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UPSC provided the UPSC Maths syllabus in the notification. Maths is one of the IAS 2024 optional subjects with 2 papers (Paper 1 and Paper 2). Candidates opting for maths as their optional subject must know the UPSC IAS syllabus for Maths before they start their IAS exam preparation. The syllabus of UPSC Maths consists of topics and sections on which the IAS Maths question paper is based in the IAS exam.
The difficulty level of UPSC Maths syllabus 2024 is of graduation level. IAS maths syllabus comprises Algebra, Real Analysis, Complex Analysis, Mechanics of Rigid Bodies, Mechanics of Deformable Bodies and much more. In this article, we have shared the detailed syllabus of UPSC IAS Maths 2024 to help IAS 2024 aspirants for quality preparation.
Download complete guide for UPSC CSE exam preparation here
How to crack UPSC Civil Services Exam: Trend Analysis, Cut off, Preparation Tips & Strategy
Both papers of the IAS Mathematics Optional paper are of 250 marks each. The UPSC IAS 2024 mains has 9 papers - GS 1, GS 2, GS 3, GS 4, Ethics and Integrity, General English, Regional subject and two optional papers.
Also read: UPSC Mock Tests Series for UPSC
From the following table, candidates can check the UPSC maths optional syllabus for paper 1.
Linear Algebra | Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimension; 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 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, straight lines, 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 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 function, particular integral and general solution. Second order linear equations with variable coefficients, Euler-Cauchy equation; determination of complete solution when one solution is known using method of 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 & 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 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. |
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From the following table, we have shared the UPSC 2024 Maths syllabus for paper 2. Candidates can check the IAS maths optional syllabus for paper 2 here in detail.
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 least upper bound property; sequences, 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 functions, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, power series representation of an analytic function, Taylor’s series; 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 system of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss-Seidel(iterative) methods. Newton’s (forward and backward) interpolation, Lagrange’s interpolation. Numerical integration: Trapezoidal rule, Simpson’s rules, 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. Algorithms and flow charts for solving numerical analysis problems. 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. |
Mechanics and Fluid Dynamics | Moment of inertia; Motion of rigid bodies in two dimensions., Generalized coordinates; D’ Alembert’s principle and Lagrange’s equations; Hamilton equations; Equation of continuity; Euler’s equation of motion for inviscid flow; Stream-lines, path of a particle; Potential flow; Sources and sinks, Two-dimensional and axisymmetric motion; vortex motion; Navier-Stokes equation for a viscous fluid. |
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Along with knowing the syllabus of UPSC Maths 2024, candidates must also know about the overall syllabus of the exam. UPSC mentions the syllabus in its official IAS notification. Candidates preparing for the exam must go through the syllabus to know the sections, subjects and topics on which the question paper of IAS will be based.
The IAS syllabus is divided into two parts - Prelims and Mains and optional subjects as well for the mains part. In this article, we have shared the UPSC IAS maths optional syllabus. Some maths concepts also appear in the CSAT exam in the preliminary round, for which candidates must refer to IAS prelims maths syllabus. There are a total of 26 optional subjects in the Civil Services Exam 2024.
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Candidates must know the IAS exam pattern along with the UPSC Maths syllabus 2024, for a sorted preparation strategy of the IAS exam 2024. UPSC mentions the IAS exam pattern in the notification. The UPSC exam pattern consists of the number of questions, stages, time duration and marking scheme of the exam.
As per the exam pattern, there are 3 stages - preliminary, mains and personality test/interview. The prelims consists of 2 papers which are MCQ-based, the mains consists of 9 papers in descriptive mode. There is negative marking in the preliminary stage of the UPSC IAS exam. Candidates qualifying each stage will be eligible to appear in the next stage.
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Candidates preparing for the IAS exam can refer to the following pointers on how to approach their preparations for IAS maths exam.
Refer to the detailed syllabus of UPSC Maths 2024 and check the subjects.
Pick the subjects from IAS Maths syllabus which are tough and keep them aside.
Enroll for subject specific classes and try to get your concepts clear for the tougher subjects.
Start solving the UPSC IAS maths optional question papers. The question papers can be downloaded from the official website of UPSC.
Keep analysing your performance from time-to-time to check the progress of the preparations.
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Apart from the IAS Maths syllabus, candidates must also get to know about the best IAS books through which they can prepare for the Maths optional subject efficiently.
Topics | Name of Books | Author/Publishers |
Analytic Geometry | Analytical Solid Geometry | Shanti Narayan and P K Mittal |
Solid Geometry | P N Chatterjee | |
Calculus | Mathematical Analysis | S C Malik and Savita Arora |
Elements Of Real Analysis | Shanti Narayan and M D Raisinghania | |
Linear Algebra | Schaum Series | Seymour Lipschutz |
Linear Algebra | Hoffman and Kunze | |
Dynamics And Statics | Me Statics & Dynamics | Vasishtha |
Ordinary Differential Equations (Ode) | Ordinary And Partial Differential Equations | M D Raisinghania |
Vector Analysis | Schaum Series | Murray R. Spiegel |
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Topics | Name of the Books | Author/ Publisher |
Complex Analysis | Complex Variable (Schaum's Outlines | Spiegel, Lipschitz, Schiller, Spellman |
Real Analysis | Elements Of Real Analysis | Shanti Narayan and M D Raisinghania |
Algebra | Contemporary Abstract Algebra | Joseph Gallian |
Partial Differential Equations | Advanced Differential Equations | M D Raisinghania |
Linear Programming | Linear Programming And Game Theory | Lakshmishree Bandopadhyay |
Mechanics And Fluid Dynamics | Fluid Dynamics | Krishna Prakashan |
Numerical Analysis And Computer Programming | Computer-Based Numerical And Statistical Techniques | M.Goyal |
Numerical Methods | Jain, Iyengar, and Jain | |
Digital-logic And Computer Design | M. Morris Mano |
Other optional subjects syllabus of IAS exam 2024
Quick links: How to prepare for UPSC IAS without coaching?
