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The Civil Services Exam (CSE) is India is a national wide Competitive examination conducted by the Union Public Service Commission (UPSC). Via the Civil Services Examination, the UPSC recruits candidates for various Central and All India Civil Service Posts in the Indian Administrative Service (IAS), Indian Foreign Service (IFS), Indian Police Service (IPS), Indian Revenue Service (IRS), Etc. it is mentioned under Article 315 of our Constitution. This Service not only provides a professional career to the youth but also provides an opportunity to do something special for the upliftment of the country and Society.

UPSC candidates often face the dilemma of choosing the right optional. Both the optional papers contain 250 marks and massively impact your selection in the final merit. Therefore, one should choose their optional subject very carefully after proper research.

  • Mathematics is basically a science that deals with the logic of quantity, shape and arrangement.
  • Mathematics is in everything all around us or everything we do.
  • Aspirants who have studied Mathematics in their graduation and have a genuine interest in maths can take it as their optional subject.
  • Maths is a technical subject. Thus, Candidates who have an academic background in mathematics consider choosing maths optional.
  • No need to link it with current affairs as most of the syllabus is static and highly scoring because it is objective in nature.
  • The marks of the optional paper play a vital role in improving your UPSC CSE all-India ranking.
  • Moreover, if you have not scored enough in the General Studies paper and you still wish to find your name in the UPSC CSE final list you should focus more to score well in the optional paper. Mathematics optional UPSC coaching fee
  • Online video lecture for both optional paper-
  • Offline classroom coaching for both optional paper-

Why is Mathematics a good optional subject to choose for?

  1. Mathematics has a high scoring potential compared with other optionals. Just prepare well for the subject and write the answer correctly and you will definitely score higher marks.
  2. The syllabus for mathematics is static and is not linked to current affairs
  3. Thus, once you are done with the syllabus there is no need to constantly get updated on your knowledge or revise.
  4. Mathematics questions are factual and not subjective or opinion based. Hence, the examiner is compelled to give you full marks if your presentation and content are up to the mark.
  5. Needless to say, We all have been connected to mathematics since our school life, maybe a table or simple calculation. Consequently, we are familiar with the subject that makes it an attractive optional subject for the students
  6. Mathematics gives the opportunity to score sure shot marks on the subject if you have managed to solve the problem successfully but it is not possible in the case with any theory-based or analytical paper.
  7. As per the recent trend, mathematics has evolved as the highest scoring optional subject to increase the overall marks in General Studies Papers and improve ranking.

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Why Mathematics as Optional Subject in UPSC-CSE?

The students who have studied Mathematics as core subject in their graduation, should take Maths as the optional in UPSC-CSE. To be more specific the students who have studied B.Tech, B.Sc-Mathematics, B.A-Mathematics or any other course in which mathematics is the core subject, can take Mathematics as optional in UPSC-CSE.

Besides this,  student should have some extra attributes as:-

  1. Student should be highly passionate and extremely devoted to subject.
  2. Eager to learn new concepts and technicalities.
  3. Ready to diversify his/her potential as per the demand of UPSC-CSE.
  4. Ready to learn the vast applications of Mathematics in the society and his career not just as a learner but as an applicant also.

Why Youth Destination for Mathematics optional

Youth destination is a well known institute for Maths optional UPSC coaching

  • Access to the best offline or online lectures anytime and anywhere.
  • Full coverage of both Mathematics paper first and second for the Mains by expert faculty.
  • The faculty is not only passionate about the subject but they also have years of experience in guiding students in mathematics optional.
  • They also teach the subject in both English as well as Hindi medium.

Online Coaching for Mathematics Optional

Candidates will have accessibility to Maths optional UPSC coaching online and the flexibility to watch a lecture 2-3 times.

The biggest advantage is that you would not miss any Mathematics optional class for UPSC and can watch them anytime on your phone or laptop.

Usually, the cost of the Mathematics Optional course for UPSC is about 50,000 but the online cost is very less. Moreover, online coaching is flexible as compared to offline coaching with a fixed schedule. It saves travelling time between home and classroom.

Mathematics - Optional Vs Other Optionals

Mathematics is the only optional subject which allowed the students to achieve 375 marks in the past while no student could get such marks in other optional subject. Out of 25 students, atleast one student has mathematics as optional subject. The popularity of Mathematics optional is increasing day by day.

Besides this,  Mathematics as an optional has many advantages over other optional as:-

  1. Most advantageous and highest scoring till date.
  2. No cramming like other optional subjects.
  3. Only conceptual understanding as per the demand of UPSC-CSE.
  4. Easy to understand and self innovative subject.
  5. Availability of lots of resources.
  6. Best expert guidance.

