences, Social Sciences and related fields.

The difference between the two courses is not difficulty so much as direction. Standard ISC Mathematics devotes 35 of its 80 theory marks to Calculus and carries Vectors and Three-Dimensional Geometry. Applied Mathematics trims Calculus to 25 marks, drops Vectors and 3D Geometry entirely, and adds two units that standard Mathematics does not have: Financial Mathematics (14 marks) and Index Numbers and Moving Averages (6 marks). Probability is also considerably heavier at 14 marks against 9, and extends beyond the binomial distribution to the Poisson and normal distributions.

This page sets out the complete unit-wise syllabus with sub-topics, the official marks weightage for each unit, the exam pattern, the project work requirements, and preparation guidance for ISC 2027 candidates.

Note: Applied Mathematics may not be taken together with Mathematics (860). Candidates choose one or the other.

Quick Facts: ISC Class 12 Applied Mathematics 2026-27

ISC Class 12 Applied Mathematics Exam Pattern 2026-27

Applied Mathematics follows the same two-paper structure as standard ISC Mathematics: a written theory paper set and marked by CISCE, and internally assessed project work.

Candidates complete two projects, each marked out of 10. For each project the mark breakdown is: overall format (1 mark), content (4 marks), findings (2 marks) and a viva-voce based on the project (3 marks). The 7 marks for format, content and findings are awarded separately by the subject teacher and by the Visiting Examiner and then averaged; the 3-mark viva-voce is conducted only by the Visiting Examiner. CISCE does not set a question paper for project work.

ISC Class 12 Applied Mathematics Syllabus 2026-27: Detailed Unit-Wise Breakdown

The theory syllabus is organised into seven units.

Unit 1: Relations and Functions (6 Marks)

1.1 Relations and Functions

• Relations on a set: identity relation, empty relation, universal relation

• Types of relations: reflexive, symmetric, transitive and equivalence relations

• Functions as special relations; writing "y is a function of x" as y = f(x)

• Types of functions: one-to-one, many-to-one, into, onto

• Real valued functions; domain and range of a function

• Conditions of invertibility

• Sketching the graph of a function and its inverse

• Composite functions and invertible functions (algebraic functions only)

1.2 Inverse Trigonometric Functions

• Definition, domain, range and principal value branch

• Graphs of inverse trigonometric functions

• Elementary properties of inverse trigonometric functions

• Principal values of sin⁻¹x, cos⁻¹x, tan⁻¹x and related functions

• Standard relations, including sin⁻¹x + cos⁻¹x = π/2 and the corresponding relations for tan⁻¹x and cot⁻¹x

Unit 2: Algebra (10 Marks)

2.1 Matrices

• Concept, notation, order and equality of matrices

• Types of matrices (up to order m × n where m, n ≤ 3): diagonal, scalar, identity, triangular

• Zero matrix and identity matrix

• Transpose of a matrix

• Symmetric and skew-symmetric matrices and their properties

• Operations: addition, subtraction, scalar multiplication, matrix multiplication and compatibility

• Non-commutativity of matrix multiplication

• Existence of non-zero matrices whose product is the zero matrix

• Singular and non-singular matrices

• Invertible matrices and the uniqueness of the inverse

2.2 Determinants

• Determinant of a square matrix (up to 3 × 3)

• Order, minors, cofactors and expansion

• Applications of determinants in finding the area of a triangle and in testing collinearity

• Adjoint and inverse of a square matrix, and properties of the adjoint and inverse

• Consistency, inconsistency and number of solutions of a system of linear equations

• Solving systems of linear equations in two or three variables using the inverse of a matrix (Martin's Rule) or Cramer's Rule

Unit 3: Calculus (25 Marks)

The largest unit in the paper, and the one that carries the commerce and economics applications.

3.1 Differentiation and Applications of Derivatives

• Derivatives of trigonometric and inverse trigonometric functions

• Derivatives of exponential and logarithmic functions

• Derivatives of implicit functions and the chain rule

• Derivatives of parametric functions

• Differentiating one function with respect to another

• Logarithmic differentiation

• Successive differentiation up to the second order

• Rate measure

• Increasing and decreasing functions

• Maxima and minima: stationary and turning points, first derivative test, second derivative test

3.2 Integrals

• Integration as the inverse of differentiation (anti-derivatives)

• Anti-derivatives of polynomials and of functions such as (ax + b)ⁿ, sin x, cos x, sec²x, cosec²x

• Integrals of sin²x, sin3x, sin4x, cos²x, cos3x, cos4x

• Integration of 1/x and eˣ

• Integration by substitution

• Integration of tan x, cot x, sec x, cosec x

• Integration by parts

• Integration using partial fractions

• Standard integral forms involving quadratic denominators

• Definite integrals: Fundamental Theorem of Calculus (without proof) and the standard properties of definite integrals

3.3 Application of Calculus in Commerce and Economics

• Cost function, average cost, marginal cost and its interpretation

• Demand function, revenue function, marginal revenue function and its interpretation

