# UPSC Maths Optional Syllabus

Ramanasri UPSC Maths Optional Coaching providing complete details of Mains IAS, UPSC, IFoS, CSE-Civil Services Exams and State PCs Mains like BPSC, HPSC, JKPSC, UPPSC, WBCS/WBPSC, OPSC Maths Optional Syllabus.

**UPSC/IAS/CSE-Civil Services Maths/Mathematics Optional Syllabus Paper-1 2****023-2024 **

### Section-A

** (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’s and similarity; Rank of a matrix; Inverse of a matrix; Solution of system of linear equations; Eigen values and Eigen vectors, characteristic polynomial, Cayley-Hamilton theorem, Symmetric, skew-symmetric, Hermitian, Skew-Hermitian, orthogonal and unitary matrices and their Eigen values.

**(2) 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.

**(3) 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.

## Section-B

**(4) 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 (Homogeneous linear equations); 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.

**(5) Dynamics & Statics:**

(i) Dynamics: 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.

(ii) Statics: 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.

**(6) 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.

**UPSC/IAS/CSE-Civil Services Maths/Mathematics Optional Syllabus PAPER-2 2023-2024**

** **** **** Section-A**

** (1) Modern 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, sub rings and ideals, homeomorphisms of rings; Integral domains, principal Ideal domains, Euclidean domains and unique factorization domains; Fields, quotient fields.

**(2) 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.

**(3) 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.

** (4) 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.

** ****Section-B**

**(5) Partial differential equations:**

Family of surfaces in three dimensions and formulation of partial differential equations; Solution of quasi linear 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.

**(6) Numerical Analysis & Computer Programming:**

**(i) Numerical Analysis**

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 Runga Kutta-methods.

** ****(ii) 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.

** (7) Mechanics and Fluid Dynamics:**

(i) **Mechanics: **Generalized coordinates; D’Alembert’s principle and Lagrange’s equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions.

**(ii) Fluid Dynamics: **Equation of continuity; Euler’s equation of motion for Inviscid flow; Stream-lines, path of a particle; Potential flow; Two-dimensional and asymmetric motion; Sources and sinks, vortex motion; Navier-Stokes Equation for a viscous fluid.

**UPSC Maths Optional Syllabus**

**About UPSC/IAS/CSE-Civil Services Maths/Mathematics Optional Syllabus 2022-2023**

Ramanasri IAS is providing a complete and updated 2022-2023 IAS Maths Optional Syllabus. We have done an analysis of UPSC Mathematics Syllabus what we observed that Civil Services Maths Optional Syllabus incredible in our IAS UPSC Mains Examinations. Maths Syllabus in UPSC is very important before confirming your one of the Optional Subject in IAS, UPSC, Civil Services, CSE, IFoS/IFS Mains Examinations. Maths Syllabus for UPSC majority people saying very easy it needs to be practice more and more number problems. Ramanasri IAS is teaching complete both Maths Optional Paper 1 and Maths Optional Paper 2. you can feel free to download complete UPSC Mathematics Optional Syllabus PDF in our mathematicsoptional.com website. After downloading the CSE Maths Optional Syllabus PDF, you can save it. Read by sentence by sentence and word by word Maths Syllabus PDF. Then you can feel most of the UPSC/IAS/CSE Maths Optional Syllabus 2022 studied in your B.tech or B.Sc. or B.A or B.Com with one subject as Mathematics.

Our Director Ramanasri Sir explaining, every detail of the UPSC Mathematics Syllabus PDF. You can realize that IAS Maths Optional Syllabus is very easy and scoring Optional Subject in the History of UPSC, IAS, IFoS, IFS, Civil Services, CSE Mains Examinations. Click on this link for IAS/CSE-Civil Service Exam/UPSC Maths Optional Syllabus in Hindi PDF. UPSC Syllabus in Hindi, we have done the analysis and Ramanasri IAS teaching both Hindi medium and English medium. IAS Maths Optional Syllabus PDF download IAS Maths Syllabus in Hindi the same as English.