The syllabus of maths is the document which consists of the subject, sections and topics on which the question paper of the maths optional subject will be based. In this article we have shared the detailed syllabus of maths UPSC 2024.
Yes, Maths is a tough subject but not for those candidates who are good at it. Maths is also a high-scoring subject and if candidates have a good command over the subject, they can get higher merit.
No, there is nothing as of such from UPSC. Optional is an integral part of the UPSC IAS selection process and candidates are advised to not believe in any rumors like this. If there is any information related to this we will be sharing it through our channels.
Apart from the addition of an extra exam centre for preliminary exams, there is nothing major change in UPSC 2024 for Civil Services recruitment.
The UPSC syllabus for Civil Services exam consists of a detailed syllabus for preliminary, mains and optional. The subject, topics and sections in it consist of the syllabus.
There are a total of 26 optional subjects in UPSC.
Books, previous year question papers, topper interviews, topper notes, newspapers, magazines and subject wise detailed explanatory videos are the best sources for UPSC IAS.
Others:28 January,2025 - 30 April,2025
Others:31 January,2025 - 30 April,2025
Application Date:28 March,2025 - 29 April,2025
Others:31 March,2025 - 30 April,2025
Application Date:07 April,2025 - 03 May,2025
Hello!
IRAS (Indian Railway Accounts Service) officers can be posted in both cities and rural areas since Indian Railways has offices and operations nationwide. They typically work in divisional offices, zonal offices, workshops, and production units. Transfers do happen but are usually not very frequent and depend on government policies and organizational needs.
Hope this helps!
To become an IAS officer, you must clear the Civil Services Examination (CSE) conducted by the Union Public Service Commission (UPSC), and then undergo training at the Lal Bahadur Shastri National Academy of Administration (LBSNAA). While any bachelor's degree is acceptable, choosing a degree that aligns with the UPSC syllabus, particularly in humanities and social sciences, can be advantageous.
UPSC Civil Services Examination (CSE):
This is the primary exam for becoming an IAS officer, involving three stages: Preliminary, Main, and Personality Test (Interview).
Any Bachelor's Degree:
A graduate degree from a recognized university is the basic requirement to be eligible for the CSE.
Beneficial Degrees:
Degrees in humanities and social sciences, such as Bachelor of Arts (BA), can be helpful due to the syllabus overlap with subjects like History, Political Science, Sociology, and Economics.
Optional Subject:
You can choose an optional subject in the Mains exam, and a BA degree can provide a wide range of options.
Hope it helps!!
Hello!!
That's a great goal—and your background as a research scholar can actually be a big asset in preparing for the UPSC Civil Services Exam (CSE), which is the path to becoming an IAS officer.
Step-by-Step Guide to Become an IAS Officer:
1. Understand the Exam Structure
The UPSC CSE has three stages:
Prelims: Objective (GS Paper + CSAT)
Mains: Descriptive (9 papers)
Interview: Personality Test
2. Check Eligibility
Age: 21 to 32 (for General category)
Degree: Any graduate degree (you already qualify)
3. Choose Your Optional Subject
Choose something you’re comfortable with (can be from your research field if it's available).
Popular optionals: Sociology, Geography, PSIR, Anthropology, etc.
4. Build a Study Plan
Since you're a research scholar, time management is key.
Daily 4–6 hours of focused study can work if you're consistent.
Focus on:
NCERTs (6th–12th)
Standard books (like Laxmikanth for Polity, Spectrum for History)
Current Affairs (The Hindu, Indian Express, or PIB + Monthly compilations)
5. Join a Test Series (Optional but Helpful)
For Prelims and Mains: to build speed, writing skills, and clarity.
6. Practice Answer Writing
Mains is all about articulation. Practice with previous year questions and mock tests.
7. Appear for the Exam
Notification comes out around February.
Prelims is in May/June, Mains in September, and Interview early next year.
How to Align This With Your Research Schedule:
Create a realistic time-table that includes 2–3 hours on weekdays, and more on weekends.
Use your research experience for deeper understanding and essay writing.
If your goal is to become an IAS officer, choose a degree that supports UPSC preparation and also provides a good backup:
BA in Political Science, History, Geography, or Economics is ideal for IAS.
BSc or BCom is fine too, but you'll need to study humanities separately for UPSC.
For backup, pick a subject with good job prospects like BBA or Computer Science.
KCET is required only if you're planning professional courses like Engineering or Agriculture. It's not needed for BA or general degrees.
Hello aspirant,
For becoming IAS , you have to first complete your graduation from any background from any recognised university or college. If you are in your final year, then also you are eligible. After that you have to give an entrance exam name UPSC . If you will clear this exam then according to your marks you may get the IAS position.
To know about UPSC entrance exam, kindly go through the given link:
https://competition.careers360.com/exams/upsc-cse
All the best!!
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