• Profit function and break-even point

• Rough sketching of AR, MR, R, C, AC and MC curves and their mathematical interpretation using maxima, minima and increasing-decreasing functions

• Identifying the regions representing consumer surplus and producer surplus graphically

• Applying the definite integral to find consumer surplus and producer surplus

• Finding total cost when marginal cost is given, and total revenue when marginal revenue is given

• Equilibrium price and equilibrium quantity

3.4 Differential Equations

• Definition, order and degree of a differential equation

• Formation of a differential equation by eliminating arbitrary constants

• General and particular solutions

• Solution by variable separable method

• Homogeneous equations

• Linear form dy/dx + Py = Q, where P and Q are functions of x or constants, and similarly for dx/dy

• Note: second-order differential equations are excluded

Unit 4: Probability (14 Marks)

The second-largest unit, and noticeably heavier than in standard ISC Mathematics.

• Independent and dependent events; conditional events

• Laws of probability: addition theorem, multiplication theorem, conditional probability

• Theorem of Total Probability

• Bayes' Theorem

• Theoretical probability distributions; probability distribution function; mean and variance of a random variable

• Binomial distribution: Bernoulli trials, the distribution, mean and variance

• Poisson distribution: definition, characteristics, mean and variance

• Normal distribution: concept of a continuous distribution, standard normal variate, mean and standard deviation, total area under the curve, area relationships between mean and standard deviation

• Applications of these distributions in real-life situations, with the advantages and disadvantages of each

Unit 5: Linear Programming (5 Marks)

• Introduction and related terminology: constraints, objective function, optimisation

• Advantages, limitations and application areas of linear programming

• Different types of linear programming problems

• Mathematical formulation of LP problems

• Graphical method of solution for problems in two variables

• Feasible and infeasible regions, bounded and unbounded

• Feasible and infeasible solutions; optimal feasible solutions (up to three non-trivial constraints), which may or may not exist

Unit 6: Financial Mathematics (14 Marks)

One of the two units unique to Applied Mathematics, and joint second in weightage. It is almost entirely application-based and highly scoring for students who learn the formulas properly.

• Perpetuity and sinking funds: meaning, real-life examples of a sinking fund, advantages of a sinking fund, and sinking fund compared with a savings account

• EMI: methods of calculating EMI by the flat rate method and the reducing balance method, with real-life examples covering various types of loans and asset purchases

• Rate of return and nominal rate of return: meaning, use and formulae

• Compound Annual Growth Rate (CAGR): meaning, use and formula

• Linear method of depreciation: meaning, formula, and the advantages and disadvantages of the method

Unit 7: Index Numbers and Moving Averages (6 Marks)

The second unit unique to Applied Mathematics.

7.1 Index Numbers

• Price index or price relative

• Simple average of price relatives

• Weighted average of price relatives: cost of living index and consumer price index

• Simple aggregate method

• All types of weighted aggregate index number methods, with their advantages and disadvantages in real-life situations

7.2 Moving Averages

• Meaning and purpose of moving averages

• Calculating moving averages for a given periodicity and plotting them on a graph

• Centred moving averages, which must be found and plotted when the period is even

Unit-Wise Overview Table

The table below summarises all seven units of the ISC Class 12 Applied Mathematics 2026-27 syllabus with their official weightage in Paper I.

Applied Mathematics (885) vs Mathematics (860): What Actually Differs

Candidates choosing between the two courses should compare the unit weightage rather than the reputation of each subject.

In short: Applied Mathematics replaces Vectors, Three-Dimensional Geometry and 10 marks of Calculus with Financial Mathematics, Index Numbers and Moving Averages, and a wider treatment of Probability. Candidates should also check the admission requirements of the undergraduate programmes they intend to apply to, since some quantitative courses specify standard Mathematics.

ISC Class 12 Applied Mathematics Marking Scheme 2026-27

Each project is marked as follows: overall format 1 mark, content 4 marks, findings 2 marks, and viva-voce 3 marks. Candidates must secure a minimum of 35% in the subject, which is the ISC passing criterion applied to each individual subject.

Suggested Project Work Topics

CISCE publishes an indicative list of project assignments for Class 12 Applied Mathematics. Useful starting points include:

• Using a graph to demonstrate a function that is one-one but not onto, or a function that is invertible

• Drawing y = sin⁻¹x from the graph of y = sin x, demonstrating the mirror line about y = x, and finding its domain and range

• Verifying the consistency of a system of three linear equations in two variables and interpreting it geometrically

• Explaining increasing and decreasing functions through the geometrical significance of dy/dx

• Illustrating local and absolute maxima and minima using graphs

• Explaining conditional probability, the theorem of total probability and Bayes' Theorem with suitable examples

• Deriving the mean and variance of a binomial probability distribution

• Finding minimum cost using a transportation problem, or minimum cost and maximum nutritional value using a diet problem

• Formulating an LPP to find the optimum cost in a manufacturing problem

• Sketching Cost, Average Cost and Marginal Cost (or Revenue, Average Revenue and Marginal Revenue) and interpreting them using maxima-minima