Mathematics Optional Pros

Mathematics is a scoring subject. There is literally no theory here to write about in the paper. You only need to understand the underlying theories and then solve problems accordingly. If you have prepared thoroughly and are able to write answers correctly and with accuracy, you can score maximum marks, even more than any humanities subject. For instance, in 2017, the highest optional marks were 375 out of 500 which went to Anubhav Singh for mathematics optional.

The questions asked in this paper are generally direct and straight-forward. If you are good in mathematics and enjoy the subject, you should go for mathematics optional as you will find it easy to solve the papers with proper preparation of syllabus.

Mathematics is a static subject, unlike some other humanities subjects. There is no linkage to current affairs here.

In this optional, you will face far less competition compared to some popular subjects.

You can assess your performance once you have given the paper. There is no scope for subjectivity and nothing is left to the interpretation of the examiner.

Mathematics Optional Cons

There is virtually zero overlap with the general studies papers.

Preparation will take more time since the syllabus is vast and to be done thoroughly and completely if you want to perform well. Moreover, maths can be studied only by practicing the sums. There is no way around it.

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Brief Overview of Mathematics Optional Syllabus

Mathematics Paper – 1

  1. Linear Algebra
  2. Calculus
  3. Analytic Geometry
  4. Ordinary Differential Equations
  5. Dynamics and Statics
  6. Vector Analysis.


  • Paper – 1

    Sec-A : Linear Algebra, Calculus, Analytic Geometry

    Sec-B: Ordinary Differential Equations, Dynamics & Statics, Vector Analysis

Mathematics Paper – 2

  1. Algebra
  2. Real analysis
  3. Complex analysis
  4. Linear programming
  5. Partial differential equations
  6. Numerical Analysis and Computer programming
  7. Mechanics and Fluid Dynamics.
  • Paper – 2

    Sec-A : Algebra, Real Analysis, Complex Analysis, Linear Programming,

    Sec-B: Partial differential equations, Numerical Analysis and Computer programming, Mechanics and Fluid Dynamics

Syllabus :- Paper - 1 : Section - A

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.

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.

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.

Syllabus :- Paper - 1 : Section - B

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.

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.

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.

Syllabus :- Paper - 2 : Section - A

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 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.

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 problems, basic solution, basic feasible solution and optimal solution; graphical method and simplex method of solutions; duality. Transportation and assignment problems.

Syllabus :- Paper - 2 : Section - B

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 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. 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.

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; Stream-lines, path of a particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.

Maths Optional

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Strategy for Mathematics Optional for UPSC Civil Services

A large number of candidates are confused in deciding whether to take mathematics as optional or not. Choosing mathematics optional for UPSC is a good decision because of the good performance of maths in recent years. Maths has been really performing well in recent years.

  1. The importance of practice cannot be underestimated in maths optional papers to cover the maths syllabus for upsc. It is a must to score well on your maths papers.
  2. Regular practice makes you faster in your problem-solving skills.
  3. Thus, answer writing practice is essential for any civil services paper.
  4. Don’t make minor mistakes like ignoring the sign, calculation mistakes or copying the numbers wrong.
  5. It will prove very costly. So, practice until you get perfection.
  6. Read the questions carefully and write down what is given in the question after this workout the answer step by step and line by line.
  7. Always, write the solution neatly because haphazardly written answers make the examiner’s work harder.
  8. Always understand the topic carefully and questions correctly before presenting the solution because if you start to write the question wrong you will lose precious space in the answer booklet and later, you will not be able to write the correct answer because of space crunch.
  9. Keep a formula sheet handy with you. It will help you to memorize certain formulas that would be needed to solve the questions.
  10. Make sure you are relaxed and calm enough before solving mathematical problems.
  11. Being nervous or panic can cause confusion and you will end up writing badly in your optional paper.

Who are the ones who should take mathematics as an optional subject?

  • The candidates coming from reputed institutions like IITs, NITs and BITs can choose mathematics as optional.
  • It is suggested only those who are very strong in Maths and have genuine interest should take mathematics as optional.
  • The competition is very high so don’t take math in haste otherwise, it will let you waste an attempt and later change optional.
  • There are so many rules regarding which optional subject is good for the UPSC exam. Simply, you should take the subject in which you have genuine interest or background.
  • If you have studied mathematics in graduation and like this subject then choosing mathematics as an optional might be a good idea.
  • There are two papers namely Paper I and Paper II in the UPSC mains exam for the optional subject.
  • Each paper is out of 250 marks and a total of 500 marks.

