Mathematics / Applied Mathematics
Subject code 319 · domain paper
Each of those rules names Mathematics / Applied Mathematics outright — no combination of the other papers reaches them. That is the whole case for spending one of five slots on it.
The paper at a glance
Mathematics / Applied Mathematics carries the subject code 319. The syllabus described here is the one published for 2026-27.
One syllabus document serves two kinds of students: those who studied Mathematics in Class 12 and those who studied Applied Mathematics. That is why it is longer than most other subject syllabi.
How the syllabus is organised
The document is laid out in three sections — Section A1, Section B1 (Mathematics) and Section B2 (Applied Mathematics). Section A is the shared ground. Section B then appears in two versions, and a candidate works with one of them: Section B1: Mathematics or Section B2: Applied Mathematics.
The syllabus itself is a list of topics only. It does not print question counts, marks, negative marking or timing, and it does not spell out how many questions come from each section. Read it alongside the exam pattern.
Section A: the common part
Section A stays close to the topics that both school courses share, and it stays at a simple level: derivatives only up to second order, integrals of simple functions, and probability described in the source as simple probability.
| Topic | What it covers |
|---|---|
| Algebra | Matrices and types of matrices; equality of matrices, transpose, symmetric and skew symmetric matrices; algebra of matrices; determinants; inverse of a matrix; solving simultaneous equations by the matrix method |
| Calculus | Higher order derivatives up to second order; increasing and decreasing functions; maxima and minima |
| Integration and its Applications | Indefinite integrals of simple functions; evaluation of indefinite integrals; definite integrals; application of integration as area under a simple curve |
| Differential Equations | Order and degree of differential equations; solving differential equations with variables separable |
| Probability Distributions | Simple probability |
| Linear Programming | Graphical method of solution for problems in two variables; feasible and infeasible regions; optimal feasible solution |
Section B1: Mathematics
This is the fuller, more theoretical half. It has six units.
| Unit | What it covers |
|---|---|
| Unit I: Relations and Functions | Types of relations — reflexive, symmetric, transitive and equivalence relations; one-to-one and onto functions; inverse trigonometric functions — definition, range, domain, principal value branches and their graphs |
| Unit II: Algebra | Matrices — concept, notation, order, equality, types, zero matrix, transpose, symmetric and skew symmetric matrices, addition, multiplication and scalar multiplication with their simple properties, non-commutativity of multiplication, non-zero matrices with zero product (restricted to square matrices of order 2), invertible matrices and proof of uniqueness of the inverse, all with real entries; Determinants — determinant of a square matrix up to 3 × 3, minors, cofactors, area of a triangle, adjoint and inverse, consistency, inconsistency and number of solutions of a linear system, solving linear equations in two or three variables having a unique solution using the inverse of a matrix |
| Unit III: Calculus | Continuity and differentiability, chain rule, derivatives of inverse trigonometric and implicit functions, exponential and logarithmic functions and their derivatives, logarithmic differentiation, parametric forms, second-order derivatives; applications of derivatives — rate of change, increasing and decreasing functions, maxima and minima with the first derivative test motivated geometrically and the second derivative test, and simple real-life problems; integrals — integration as the inverse of differentiation, integration by substitution, partial fractions and by parts, a listed set of standard integral forms, Fundamental Theorem of Calculus without proof, basic properties and evaluation of definite integrals; applications of integrals — area under simple curves, especially lines, circles, parabolas and ellipses in standard form; differential equations — definition, order and degree, general and particular solutions, separation of variables, homogeneous equations of first order and first degree, and linear differential equations of the type dy/dx + Py = Q and dx/dy + Px = Q where P and Q are functions or constants |
| Unit IV: Vectors and Three-Dimensional Geometry | Vectors and scalars, magnitude and direction, direction cosines and direction ratios, types of vectors (equal, unit, zero, parallel, collinear), position vector, negative of a vector, components, addition, multiplication by a scalar, position vector of a point dividing a line segment in a given ratio, scalar (dot) and vector (cross) products with geometrical interpretation, properties and applications; three-dimensional geometry — direction cosines and ratios of a line joining two points, Cartesian and vector equations of a line, skew lines, shortest distance between two lines, angle between two lines |
| Unit V: Linear Programming | Introduction and related terminology such as constraints, objective function and 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 | Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes' theorem |
The calculus unit is where the syllabus is most specific: it names a fixed list of standard integral forms and two standard types of linear differential equation, so you can see exactly what kind of manipulation is expected.