• Finding regression equations by the method of least squares and the angle between the regression lines

• Comparing methods of calculating depreciation, with a real-life example of the straight-line method

• Identifying purchasing power using the cost of living index or a weighted aggregate price index

• Calculating moving averages with even periodicity and plotting them alongside the original data

• Real-life applications of binomial, Poisson and normal distributions in medicine, banking, games or election results

Preparation Tips for ISC Class 12 Applied Mathematics 2026-27

Give Calculus the Time Its 25 Marks Deserve

Calculus is the largest unit, and within it the commerce and economics applications are the part students most often underprepare. Make sure you can move fluently between cost, revenue and profit functions, find marginal values, sketch AR/MR/AC/MC curves, and use the definite integral for consumer and producer surplus. These questions are predictable and heavily weighted.

Treat Financial Mathematics as a Scoring Unit

Fourteen marks sit in Financial Mathematics, and the content is finite: perpetuity, sinking funds, EMI by both the flat rate and reducing balance methods, rate of return, CAGR and linear depreciation. Learn each formula with a worked real-life example attached, and be ready to explain concepts in words as well as compute them, because the syllabus asks for meanings, advantages and comparisons, not just numbers.

Do Not Underestimate the Probability Unit

At 14 marks, Probability in Applied Mathematics goes well beyond the binomial distribution. Poisson and normal distributions are both prescribed, along with Bayes' Theorem and the theorem of total probability. Practise reading standard normal tables and identifying which distribution a situation calls for, since choosing the wrong model costs the whole question.

Practise Index Numbers with Real Data

Index Numbers and Moving Averages is only 6 marks but it is self-contained and quick to master. Work through price relatives, the simple aggregate method and the weighted aggregate methods on real price data, and practise centring moving averages when the periodicity is even. That is the step most commonly dropped.

Show Every Step

Applied Mathematics is step-marked. Write the formula, substitute with units where relevant, then compute. In financial questions, state the rate and the period explicitly. A correct final answer with no visible working forfeits most of the available marks.

Start the Two Projects Early

Both projects carry a viva-voce conducted by the Visiting Examiner, which means you need to understand your own work rather than reproduce it. Choose topics early, keep the format tidy with a clear title page and headings, and be ready to explain your findings aloud.

Use the Specimen Papers Once the Syllabus Is Covered

Because 2027 is the first examination in Applied Mathematics, there is no bank of past papers. Attempt ISC Class 12 Applied Mathematics specimen papers under timed conditions after a full revision, then mark them against the answer key to find the units where you are losing marks.

Frequently Asked Questions (FAQs)

What is the subject code for ISC Class 12 Applied Mathematics?

The subject code for ISC Class 12 Applied Mathematics is 885, as assigned by the Council for the Indian School Certificate Examinations (CISCE).

Is Applied Mathematics a new ISC subject?

Yes. Applied Mathematics (885) was introduced by CISCE as a new subject for the ISC Year 2027 examination, intended for candidates pursuing Humanities, Commerce, Economics, Biosciences, Social Sciences and related fields.

Can I take both Mathematics and Applied Mathematics at ISC?

No. The CISCE syllabus states that Applied Mathematics may not be taken with Mathematics. Candidates choose one of the two.

How many marks is the ISC Class 12 Applied Mathematics exam?

The subject carries 100 marks in total: 80 marks for Paper I (Theory) and 20 marks for Paper II (Project Work). The theory paper is three hours long, with an additional 15 minutes of reading time.

How many units are in the ISC Class 12 Applied Mathematics syllabus?

Seven: Relations and Functions (6), Algebra (10), Calculus (25), Probability (14), Linear Programming (5), Financial Mathematics (14), and Index Numbers and Moving Averages (6).

Which unit carries the highest weightage?

Calculus, at 25 marks out of 80. Probability and Financial Mathematics follow jointly at 14 marks each.

How is Applied Mathematics different from standard ISC Mathematics?

Applied Mathematics does not include Vector Algebra or Three-Dimensional Geometry and allocates fewer marks to Calculus. In their place it adds Financial Mathematics (14 marks) and Index Numbers and Moving Averages (6 marks), and gives Probability a wider scope including the Poisson and normal distributions.

Does Applied Mathematics include Vectors and 3D Geometry?

No. Neither Vector Algebra nor Three-Dimensional Geometry appears in the Applied Mathematics syllabus. Both are prescribed in standard Mathematics (860).

How many projects are required for ISC Applied Mathematics?

Two projects, each worth 10 marks. Each is assessed on overall format (1), content (4), findings (2) and a viva-voce (3). The viva-voce is conducted only by the Visiting Examiner.

What is the passing mark for ISC Class 12 Applied Mathematics?

Candidates must secure a minimum of 35% in the subject, which is the ISC passing criterion applied to each individual subject.

Where can I download the official ISC Class 12 Applied Mathematics syllabus?

The official syllabus PDF for Applied Mathematics (885) can be downloaded from the CISCE website at www.cisce.org, under the Publications or Syllabus section for the relevant ISC examination year.