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Mathematics has always been the most chosen subject among UPSC candidates despite being the toughest subject. Candidates will have a chance to secure full marks by the point to point marking of math's. This is the reason it is the most scoring subject.
In this article, we will discuss all that you need to know about mathematics optional for the UPSC exam.
There are two UPSC math's optional question papers  paper 1 and paper 2 with a total of 500 marks.
Syllabus of Math's optional for UPSC
Aspirants need to download the Math's syllabus for UPSC to start their preparation. We have provided the detailed math's UPSC syllabus released by UPSC to make this process easier for students.
Paper I

Linear algebra

Calculus

Analytic Geometry

Ordinary differential equations

Dynamics and statics

Vector analysis
PaperII

Algebra

Real analysis

Complex analysis

Linear programming

Partial differential equations

Numerical analysis and computer programming

Mechanics and fluid dynamics
Linear Algebra:
Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions, Linear transformations, rank and nullity, matrix of a linear transformation. Algebra of Matrices; Row and column reduction, Echelon form, congruence and similarity; Rank of a matrix; Inverse of a matrix; Solution of a system of linear equations; Eigenvalues and eigenvectors, characteristic polynomial, CayleyHamilton theorem, Symmetric, skewsymmetric, Hermitian, skewHermitian, orthogonal and unitary matrices and their eigenvalues.
Calculus:
Real numbers, functions of a real variable, limits, continuity, differentiability, meanvalue 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 integral; Double and triple integrals (evaluation techniques only); Areas, surface and volumes.
Analytic Geometry:
Cartesian and polar coordinates in three dimensions, seconddegree equations in three variables, reduction to Canonical forms; straight lines, the shortest distance between two skew lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.
Ordinary Differential
Equations: Formulation of differential equations; Equations of the first order and first degree, integrating factor; Orthogonal trajectory; Equations of first order but not of the first degree, Clairaut’s equation, singular solution. Second and higherorder linear equations with constant coefficients, complementary function, particular integral and general solution. Section order linear equations with variable coefficients, EulerCauchy equation; Determination of complete solution when one solution is known using the 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 and Statics:
Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; constrained motion; Work and energy, conservation of energy; Kepler’s laws, orbits under central forces. Equilibrium of a system of particles; Work and potential energy, friction, Common catenary; Principle of virtual work; Stability of equilibrium, equilibrium of forces in three dimensions.
Vector Analysis:
Scalar and vector fields, differentiation of vector fields 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; SerretFurenet's formulae. Gauss and Stokes’ theorems, Green's identities.
Mathematics Paper I
Algebra:
Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley’s theorem. Rings, subrings and ideals, homomorphisms of rings; Integral domains, principal ideal domains, Euclidean domains and unique factorization domains; Fields, quotient fields.
Real Analysis:
Real number system as an ordered field with the least upper bound property; Sequences, the limit of a sequence, Cauchy sequence, completeness of real line; Series and its convergence, absolute and conditional convergence of series of real and complex terms, rearrangement of series. Continuity and uniform continuity of functions, properties of continuous functions on compact sets. Riemann integral, improper integrals; Fundamental theorems of integral calculus. Uniform convergence, continuity, differentiability and integrability for sequences and series of functions; Partial derivatives of functions of several (two or three) variables, maxima and minima.
Complex Analysis:
Analytic function, CauchyRiemann 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, RegulaFalsi and NewtonRaphson methods, solution of a system of linear equations by Gaussian Elimination and GaussJorden (direct), GaussSeidel (iterative) methods. Newton’s (forward and backwards) and interpolation, Lagrange’s interpolation. Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula. Numerical solution of ordinary differential equations: Euler and Runga Kutta methods. Computer Programming: Binary system; Arithmetic and logical operations on numbers; Octal and Hexadecimal systems; Conversion to and from decimal systems; Algebra of binary numbers. Elements of computer systems and concept of memory; Basic logic gates and truth tables, Boolean algebra, normal forms. Representation of unsigned integers, signed integers and reals, double precision reals and long integers. Algorithms and flow charts for solving numerical analysis problems.
Mechanics and Fluid
Dynamics: Generalized coordinates; D’Alembert’s principle and Lagrange’s equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions. Equation of continuity; Euler’s equation of motion for inviscid flow; Streamlines, the path of a particle; Potential flow; Twodimensional and axisymmetric motion; Sources and sinks, vortex motion; NaiverStokes equation for a viscous fluid.
Books for UPSC mathematics optional
Choosing the right study material is a must while preparing the UPSC math's optional. Questions will be a bit difficult to deal with in the optional subject. Therefore, you need the right UPSC math's optional books that would not only help you practice but also make your concepts clear. These books are very helpful to get a strong grasp of the UPSC Math's optional syllabus.

Linear algebra Hoffman and Kunze

Calculus and real analysis S.C. Malik and Savita Arora, Shanti Narayan

Analytic Geometry Shanti Narayan and P.K. Mittal

Ordinary differential equations M.D. Raisinghania

Dynamic and statistics A.R. Vasista, M. Ray

Vector analysis A.R. Vasista

Algebra Shramik Sen Upadhyay, Joseph A. Galiyan

Complex analysis J. N. Sharma, G.K. Rangnath

Linear programming and game theory Laxmi Shree Bandopadhyay

Ordinary and Partial differential equations M.D. Rai Singhania

Numerical analysis Jain and Iyengar, K. Shankar Rao

Computer programming Raja Raman

Mechanics and fluid dynamics M.D. Rai Singhania, J.K. Goyal and KP Gupta, RK Gupta.
Benefits of taking Math's an optional
Advantage

Civil service candidates often face a dilemma of choosing the right Optional since both the papers consist of 500 marks so they have an immense impact on your selection in the final merit list. Therefore, you should choose optional subjects very carefully after proper research.