## How to Cover the IAS Maths Optional Syllabus of the UPSC Exam

The maths optional syllabus is a huge one and can seem daunting at first for the candidates planning to take it in the mains exam. However, with proper preparation and the right strategy it can be covered easily. Here are some key tips to help you cover the syllabus more effectively.

Understand the Syllabus – This is the first step in any UPSC preparation and it must be followed rigorously. Candidates should start by understanding the syllabus thoroughly and then dividing it into smaller sections. They should also read all the important topics from the previous years papers to get an idea of what type of questions are asked in each section. Once this is done, they should begin with their preparation.

Make a Time Table – This will help them plan their study schedule in a more effective way and make sure they do not miss out on any important topic. Candidates should divide the syllabus into small units and try to complete one unit every day. This will help them cover the syllabus at a faster pace and increase their speed in solving problems. They should also keep revising the syllabus regularly to ensure they do not forget any important concepts.

Practice – This is the most crucial part of any preparation for Maths. It is important to solve as many problems as possible, as this will help them improve their speed and accuracy. This will also help them reduce the chances of making silly mistakes that can cost them marks in the exam.

Choose the Right Books – It is essential for a candidate to choose the right Mathematics books for UPSC as these will help them prepare in a better way. They should avoid choosing any book based on the recommendation of other candidates and instead go for the ones that best suit their level of understanding. They should also ensure they practise as many questions as they can from these books to get a good feel for the subject.

Know How to Score – Conclusion: Maths is one of the most scoring subjects in the Civil Services Examination and can give you high ranks if you prepare well for it. The reason behind this is that it is a very objective subject and the examiners are compelled to give you full marks if your content and presentation are up to the mark. This makes it a very safe option for any student who wants to increase their overall marks and secure a good rank. Moreover, unlike the subjective and opinion-based subjects like humanities, maths is more factual and requires a lot of logical reasoning. This is why it has become the preferred subject for students who want to clear IAS.

## How to Complete UPSC Maths Optional Syllabus

Upsc maths optional syllabus is a huge subject and can be time consuming to finish. Aspirants must remember that completing the syllabus is just the first step in the process of IAS preparation. The next step is to practice and keep revising your concepts. This is the only way you can ensure that you will not miss a question in the exam hall. IAS toppers recommend maintaining a formula sheet which will help you to review the key formulas at any moment. This will make sure that you do not lose marks by missing a simple solution.

Mathematics is a scoring subject, and is the highest scoring optional paper when compared to other subjects such as Political Science or Anthropology. As the questions are factual and not opinion based, the examiner is compelled to give you marks if you have prepared well. This is not possible in case of other subjects.

It is important to understand the logical flow of the subject and build a strong foundation. This will make it easier for you to solve all types of questions. It is also advisable to do the easy units first and then move on to the difficult units. This way you will be able to finish the entire syllabus in less time.

Do not cram, remember that Maths is a logical subject and requires a lot of understanding and retaining the concepts. Doing this will save you a lot of time in the long run and help you to score better.

Try to make notes and take a lot of practice tests while studying maths. This will help you to identify your mistakes and improve yourself. Also, it will help you to know which topics are the most difficult for you and the ones you need to focus on more. This will help you to save time in the exam hall and avoid getting stuck on a single question.

Many students feel that once they have finished the syllabus, their preparation is complete and they will be able to get good marks in the exam. However, it is important to spend the next 50% of your time for Practice and exploring new questions. The best way to do this is by attempting Previous Year Papers. These papers will help you to improve your speed and accuracy. You can check out Ramana Sri IAS/IFoS INSTITUTE’s Official Question Bank for more question papers.

You must also focus on your presentation skills as it will play a major role in improving your score. Ensure that you write your answers correctly and clearly. This will make the examiner happy and he will reward you with higher marks. Also, try to cover all the important points of a particular topic. Hence, you should not miss anything.

