UPSC Maths Optional Syllabus- Preparation Strategy, Syllabus Paper-1, and Paper-2

The Union Public Service Commission (UPSC) Exam is one of the toughest exams and has two papers. It is conducted to choose central government public servants. Due to this, it is important to select the optional subject wisely. Both the papers account for 500 marks out of 1750 marks in the UPSC Main Examination. Hence, it should be chosen with utmost care.  The optional subject should be the one which is of the candidate’s interest. Maths is one of the most popular optional subjects because it is extremely scoring. But only those candidates who had studied Mathematics in their graduation should consider opting for it.


UPSC Maths Optional Syllabus

UPSC Mathematics Optional Syllabus: Paper 1

Name of the SubjectSyllabus
Analytic GeometryStraight lines, second-degree equations in three variables, reduction to canonical forms,  the shortest distance between two skew lines, cartesian and polar coordinates in three dimensions, Cylinder, Plane, Paraboloid, Sphere, Cone, ellipsoid, hyperboloid of one and two sheets and their properties
Linear AlgebraLinear independence and dependence, Matrices Algebra,  Linear Transformations, subspaces, bases, dimensions, Vector spaces over R and C, Matrix’s Row and column reduction, Echelon form, congruence and similarity, Matrix Ranking and Inverse, Eigenvalues and eigenvectors, Characteristic polynomial, symmetric, skew-symmetric, hermitian, orthogonal, and unitary matrices and their eigenvalues, Cayley Hamilton theorem, solution of a system of linear equations
CalculusFunctions of two or three variables, Real numbers, functions of a real variable, Limits, continuity, partial derivatives, maxima and minima, Riemann’s definition of definite integrals, differentiability, mean value theorem, Lagrange’s method of multipliers jacobian, Taylor’s theorem with remainders, indeterminate forms, maxima and minima, asymptotes, indefinite integrals, double and triple integrals (evaluation techniques only), infinite and improper integral, areas, surface, and volume, curve tracing
Ordinary differential equationsFormulation of differential equations, equations of the first order and first degree, integrating factor, orthogonal trajectory, equations of first order but not of the first degree, Clairaut’s equation, Laplace and inverse Laplace transforms and their properties, singular solution, second and higher-order linear equations, with constant coefficients, complementary function, particular integral and general solution, second-order linear equations with variable coefficient, determination of complete solution when one solution is known using the method of variations of parameter, Euler Cauchy’s equation, Laplace transforms of elementary functions application to initial value problems, of second-order linear equations with constant coefficients.
Vector AnalysisScalar and vector fields, differentiation of vector field of a scalar variable, gradient, divergence, curl in cartesian and cylindrical coordinates. Higher-order derivatives, Vector identities, vector equations, Application to geometry, curves in space, curvature and torsion, Serret-Furenets formula, Gauss and stokes theorem, Greens identities.
Dynamics and StaticsRectilinear motion, simple harmonic motion, motion in a plane, projectiles, work and energy, conservation of energy, the principle of virtual work, Kepler’s law, orbits under central forces, work and potential energy, friction, common catenary, equilibrium of a system of particles, stability of equilibrium, equilibrium of forces in three dimensions .constrained motion.