Section B2: Applied Mathematics
This version has eight units and reaches into statistics and finance, areas that do not appear in B1 at all.
| Unit | What it covers |
|---|---|
| Unit I: Numbers, Quantification and Numerical Applications | Modulo arithmetic; congruence modulo; alligation and mixture, including the mean price of a mixture; solving real-life numerical problems; boats and streams, upstream and downstream; pipes and cisterns; races and games; numerical inequalities |
| Unit II: Algebra | Matrices and types of matrices; equality of matrices, transpose, symmetric and skew symmetric matrices; algebra of matrices — addition, subtraction, multiplication and scalar multiplication; determinant of a square matrix, elementary properties of determinants, singular and non-singular matrices and the result |AB| = |A||B|; inverse of a square matrix using cofactors and properties of inverses such as (AB) inverse = B inverse times A inverse; solving systems of simultaneous non-homogeneous equations in up to three variables |
| Unit III: Calculus | Higher order derivatives up to second order, differentiation of parametric and implicit functions; application of derivatives to rate of change; marginal cost and marginal revenue using derivatives; increasing and decreasing functions and the conditions for each; maxima and minima — critical points, local maxima and minima, absolute maximum and minimum and applied problems; integration — indefinite integrals of simple functions as anti-derivatives; indefinite integrals as a family of curves, evaluated by substitution, partial fractions and by parts; definite integral as area under a non-trigonometric curve, fundamental theorem of integral calculus and properties of definite integrals; application of integration to consumer surplus and producer surplus; differential equations — recognising a differential equation, its order and degree, formulating differential equations, verifying a solution and solving simple differential equations |
| Unit IV: Probability Distributions | Random variables and their probability distributions, including the distribution of a discrete random variable; mathematical expectation using the arithmetic mean of a frequency distribution; variance and standard deviation of a random variable; binomial distribution — Bernoulli trials and its mean, variance and standard deviation; Poisson distribution — its conditions, mean and variance; normal distribution as a continuous distribution, the standard normal variate and the area relationship between mean and standard deviation |
| Unit V: Time Based Data | Time series as chronological data; components of a time series; time series analysis for univariate data and interpretation of statistical data; secular trend as a long term tendency; methods of measuring trend |
| Unit VI: Inferential Statistics | Population and sample, representative and non-representative samples, drawing a sample by simple random and systematic random sampling; parameter and statistic and the relation between them, limitations of a statistic in generalising to a population, statistical significance and statistical inference, Central Limit Theorem, the population-sampling distribution-sample relation; one sample t-test for a small group — hypothesis, null and alternate hypothesis, degrees of freedom, testing the null hypothesis and drawing inferences |
| Unit VII: Financial Mathematics | Perpetuity and sinking funds, and how a sinking fund differs from a savings account; concept and calculation of EMI; rate of return and nominal rate of return; compound annual growth rate and how it differs from annual growth rate; linear method of depreciation with cost, residual value and useful life of an asset; valuation of bonds and related terms using the present value approach |
| Unit VIII: Linear Programming | Introduction and related terminology; mathematical formulation of a linear programming problem; different types of linear programming problems; graphical method of solution for problems in two variables; feasible, infeasible and bounded regions; feasible and infeasible solutions and the optimal feasible solution |
Choosing between B1 and B2
The honest answer is that the school subject you actually studied decides this for you. Compare the two lists:
- Only in B1: relations and functions, inverse trigonometric functions, vectors, three-dimensional geometry, conditional probability and Bayes' theorem.
- Only in B2: modulo arithmetic, alligation and mixture, boats and streams, pipes and cisterns, races and games, marginal cost and marginal revenue, consumer and producer surplus, probability distributions such as binomial, Poisson and normal, time series, inferential statistics with the t-test, and financial mathematics.