Benefits of taking Mathematics as an optional

  • The first advantage associated with keeping maths as an optional subject is that you can score higher marks provided you have solved the question properly.
  • Mathematics is the subject to understand the core principle of the subject and the functioning of the mathematical applications.
  • If you take mathematics as an optional paper you will have to face less competition and there are more chances to succeed.
  • The biggest advantage attached to keeping mathematics as an optional paper in the UPSC examination is being easier to understand and comprehend.
  • The basic concept of mathematics involves the practical orientation of the mathematical principles associated with the subject like understanding the mathematical problems, analysing the solution and finding out the right answer.
  • You are sure and confident about your preparation. You can also have the idea about getting the marks as you are aware of the answer or the solution to the problem provided in the answer sheet.
  • Keeping mathematics as an optional paper in your UPSC mains exam would help you in analysing the problems and understanding principle-oriented study since mathematics is a static subject and revolves around the basic idea of understanding the principles of application and guiding the functioning of mathematical applications.
  • Mathematics has always been a scoring subject as the questions in this paper are generally taken directly from the textbooks.
  • Thus if you have prepared well and written the answer correctly you will definitely score high marks.
  • The Maths UPSC Syllabus is static and not linked to current affairs so once you have completed the syllabus, there is no need to update your knowledge. Simply, you can revise before the exam.

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Booklist for Mathematics optional

Candidates who have opted for Maths optional for UPSC must have to dedicate their time to practice with the help of the best UPSC maths optional books.

Books for Maths optional are a crucial part of UPSC Mains preparation.

Mathematics is one of the most scoring optional subjects in the UPSC Mains examination and this subject is always fun. Through the right strategy and practice in Maths Optional, candidates can crack this exam.

  • For a subject like mathematics, there is an absolute answer to each question. Maths is a quite scoring subject.
  • The more you practice the more you will be able to ace the Maths optional. Aspirants from Science backgrounds mostly choose this as their optional subject.
  • Those who have opted for Maths optional or beginner must choose the best maths optional books for UPSC Mains preparation as there are a plethora of UPSC Maths optional books available in the market. Therefore choosing the right one may be a bit confusing for the students.
  • We have come up with a list of highly recommended Maths Optional books to make the process easier. It will help you have a solid grasp of mathematics and preparation.

UPSC Maths optional books

The syllabus of Maths Optional for Paper I is advanced so you must have the best UPSC Maths optional books that will help you to prepare for this section. This paper covers topics like Linear Algebra, Analytic Geometry, Calculus, Vector Analysis, Dynamics and Statics.

Books for Paper I

  • Linear Algebra- K.C. Prasad, K.B. Dutta
  • Analytic Geometry- Shanti Narayan, DK Jha, HC Sinha
  • Differential calculus- PK Mittal, Shanti Narayan
  • Dynamics and Statics- M Ray
  • Differential Equations- Golden series- NP Bali
  • Vector Analysis- PK Mittal, Shanti Narayan

Books for Paper II

UPSC maths optional paper II consists of a wide range of topics such as Algebra, Real Analysis, Linear Programming, Partial Differential Equations, Mechanics and Fluid Dynamics.

The books mentioned below would give you crisp information about these topics.

  • Textbook of Coordinate Geometry – Rai Ballabh
  • Analytical Solid Geometry – Shanti Narayanan, PK Mittal
  • Modern Algebra – A. R. Vashistha
  • Differential Equations – Schaum Series (Frank Ayres)
  • Operations Research – Kanti Swarup, PK Gupta, Man Mohan
  • Statistical Mechanism – Carson Horg
  • Schaum’s Outline of Matrices – Frank Ayres
  • Matrices – A R Vashistha
  • Coordinate Geometry – R. L. Soni
  • Mathematics Dictionary – RC James
  • Algebra – J. N. Sharma and S. N. Goyal
  • Problems in Calculus of One Variable – I. A. Maron
  • Text Book on Differential Calculus – Gorakh Prasad
  • Linear Algebra – Finite Dimension Vector Spaces – Sharma and Vashishta

Moreover, there are some reference books apart from the Maths books mentioned above, you can also add them. Don’t pick multiple books at one time to cover one topic.