UPSC Mathematics Optional Syllabus: Paper 2

Name of the SubjectSyllabus
Real AnalysisSequences, limit of a sequence, Cauchy sequence, completeness of real line, Real number system as an ordered field, with least upper bound property, series and its convergence, absolute and conditional convergence of series of real and complex term, rearrangement of series.Uniform convergence, continuity, differentiability, and integrability for sequences and series of functions, Partial derivatives of functions of several (two or three) Variables, Maxima and Minima. Riemann Integral, improper integrals, Fundamental theorems of integral calculus. Continuity and uniform continuity of functions, properties of continuous functions on compact sets.
Partial Differential EquationsFamily of surfaces in three dimensions and formulation of partial differential equations, Equation of a vibrating string, heat equation, Laplace Equation, and their solutions. Solution of quasilinear partial first-order differential equations, Cauchy’s method of characteristics, Second-order Linear Partial Differential Equations having constant coefficients, canonical form.
Linear ProgrammingLinear Programming problems, basic solution, basic feasible solution and optimal solution, Graphical Method and simplex method of solutions, Duality. Transportation and assignment problems. 
AlgebraRings, subrings and ideals, homomorphism of rings, Integral Domains, Principal Ideal Domains, Euclidean Domains and unique factorization domains, Fields, Quotient Fields.Groups, Cyclic Groups, cosets, Subgroups, Normal Subgroups, Quotient Groups, Lagrange’s Theorem, Homomorphism of groups.Basic Isomorphism theorems, permutation groups, Cayley’s theorem.
Mechanics and Fluid dynamicsGeneralized coordinates, D Alemberts Principle and Lagrange’s equations, moment of inertia, Hamilton’s equations, Equation of continuity, Motion of rigid bodies in two dimensions, streamlines, the path of a particle, Euler’s equation of motion for inviscid flow, potential flow, sources, and sinks, vortex motion, two dimensional and axisymmetric motionNanier stokes equation for a viscous fluid
Numerical Analysis and computer programmingNumerical methods: solution of algebraic and transcendental equations of one variable by bisection, Regula falsi, and newton raphson’s method, solution of a system of linear equations by Gaussian elimination and gauss jorden (direct), gauss seidel (iterative) methods, newtons( forward and backward ) and interpolation, Lagrange’s interpolation, Numerical solution of ordinary differential equations: Euler and Ranga Kutta methods. Numerical integration: Simpson’s rule, trapezoidal rule, Gaussian quadrature formula Computer programming: binary system, arithmetic, and logical operations on numbers, octal and hexadecimal system, basic logic gates and truth tables, Boolean algebra, normal forms, conversions to and from decimal systems, algebra of binary numbers, elements of computer systems, and concept of memory, representation of unsigned integers, signed integers and reals, double precision reals and long integers, algorithms and flow charts for solving numerical analysis problems.
Complex AnalysisCauchy-Riemann equations, Cauchy’s Theorem, Cauchy’s Integral Formula, Power Series, Analytic Function, representation of an analytic function, Taylor’s Series, Singularities, Laurent’s Series, Cauchy’s residue theorem, Contour Integration.

Preparation Strategy for UPSC Maths Optional For UPSC Main Exam

UPSC Maths Optional is one of the preferred optional subjects, especially for engineering students. Few things which will help you in the preparation are as follows:

  1. Conceptual Clarity- For Maths, it is essential to have a proper conceptual understanding. Thus have an excellent knowledge of every topic of the UPSC Mathematics Optional Syllabus.
  2. Revising again and again- Revising what you have studied is important. Allocation of the proper time for revision is necessary to retain the maximum part studied. 
  3. Proper Presentation- Presentation is very important in the UPSC Answer writing. Thus practice more and more in the best presentation possible and get them checked from mentors.
  4. Go for logic and not cramming- Maths is not a subject for cramming. It is based on logic. Proper logic will help you in solving all the questions properly in the exam.
  5. Formula sheet preparation- It is important to learn formulae and theorems to solve questions. So, maintain a proper formula sheet that is handy and keep revising it to remember all the formulae.
  6. Practice mock tests and past year papers- Practise more and more mock tests and past year exams to avoid making silly mistakes in the exams.


We hope this article has given you the required information regarding the UPSC Mathematics Optional Syllabus for both the Papers – Paper 1 and Paper 2. Follow this preparation strategy for the UPSC Maths Optional and study meticulously. For any queries, contact us at Oliveboard.


How many marks do the optional subject have in UPSC  Main Exam?

The optional Subject carries 500 out of 1750 marks in the UPSC Main Exam.

What are the topics in Maths Optional Syllabus Paper 1?

The UPSC Maths Optional Syllabus Paper 1 contains the following topics, namely Linear Algebra, Calculus, Analytic Geometry, Ordinary Differential Equations, Dynamics and Statistics, and Vector Analysis.

What are the topics in Maths Optional Syllabus Paper 2?

The UPSC Maths Optional Syllabus  Paper 2 contains the following topics: Algebra, Real Analysis, Complex Analysis, Linear Programming, Partial Differential Equations, Mechanics and Fluid Dynamics, and Numerical Analysis and Computer Programming.


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