- In both, though treated differently: matrices, determinants, derivatives and their applications, integration, differential equations and linear programming.
If your Class 12 course included vectors, three-dimensional geometry and Bayes' theorem, B1 is your section. If it included EMI, depreciation, time series and the t-test, B2 is. Either way, Section A draws on the ground the two courses share, so it needs no separate decision.
What this syllabus does not tell you
The document is a topic list and nothing more. From it you cannot learn:
- how many questions come from Section A and how many from your Section B, or the marks, negative marking and duration — those are on the exam pattern;
- the weightage of any unit, so treat every unit as examinable rather than ranking them;
- how difficult the questions will be, or whether they will be direct one-step problems or longer ones;
- which books to use; the syllabus names no textbook;
- how much weight a university gives this paper once you qualify — the programmes that accept it are listed below, but the marks or ranking rule each one applies sits in its own admission bulletin.
For anything about attempt rules or numbers of questions, go to the NTA information bulletin and the exam pattern rather than to this list.
Where this paper can take you
Programmes whose rule can use Mathematics / Applied Mathematics, from the 23 universities whose documents we have read. 144 of them cannot be reached any other way.
B Tech (CE) [ 2+2 with international credit transfer option] · B Tech (CSE) [ 2+2 with international credit transfer option] · B Tech (ECE) [ 2+2 with international credit transfer option] · and 152 more
B.A. (Hons.) Economics · B.Sc. (Hons.) Statistics · B.Sc. (Hons.) Mathematics · and 60 more
Bachelor of Planning · Bachelor of Science (Actuarial Science) · Bachelor of Science (Actuarial Science) (Honours/Honours with Research) · and 48 more
Bachelor of Fine Arts (BFA) · B.A. LL.B. (Hons.) · Five Year Integrated Programme in Management (BBA & MBA) · and 13 more
Bachelor of Arts (Global Studies) · Bachelor of Arts (Hons.) - Economics · Bachelor of Arts (Hons.) - English · and 11 more
5-Year Integrated B.Tech. + MBA (Computer Science and Business Systems) · B.Sc. (Hons.) Applied Geology · B.Sc. (Hons.) Mathematics (Puducherry Main Campus) · and 10 more
B.Tech. — Biotechnology · B.Tech. — Chemical Engineering · B.Tech. — Civil Engineering · and 9 more
5-Year Dual Degree B.Tech.–M.Tech. (Artificial Intelligence & Data Science) · 5-Year Dual Degree B.Tech.–M.Tech. (Internet of Things) · 5-Year Integrated Dual Degree B.Tech.–M.Tech. (Computer Science) · and 7 more
B.S.–M.S. in Data Science · B.Tech. (Lateral Entry) — thirteen engineering and chemical-technology branches · B.Tech. Biotechnology · and 4 more
Chemistry (Major) · Computer Science (Major) · Economics (Major) · and 4 more
B.Sc. (Honours) — Mathematics, Statistics, Computer Science, Physics, Chemistry, Geography, Earth Science (Maths group) · B.Tech. Food Technology · B.Tech. Dairy Technology · and 3 more
B.Tech. Artificial Intelligence and Data Science · B.Tech. Biotechnology · B.Tech. Computer Science & Engineering · and 3 more
B.Sc. in Computer Science · B.Sc. in Forensic Science · Bachelor of Arts (Journalism and Mass Communication) · and 2 more
5 Year Integrated M.Sc. (Applied Geology) · 5 Year Integrated M.Sc. (Mathematical Sciences) · 5 Year Integrated M.Sc. (Physics) · and 1 more
B.Sc. (Hons.) Community Science · B.Sc. (Hons.) Geography · B.Sc. (Hons.) Statistics
Integrated B.Sc.–M.Sc. (Mathematics) · Integrated B.Sc.–M.Sc. (Chemistry) · Integrated B.Sc.–M.Sc. (Physics)
B.Tech. (School of Engineering) · Integrated B.Sc. B.Ed. (Major in Mathematics) · Integrated M.Sc. in Mathematics
Five Year Integrated Post Graduate Programme in Mathematics · Five Year Integrated Post Graduate Programme in Statistics