  • Linear Algebra – Finite Dimension Vector Spaces – Sharma and Vashishta
  • Schaum’s Outline of Matrices – Frank Ayres
  • Problems in Calculus of One Variable – I. A. Maron
  • Coordinate Geometry – R. L. Soni
  • Matrices – A R Vashistha
  • Modern Algebra – A. R. Vashistha
  • Differential Equations – Schaum Series (Frank Ayres)
  • Textbook of Coordinate Geometry – Rai Ballabh
  • Statistical Mechanism – Carson Horg
  • Analytical Solid Geometry – Shanti Narayanan, PK Mittal
  • Text Book on Differential Calculus – Gorakh Prasad
  • Operations Research – Kanti Swarup, PK Gupta, Man Mohan
  • Algebra – J. N. Sharma and S. N. Goya
  • Mathematics Dictionary – RC James

Why choose Mathematics as UPSC optional subject?

  • Since mathematics is a science subject, there is no scope for interpretation. Every question has a definite answer and no multiple perspectives so you can predict your score after attempting the question paper. This makes Mathematics a predictable optional subject in the UPSC Mains exam.
  • Previous year toppers in mathematics have scored more than 350 marks which are a way ahead of any other Optionals highest marks. Therefore, mathematics is a good scoring subject.
  • There are a lot of good books available in the market relevant to mathematics optional
  • Here, at the Youth Destination IAS academy, you have the opportunity to learn from the best faculty.
  • Candidates who are engineering graduates have already covered approx 60% of the syllabus i.e. Calculus, Vector Numerical Analysis, Geometry, Ordinary and Partial Differential Equations, Dynamics Statistics and Mechanics. Mathematics is the best option for you if you are a science graduate or engineering graduate.

Preparation strategy to cover Maths books

  • Mathematics is a scoring subject if prepared with the right strategy and in the right way. These are some key tips that would help you to cover the Maths books in a more efficient manner and make the most out of your preparation.
  • Don’t choose any book with the recommendation of other candidates but give preference to your level of understanding and choose a book as per your comfort..
  • Choose the books that help to clear the basic concepts and terms such as Linear Algebra, Calculus, Analytic Geometry and so on.
  • A complete understanding of the basic concept is a must to score well in the exam. Make sure not to choose multiple books for covering one particular topic. You can also make notes of important formulas and theorems while studying these books.

Frequently Asked Question

Well, it is only a myth. Any candidate can crack the UPSC exam with Maths optional with timely preparation and proper guidance. The candidates who have an interest in Maths, opt for it. Mathematics optional is more subjective and requires a little bit of cramming. Engineering students can cover the entire syllabus in college itself by choosing Maths department electives.

If you are able to attempt the majority of questions, you will surely score much more than other optional subjects like History, Anthropology and Sociology.

Candidates should focus on conceptual clarity and hard work through consistent practice in Mathematics Paper. There is no need to memorize much, making it a preferred choice for optional among the students.

You can ignore proofs in ODE, PDE and Real Life Analysis but they should be done in some topics like Linear Algebra. However, if time allows you should cover each and every topic of the syllabus.

Focus on preparing well and you will definitely score more than 300 + easily. Don’t think about scaling too much.

Dedication, hard work and proper guidance are needed to clear the IAS examination. If you know the demand of the UPSC exam or you have someone to guide you and discuss queries then no need to join any coaching institute but if you don’t have this luxury you should join a coaching institute.

It plays a vital role in the preparation and provides the proper mentorship which is required to clear the examination.

It will take approx. 6 months to cover the entire syllabus if you are investing 5 to 6 hours every day.

If you are planning to cover the entire syllabus within 6 months then you might have to study 5 to 6 hours every day depending on your target.

Solving the previous year’s question papers is a must. It will help to understand the syllabus in a much better way.

Joining the test series is a must and if you have not joined any coaching institute then at least join the test series. It will keep your brain sharp and preparation on track throughout the year. Regular evaluation will help you in identifying your strengths and weaknesses as well.

All scientific non-programmable computers are allowed in the UPSC exam hall.

Candidates who have studied Mathematics at their PG or UG level. They will find the UPSC Maths syllabus easier than other ones but for those who have not studied Mathematics their graduation may face difficulties in dealing with this subject.

No, they are not the same. The CSAT syllabus includes reasoning and analytical sections while the Maths syllabus covers topics that are pure mathematics. So, they are different from each other.

UPSC mains exam includes a total of 48 optional subjects and Maths is one of them, popular as well as scoring.

The mathematics syllabus includes topics like Calculus, Algebra, Vector Analysis, Partial Differential Equations etc.

You can fetch good marks in Mathematics as every question has an absolute answer.

Mathematics is a concept-focused subject and you can easily cover the whole syllabus in 6 months.

Follow a systematic approach, time management, tricks and the right study plan. Also, practice questions as much as possible.

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