## Learn more important and vital points About UPSC/IAS/CSE-Civil Services Exam Maths Optional Syllabus

For UPSC/IAS/CSE-Civil Services Exam mathematics optional, you can read about the previous years' exam question papers. Taking a look at the question papers will help you identify the topics that you need to study the most. Besides studying the UPSC/IAS/CSE-Civil Services Exam maths optional syllabus, you should also take notes for difficult concepts and read relevant parts of the study material. The following tips will help you with UPSC mathematics optional. Read on! They'll come in handy!

## Complete and detailed UPSC/IAS/CSE-Civil Services Exam Maths Optional Syllabus with in 6 months

The UPSC/IAS/CSE-Civil Services Exam maths optional syllabus covers two papers with each paper having 250 marks. The total number of marks is 500. The exam pattern is available online. You can download the syllabus for UPSC maths paper 1 as a PDF. The syllabus contains the key concepts that you need to know. Moreover, you should take notes on the concepts that you have to learn. Practice solving sample questions and taking notes can help you to answer the test questions with a logical flow.

The UPSC mathematics optional syllabus is well-balanced and includes topics related to theorems, proofs, and application. The proofs part is a significant portion of each paper, accounting for around 20-30% of the total mark. To solve a proof, you need to have some basic understanding of the theory, and then follow the steps logically. The syllabus breaks up problems so that you don't get confused between the theory and the proof. It also helps you reduce the chances of making mistakes.

## maths optional syllabus for upsc

The UPSC/IAS/CSE-Civil Service Exam Maths Optional Syllabus requires conceptual clarity and proper time management to prepare for the examination. The syllabus is divided into two parts. Part 1 focuses on Mechanics and Part 2 focuses on Calculus. Mechanics is a good subject to choose if you are interested in a career in engineering. This syllabus will be useful in all levels of engineering, including the Civil Service.

For preparing for paper I, you must know what the exam covers. Mechanics will cover topics like generalized coordinates, homomorphism, and Hamilton equations. The question paper also covers motion of rigid bodies in two dimensions and the Navier-Stokes equation for viscous fluid. It is recommended to study all the required material from standard textbooks, including Schaum series.

## UPSC/IAS/CSE-Civil Service Exam maths optional syllabus pdf

There is no common portion between the UPSC Mathematics Optional Syllabus and the GS Paper. This provides the students with ample opportunity to formulate a study plan that addresses all aspects of this optional paper. Practice is the key to success for this subject. Practice solving questions and re-writing them, so that your answers are clear and accurate. The main purpose of this practice is to get the most out of the limited time you have for preparing for the UPSC Mains Exam.

The UPSC Mathematics Optional syllabus includes topics such as Analytical Geometry, Linear Algebra with matrices, Calculus, Ordinary and Partial Differential Equations, Complex Analysis, Numerical Analysis, Vector Analysis, and Applied & Mathematical Physics. Fluid Dynamics and Mechanics are two topics in UPSC/IAS/CSE-Civil Service Exam Maths Optional Syllabus Paper 2.

## UPSC/IAS/CSE-Civil Service Exam Maths Optional Syllabus

UPSC/IAS/CSE-Civil Service Exam Maths optional syllabus contains a number of topics that can be quite challenging to crack. There are formula sheets and a static syllabus, making it an extremely difficult subject to crack. Thankfully, this article will give you some tips to overcome this. You will learn how to make your study time go by much quicker. Here are some helpful tips to help you pass this exam with flying colors. Read on to discover how to make your time count in the exam hall!

## Mathematics is a hard science subject

While many students dread mathematics, many take it for its practical application. Although it is considered difficult, it is a subject that many aspirants have aced in the IAS exam. In addition to its difficulty level, mathematics also has a reputation for being a difficult subject in school. Although many aspirants have gotten high marks in maths optional papers, it has been difficult to gauge the effect of the optional on overall performance.

