UPSC Maths Optional Syllabus 2025: Mains Paper 1 & 2 for UPSC, Download PDF

The UPSC Maths optional syllabus is the most opted-up optional subjects for candidates who have a sound background in mathematics. Because of its structured approach, objective scores, and a clear-cut approach to solving problems, the mathematics optional subject is very rewarding. The UPSC mathematics optional syllabus is broad and expands into algebra and calculus, differential equations, and numerical analysis. This subject can significantly enhance a candidate's score due to its high scoring potential and the precision required in solving mathematical problems.

Mathematics is that one subject that gives an edge to the UPSC aspirant, especially if a candidate has an engineering or maths background. The UPSC maths syllabus is as relevant for getting theoretical knowledge as it is in enhancing analytical and logical reasoning. This skill is taken up by the candidates while attempting all other parameters of the civil services examination. Moreover, the detailed UPSC Mathematics optional syllabus PDF download assures that the candidate would obtain all the syllabus material for proper preparation.

Check the detailed UPSC IAS Mains Syllabus here.

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Overview of UPSC Maths Optional Syllabus 2025

The UPSC Mathematics optional syllabus is covered in two papers - Paper 1 and Paper 2. These two papers test a broad range of mathematical concepts. This divides the candidates within the syllabus for making sure they are well equipped with both fundamental and advanced topics. So, the maths optional UPSC syllabus covers areas as broad as these can get, and one needs to understand each section to stand out.

There will be 2 papers for the UPSC Maths syllabus. Each paper of the Maths syllabus UPSC would comprise 250 marks, which makes it 500 marks in total. The time duration to attempt the UPSC Maths paper is 3 hours. Each paper would have objective answer-type questions. There is no negative marking.

Overview of UPSC Maths Syllabus 2025
Sl. No.	UPSC IAS Mains Papers	Subject	Marks
1.	Paper VI	Optional Subject Paper-I	250
2.	Paper VII	Optional Subject Paper-II	250
TOTAL			500
Time Duration			3 hours

Check the detailed UPSC IAS exam pattern here.

UPSC Maths Optional Syllabus for Papers 1 & 2

The mathematics optional syllabus for UPSC is divided into two papers. Core subjects like linear algebra, calculus, and analytic geometry form paper 1, while more complex topics of complex analysis, numerical analysis, and mechanics have to be covered under paper 2. Such bifurcation would eventually allow an in-depth study of each area and thereby render the UPSC CSE maths optional syllabus strong and comprehensive.

UPSC Maths Optional Syllabus PDF Download!

UPSC Mathematics Optional Syllabus for Paper 1

The UPSC maths optional syllabus for paper 1 contains the following fundamental topics that include - Linear Algebra, Calculus, Analytic Geometry, Ordinary Differential Equations, Dynamics, Statics, and Vector Analysis. These are for a basic understanding of the mathematics and problem-solving in the IAS maths syllabus.

The table below gives a detailed view of the the key topics of the UPSC Optional Maths syllabus for Paper 1:

UPSC Maths Optional Syllabus for Paper 1
Topics of the Maths Syllabus for UPSC Optional	Details
Linear Algebra	Vector spaces over R\mathbb{R}R and C\mathbb{C}C, linear dependence and independence, subspaces, bases, dimensions.
	Linear transformations, rank and nullity, matrix of a linear transformation.
	Algebra of matrices; Row and column reduction, Echelon form, congruence and similarity.
	Rank of a matrix; Inverse of a matrix; Solution of a system of linear equations.
	Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem.
	Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues.
Calculus	Real numbers, functions of a real variable, limits, continuity, differentiability.
	Mean-value theorem, Taylor’s theorem with remainders, indeterminate forms, maxima and minima, asymptotes.
	Curve tracing; Functions of two or three variables; Limits, continuity, partial derivatives, maxima and minima.
	Lagrange’s method of multipliers, Jacobian.
	Riemann’s definition of definite integrals; Indefinite integrals; Infinite and improper integrals.
	Double and triple integrals (evaluation techniques only); Areas, surface and volumes.
Analytic Geometry	Cartesian and polar coordinates in three dimensions, second-degree equations in three variables, reduction to Canonical forms.
Analytic Geometry	Straight lines, the shortest distance between two skew lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.
Ordinary Differential Equations	Formulation of differential equations; Equations of the first order and first degree, integrating factor.
	Orthogonal trajectory; Equations of first order but not of first degree, Clairaut’s equation, singular solution.
	Second and higher-order linear equations with constant coefficients, complementary functions, particular integrals and general solutions.
	Linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using variation of parameters.
	Laplace and Inverse Laplace transforms and their properties, Laplace transforms of elementary functions.
	Application to initial value problems for 2nd order linear equations with constant coefficients.
Dynamics and Statics	Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; Constrained motion; Work and energy, conservation of energy.
	Kepler’s laws, orbits under central forces.
	Equilibrium of a system of particles; Work and potential energy, friction, Common catenary.
	Principle of virtual work; Stability of equilibrium, equilibrium of forces in three dimensions.
Vector Analysis	Scalar and vector fields, differentiation of the vector field of a scalar variable.
	Gradient, divergence and curl in Cartesian and cylindrical coordinates; Higher order derivatives.
	Vector identities and vector equations.
	Application to geometry: Curves in space, curvature and torsion; Serret-Frenet’s formulae.
	Gauss and Stokes’ theorems, Green’s identities.

