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UPSC Maths Syllabus 2024 - Check Detailed UPSC Maths syllabus PDF here

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UPSC Maths Syllabus 2024 - Check Detailed UPSC Maths syllabus PDF here

Edited By Vishnukumar | Updated on Feb 14, 2024 06:08 PM IST | #UPSC CSE

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.

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How to crack UPSC Civil Services Exam: Trend Analysis, Cut off, Preparation Tips & Strategy

IAS Maths syllabus

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

UPSC Maths syllabus 2024 - For Paper 1

From the following table, candidates can check the UPSC maths optional syllabus for paper 1.

UPSC Maths Syllabus 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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UPSC Maths Syllabus 2024 - Paper 2

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.

UPSC Maths syllabus - Paper 2

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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UPSC IAS Syllabus 2024

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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UPSC IAS Exam Pattern 2024

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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UPSC IAS Maths Preparation Tips 2024

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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UPSC IAS Maths Books 2024 - For Paper 1 and 2

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.

UPSC Maths Books for Paper 1 (topic-wise)

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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Topic-wise UPSC Maths Books for Paper 2

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?

Frequently Asked Question (FAQs)

1. What is the syllabus of maths in UPSC?

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.

2. Is Upsc maths optional tough?

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.

3. Will Optional be removed from UPSC 2024?

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.

4. Is there any change in UPSC 2024?

Apart from the addition of an extra exam centre for preliminary exams, there is nothing major change in UPSC 2024 for Civil Services recruitment.

5. What is the UPSC Syllabus for the Civil Services exam?

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.

6. How many optional subjects are there in the UPSC Syllabus?

There are a total of 26 optional subjects in UPSC.

7. What are the sources to prepare the UPSC IAS Syllabus?

Books, previous year question papers, topper interviews, topper notes, newspapers, magazines and subject wise detailed explanatory videos are the best sources for UPSC IAS.

8. When will the the UPSC IAS 2024 notification released?

The UPSC IAS notification 2024 has been released on February 14, 2024.

9. When is the IAS 2024 prelims exam?

UPSC IAS prelims exam 2024 is scheduled to be held on May 26, 2024.

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Questions related to UPSC CSE

Have a question related to UPSC CSE ?

Dear Aspirant !

Hope you are fine!

The main subjects for the UPSC Exam are Indian Politics, Indian Economy, International Relations, Science and Technology, Geography, History, Environment and Ecology, and related Current Affairs . There are 25 subjects and 23 literature optional that one can choose from in addition to this..

Thank you

Hello aspirant,

With the help of the IAS Answer Key 2024, applicants can estimate their potential score for the IAS preliminary exam, learn about the various IAS question types, gauge the exam's difficulty, and much more.

To get the answer key, you can visit our website by clicking on the link given below.

https://competition.careers360.com/articles/upsc-ias-answer-key

Thank you

Hope this information helps you.

Hii There,

You will meet the eligibility criteria for the UPSC exam in 2026. This is because the UPSC requires candidates to be at least 21 years old as of August 1st in the year they intend to take the examination. According to your provided date of birth, you will turn 21 on March 15, 2026, which is before the August 1, 2026, deadline for eligibility. Therefore, you will become eligible to take the UPSC exam in 2026, specifically for the UPSC 2026 examination cycle.


For detailed  information related to UPSC eligibility criteria ,  you can visit our page at:

https://competition.careers360.com/articles/upsc-ias-eligibility-criteria/amp


For detail examination pattern check out :- https://competition.careers360.com/articles/upsc-ias-exam-pattern


Thank you

Hello candidate ,

I am hoping that you are absolutely fine and doing well. As per your mentioned query, you will be glad to know that if you are a diploma holder still you can give the UPSC civil services exam. As per the eligibility criteria of this exam , candidate of any graduation course in any discipline from a recognised university as laid down under the law can give the exam of IAS. This exam is very reputed exam. And also very challenging one if you want to crack this exam you have to work very hard.

To know more about this exam , kindly go through the given link:

https://competition.careers360.com/articles/upsc-ias-eligibility-criteria

I hope this will be helpful for you!!

you can choose any undergraduate course of your choice, but it is recommended to choose a course that aligns with the UPSC syllabus. Subjects such as history, public administration, economics, political science and sociology can be helpful in preparing for the exam.

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For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

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Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. 

Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article. 

3 Jobs Available
Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

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Linguist

Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning). 

Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian

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Public Relation Executive
2 Jobs Available
Travel Journalist

The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.

2 Jobs Available
Welding Engineer

Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues. 

5 Jobs Available
QA Manager
4 Jobs Available
Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product. 

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
Production Manager
3 Jobs Available
Merchandiser
2 Jobs Available
QA Lead

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans. 

2 Jobs Available
Metallurgical Engineer

A metallurgical engineer is a professional who studies and produces materials that bring power to our world. He or she extracts metals from ores and rocks and transforms them into alloys, high-purity metals and other materials used in developing infrastructure, transportation and healthcare equipment. 

2 Jobs Available
Azure Administrator

An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems. 

4 Jobs Available
AWS Solution Architect

An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party. 

4 Jobs Available
QA Manager
4 Jobs Available
Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
ITSM Manager
3 Jobs Available
Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.  

3 Jobs Available
Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack 

3 Jobs Available
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