While math is not a particularly popular subject, it is still a popular choice for UPSC examinations. Although it has less competition than subjects such as English, physics, and social science, its syllabus is static and is not connected to current affairs. Despite its low popularity, however, it will never go out of style and the marks scored by students chosen for the exam will speak volumes about their mathematical skills.

## It is a static syllabus

The UPSC/IAS/CSE-Civil Service Exam maths optional syllabus follows the same pattern of most institutes and universities in India. The focus is on building concepts, solving problems, and applying mathematics. Unlike other optional subjects, the maths syllabus does not link current affairs, so students are required to reread the topics and revise them for the examination. This means a less intensive preparation time. The syllabus also focuses on the fundamentals of mathematics rather than current affairs.

The UPSC/IAS/CSE-Civil Service Exams Maths Optional syllabus is static and does not change with the changing news and current affairs. It is important to know that you do not have to update your syllabus frequently. However, this static syllabus does not mean that you cannot study and prepare. You can prepare for the UPSC maths optional exam by studying books or practice question papers. For help, you can download sample papers for UPSC maths optional. You can also find some question papers on the UPSC website.

## It has formula sheets

The UPSC/IAS/CSE-Civil Service Exam maths optional syllabus follows the curriculum of most Indian institutes and universities. Students are expected to understand the concept very well, revise extensively, and practice solving problems with the help of formula sheets. This syllabus does not involve much of current affairs, so it is a good choice for students who are interested in the subject but do not wish to spend much time studying it. The formula sheets are useful for revision and are available in the exam center.

There are many UPSC aspirants who choose mathematics as their optional subject, and most of them qualify with excellent scores. Toppers usually focus on standard books for static subjects and on the internet for dynamic subjects. Toppers suggest that students choose maths optional if they are strongly interested in the subject. It is not an easy subject to pass, but with hard work, you can achieve good results. If you are passionate about mathematics, you should opt for it.

## It is a hard subject to crack

While UPSC has a tough syllabus, the maths optional exam is relatively easier than other subjects. By following a few simple strategies, you can prepare well for the maths optional examination. To prepare for the maths optional syllabus, you can get ready by reading books and getting ready-made question banks. It is also beneficial to practice solving previous year's mock tests and full-length papers.

One of the biggest mistakes candidates make is not preparing adequately for the exam. Most students do not have the time or interest to study for the maths optional. Practicing previous year questions is a great way to improve your accuracy and score. Practice previous year's questions until you're comfortable answering them. Then, try to crack the exam in your first attempt. Make sure you have ample time to study for the maths optional syllabus.

## UPSC/IAS/CSE-Civil Service Exam Maths Optional Syllabus

UPSC/IAS/CSE-Civil Service Exam Mathematics Optional Syllabus comprises of the following topics: Cauchy-Riemann equations, Taylor's Series, Laurent's Series, contour integration, and Cauchy's Theorem. Conceptual Clarity is important for acing UPSC exams, as well as proper revision to retain the maximum part of the syllabus studied. You can also practice writing UPSC Answers by seeking help from your mentor.

## Static syllabus

UPSC/IAS/CSE-Civil Service Exam maths optional is one of the best scoring subjects for the exam. Applicants must understand the concepts, solve problems, revise and remember them to perform well in the examination. The syllabus is composed of two papers that are three hours each, and contain four questions for each section. The questions are divided into four parts, and there are 250 marks to be earned for each paper. The syllabus is written with a balance between theoretical and applied math.

## Less emphasis on memory

One of the most important UPSC exams is the numerical aptitude test and maths is one of the highest scoring subjects. Candidates must understand concepts and solve questions to score well. It is recommended to prepare your maths questions beforehand. The syllabus has less emphasis on memory compared to other optional subjects, and the exam requires less time to prepare. It is important to practice as much as possible. Most questions in maths require you to recall formulas and theorems.