UPSC Mathematics Optional Syllabus for Paper 2

The UPSC maths optional syllabus contains paper 2 that deals with specialized topics comprising algebra, real analysis, complex analysis, linear programming, partial differential equations, numerical analysis, computer programming, mechanics, and fluid dynamics. The subjects demand an in-depth understanding of them; hence, the UPSC mains maths syllabus may be scored best if proper dedication is provided to such advanced study.

The table below highlights the key topics of the Maths Optional UPSC syllabus for Paper 2:

UPSC Mathematics Optional Syllabus for Paper 2
Topics of the UPSC Mains Maths Syllabus	Details
Algebra	Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley’s theorem.
	Rings, subrings and ideals, homomorphisms of rings; Integral domains, principal ideal domains, Euclidean domains, and unique factorization domains.
	Fields, quotient fields.
Real Analysis	Real number system as an ordered field with the least upper bound property.
	Sequences, the limit of a sequence, Cauchy sequence, completeness of real line.
	Series and its convergence, absolute and conditional convergence of series of real and complex terms, rearrangement of series.
	Continuity and uniform continuity of functions, properties of continuous functions on compact sets.
	Riemann integral, improper integrals; Fundamental theorems of integral calculus.
	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.
Complex Analysis	Analytic function, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, power series, representation of an analytic function, Taylor’s series.
Complex Analysis	Singularities; Laurent’s series; Cauchy’s residue theorem; Contour integration.
Linear Programming	Linear programming problems, basic solution, basic feasible solution, and optimal solution.
	Graphical method and simplex method of solutions; Duality.
	Transportation and assignment problems.
Partial Differential Equations	Family of surfaces in three dimensions and formulation of partial differential equations.
	Solution of quasilinear partial differential equations of the first order, Cauchy’s method of characteristics.
	Linear partial differential equations of the second order with constant coefficients, canonical form; Equation of a vibrating string, heat equation, Laplace equation and their solutions.
Numerical Analysis and Computer Programming	Numerical methods: Solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi, and Newton-Raphson methods.
	Solution of a system of linear equations by Gaussian Elimination and Gauss-Jordan (direct), Gauss-Seidel (iterative) methods.
	Newton’s (forward and backwards) and interpolation, Lagrange’s interpolation.
	Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula.
	Numerical solution of ordinary differential equations: Euler and Runge-Kutta methods.
	Computer Programming: Binary system; Arithmetic and logical operations on numbers; Octal and Hexadecimal Systems; Conversion to and from decimal Systems.
	Algebra of binary numbers. Elements of computer systems and concept of memory; Basic logic gates and truth tables, Boolean algebra, normal forms.
	Representation of unsigned integers, signed integers and reals, double precision reals and long integers. Algorithms and flow charts for solving numerical analysis problems.
Mechanics and Fluid Dynamics	Generalized coordinates; D’Alembert’s principle and Lagrange’s equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions.
	Equation of continuity; Euler’s equation of motion for inviscid flow; Streamlines, path of a particle; Potential flow; Two-dimensional and axisymmetric motion.
	Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.

Prepare for the exam using the UPSC IAS mains strategy here.

How to Prepare for Maths Optional Syllabus for UPSC?

To prepare for the Maths optional syllabus for UPSC, a proper strategic approach is required:

Understand the Syllabus: Download and read the UPSC mathematics optional syllabus PDF to understand the topics comprehensively.
Create a Study Plan: Structure a detailed timetable covering all sections of the maths optional syllabus UPSC.
Standard Textbooks: Use recommended books to cover each topic in the UPSC optional subject mathematics syllabus thoroughly.
Regular Practice: Solve problems regularly to strengthen concepts and improve problem-solving speed.
Previous Year Papers: Solve previous year question papers for a feel of the exam pattern and find out what questions have been repeated.
Group Discussions with Peers: Discuss your doubts with friends and gain new insights.
Continuous Revision: Concept/technique retention is possible only through regular revision.
Mock Tests: Take mock tests to evaluate your preparation level and enhance your examination strategy.