## More emphasis on formulas

Unlike GS papers, the UPSC/IAS/CSE-Civil Service Exam maths optional syllabus has no common portion. However, students can devise a study plan to get ahead of the competition. For effective preparation, practice is the key. Practice helps in improving accuracy and speed. Here are some tips to improve your performance:

## No current affairs knowledge

A good reason to choose mathematics for your optional syllabus is the fact that it is less competitive than other subjects. While humanities subjects are more prone to examiner prejudice and subjectivity, math answers are more objective and are either right or wrong. Additionally, the syllabus for mathematics is static and doesn't incorporate current affairs in any way. Candidates can focus on revising instead of learning current affairs. This is great news for candidates who wish to pass the exam without spending too much time studying current affairs.

## Predictability of paper

The UPSC/IAS/CSE-Civil Service Exam mathematics optional syllabus contains topics from various branches of science. The syllabus covers topics like Computer Programming, Physics, Digital Electronics, K-MAPS, Flowcharts, Algorithms, Logic, Analytical(solid or 3-D)Geometry, and Modern(Abstract) Algebra. In the exam, the questions asked will mainly be based on the concepts that students learn in class.

## UPSC/IAS/CSE-Civil Service Exam maths optional syllabus

a few of the ias aspirants UPSC/IAS/CSE-Civil Service Exam maths optional syllabus as upsc cse maths optional syllabus

For UPSC exam preparation, UPSC/IAS/CSE-Civil Service Exam maths optional syllabus is an important aspect to be studied. This subject is less opinion-based and has no current affairs, making it easier for students to focus on the material in order to score well in the exam. It also requires less preparation time compared to other subjects.

The UPSC/IAS/CSE-Civil Service Exam maths optional syllabus consists of two papers, Paper-1 and Paper-2 and each of which carries 250 marks. It is recommended for candidates who have a solid background in mathematics, whether through education or employment. If you are studied intermediate maths then you can choose this optional. This subject will help them develop their analytical and problem-solving skills. While this subject may seem boring to those without a solid math background, it is an important subject for UPSC exams.

While the mathematics optional syllabus is not as competitive as the other subjects, it consistently produces good ranks, such as Anubhav Singh. However, it is also important to note that the syllabus of this subject is relatively static, and does not contain a lot of current affairs. Hence, maths questions tend to be factual, not subjective, and may not be as challenging as those in other subjects.

The UPSC/IAS/CSE-Civil Service Exam maths optional syllabus is divided into two sections: theorems and application. This syllabus has the theory and application sections, and includes a proofs section, which makes up only three percent of the paper. Students are expected to understand the theory and write it in a logical way, and the syllabus is divided into questions to make it easy to answer the questions and make a good score.

The UPSC/IAS/CSE-Civil Service Exam maths optional syllabus includes two papers: ordinary differential equation and real analysis. Both papers carry 250 marks. While the mathematics optional syllabus does not require current affairs, it can help develop analytical skills and problem-solving skills. It is also a useful way to brush up on your knowledge and prepare for the UPSC exam.

Regardless of the subject, math can be a difficult subject to prepare for, but it can be done effectively and with a great strategy. The right preparation will ensure you score high marks in the exam. Choosing the right book for the right topic is crucial. Read the maths optional syllabus thoroughly and understand how the topics are graded. Make notes of important formulas and theorems as you go along with your studies.

The UPSC/IAS/CSE-Civil Service Exam maths optional syllabus is made up of two separate papers worth 250 marks. The question papers in maths are designed to provide a deeper understanding of the subject and its applications. By studying maths as an optional subject, you can boost your chances of success by answering questions accurately and correctly.

While the UPSC syllabus for the IAS mains exam is comprehensive, candidates need to choose the optional subjects wisely. This will help them to score higher in the main exam and to secure a better rank in the final UPSC merit list.

## UPSC/IAS/CSE-Civil Service Exam maths optional syllabus in Hindi

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## UPSC/IAS/CSE-Civil Service Exam maths optional syllabus in Hindi pdf

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