UPSC Maths Previous Year Question Papers

Practicing previous year question papers is very important regarding maths UPCS optional syllabus. It helps one understand the nature of questions, the pattern of the paper, and the level of difficulty. Regular practice also helps in managing time properly in the actual examination.

Maths Optional Previous Year Question Papers
Year	Paper	Download Link
2021	Maths Paper I	Click Here
2021	Maths Paper II	Click Here
2022	Maths Paper I	Click Here
2022	Maths Paper II	Click Here
2023	Maths Paper I	Click Here
2023	Maths Paper II	Click Here
2024	Maths Paper I	Click Here
2024	Maths Paper II	Click Here

Check some more UPSC IAS Previous Year’s Question Papers for other subjects here!

List of Books for Maths UPSC Optional Syllabus

The UPSC maths optional syllabus requires tremendous preparation, which cannot be made unless referred from standard textbooks for each of the topics. Certain standard works that are recommended both for Paper 1 and Paper 2 are listed as follows:

Books for UPSC Mathematics Syllabus Paper 1

Look into the list of books for the mathematics optional syllabus for upsc Paper 1:

For Linear Algebra, the student should refer to "Schaum's Outline of Linear Algebra" by Seymour Lipschutz and "Linear Algebra" by Hoffman and Kunze.
While studying Calculus, "Mathematical Analysis" by S. C. Malik and Savita Arora and "Elements of Real Analysis" by Shanti Narayan and M. D. Raisinghania are recommended.
While studying Analytic Geometry, two most important books are "Analytical Solid Geometry" by Shanti Narayan and P. K. Mittal and "Solid Geometry" by P. N. Chatterjee.
For Ordinary Differential Equations, "Ordinary and Partial Differential Equations" by M. D. Raisinghania is very useful.
For Dynamics and Statics, Krishna Series is a good collection.
Lastly, for Vector Analysis, one may refer to Schaum's Outline of Vector Analysis by Murray R. Spiegel.

Books for UPSC Mathematics Syllabus Paper 2

The important books to refer for UPSC Mains syllabus Mathematics optional paper 2 include the following:

For Algebra, "Contemporary Abstract Algebra" by Joseph Gallian should be referred to.
For Real Analysis, the same books recommended in Paper 1 for calculus can be used.
For Complex Analysis, "Schaum's Outline of Complex Variables" by Spiegel, Lipschutz, Schiller, and Spellman should be kept as reference.
For Linear Programming, "Linear Programming and Game Theory" by Lakshmishree Bandopadhyay is a good textbook.
For Partial Differential Equations, you can refer to the same book for ODE in Paper 1, and you may also refer to "Advanced Differential Equations" by M. D. Raisinghania.
When it comes to Numerical Analysis and Computer Programming, for Numerical Analysis, use "Computer Based Numerical and Statistical Techniques" by M. Goyal and "Numerical Methods" by Jain, Iyengar, and Jain.
For Computer Programming, "Digital Logic and Computer Design" by M. Morris Mano is recommended.

Check Related UPSC Syllabus for Other Subjects
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This was a detailed article about the UPSC Maths Syllabus. To start the preparation for your examination, download the Testbook App and get access to world-class educations, mock tests, etc.

UPSC Mathematics Syllabus 2025 FAQs

How important is the syllabus of mathematics optional for UPSC mains?

It forms the backbone of study as it keeps track of which topics to cover for the exams. In doing so, it helps in streamlining and planning your study.

What is included in the UPSC mains mathematics optional syllabus PDF?

It includes in-depth study material for Paper 1 and Paper 2, including topics like linear algebra, calculus, complex analysis, and numerical analysis.

Are the topics in the maths UPSC optional syllabus challenging?

They are challenging but are manageable provided you follow a proper study and practice schedule. If you have a background in mathematics, they will be easy for you to manage and comprehend.

Can I use the same books for the UPSC mains syllabus mathematics as other mathematical exams?

Yes, most such standard books are common to all the competitive exams but ensure they cover the entire syllabus of the UPSC.

How should I approach the UPSC optional subject mathematics syllabus?

Systematically tackle the syllabus, standard textbooks should be used and the previous years question papers must be practiced regularly.

How often should I revise the mathematics syllabus in UPSC?

Revision is essential and best done every month or two so that concepts and problem solving techniques do not fade from the mind.

Is the IAS maths syllabus different from other competitive exams?

Maths Syllabus for IAS is designed for civil services and involves all mathematical skills needed in administrative functions.

Where can I get the UPSC maths optional syllabus PDF download?

The PDF can be accessed on the UPSC official website or other reliable sources online that provide UPSC mathematics optional preparation material.

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