Math is all around us in everything we do. It is the science that deals with logic, quantity and arrangement. Math is the building block for everything in our lives for example architecture, mobile devices, money, sports, engineering etc.

Students who have studied mathematics in their graduation can choose math's optional for UPSC. Since math is a technical subject, candidates that have some academic background in mathematics should choose math as optional. There is no need to link it with current affairs as most of the syllabus is static in nature. It is extremely scoring because it is objective in nature.

If you have not scored well in General Studies papers then you should try to score well in optional papers to find your name in the final merit list. Optional subject marks play a vital role in improving your rank.

Since the UPSC Mathematics optional syllabus is static, you don't have to constantly update your knowledge or be linked to current affairs. Once you are done with the syllabus you just need to revise it.

Math's is a high scoring optional subject compared to other subjects. If you write your answer correctly you will definitely score high marks.

Examiner will have to give the full marks if your content and presentation is okay because the questions are not subjective or opinion based but factual.

The math's syllabus needs less time to prepare as it is short compared to other optional subjects. You will just need to make the right study plan and prepare accordingly.

The short syllabus of math's needs less time to cover and you can utilize this time in the preparation of other subjects.

There is no need to prepare long theories. You will have to understand the concepts and solve the problems accordingly. If you can solve problems with accuracy and also have good knowledge of mathematics then you can score maximum marks in the exam.

The questions in this subject are taken directly from the textbooks so if you are a math's graduate you will find the question of an average medium difficulty level.

Moreover, the competition is limited because many UPSC aspirants don't take up this optional compared to other subjects like Sociology, History, Geography.

Humanities subjects have an element of subjectivity involved therefore two examiners might grade differently for the same answer because examiners' personal prejudice comes into play. But in mathematics, the answer is either right or wrong. You will have a clear idea of how you have fared after taking the exam.

Our school life has been involved with mathematics from class 1 to class 12th. Maybe it is the table or simple calculation. We have been associated with math's right from childhood. Therefore, our familiarity has grown over and over with the subjects which is why it is attractive and easy to grasp an optional subject for students.
Disadvantage

You should study mathematics along with a pen and paper because you can prepare it only by working out the solution. You cannot just read through all the study material in mathematics. The best way to learn mathematics is to do mathematics.

Mathematics is not helpful for General Studies papers; it will add nothing to your general studies preparation like History, Geography, Polity at the syllabus does not overlap.

Candidates will not get marks for wild attempts because mathematics involves either right or wrong answers but you can expect some marks for attempting if the solution provided by you is in the right direction.
"Mathematics is the most beautiful and most powerful creation of the human spirit"
 Stephen Banach, (Polish mathematician)
"Without mathematics, there is nothing you can do. Everything around you is mathematics. Everything around you is numbers". ( Shakuntala Devi)
Why you should join Youth Destination for mathematics optional

You will get access to the best math's optional classes for UPSC anytime and anywhere.

Full coverage of both mathematics paper I and II for UPSC mains exam by expert faculties

Online interactive video lectures by Delhi's best faculties.

The faculty has years of experience in providing math's optional coaching for UPSC teaching candidates of mathematics optional. They are very concerned and passionate about the subject.

Youth Destination provide the subject in both Hindi and English medium.

You can watch videos later in case you miss any class.

Facility to watch a lecture three times.
Preparation strategy for mathematics optional paper
Know the syllabus
First of all, you need to understand the syllabus and collect the updated syllabus from authentic sources. After collecting the syllabus you should read it carefully.
Make a right study plan
After going through the syllabus, you need to prepare a suitable study plan at your convenience. Furthermore, your study plan should have a slot for proper revision after every topic. Try to align your study plan for the optional paper with other parts of the main exam because you have limited time to prepare for the Mains exam.
Practice previous year's question papers
Solving the previous year's question paper is a good strategy to prepare for UPSC mathematics Optional. Toppers’ advice is to solve and analyse previous year's papers in an exam like environment so that you can have an idea of the pattern and type of questions. It will also help to develop time management skills.
Focus on important topics
Once you go through the entire syllabus then you will have the idea that some topics have repetitive patterns so there are higher chances to come into the exam again. You should focus on these important topics and practice them regularly. It will lead to a higher chance to solve similar questions easily.
Join a good coaching institute
Undoubtedly, coaching institutes play an important role in UPSC exam preparation. If you have sufficient knowledge and ability, you can prepare for the IAS exam by selfstudy but if you need guidance then you should join a coaching institute. There are many UPSC Math's optional coaching institutes in Delhi that provide coaching for optional subjects. If you have no time to join offline coaching, you can also go for an online one. Youth Destination IAS Academy is the best IAS coaching in Delhi. We provide the best online and offline classes for mathematics optional for UPSC.
Read the best books
Books are one of the important factors in the success of students. Therefore, they must refer to genuine books. They must have some reference books to study. Don't collect too many books but always study with the best books and study well from the selected books because reading too many books may confuse you.
Some tips to score high in math's Optional for UPSC

The importance of practice is immense to score well on your math paper. No doubt, answer writing is essential for the UPSC exam and it gets double in mathematics optional UPSC. Regular practice makes you faster in your problemsolving skills.

Don't make silly mistakes like copying numbers wrong, Calculation mistakes or ignoring the sign because it will cost you very high. Practice until you are not perfect.

Go through the questions carefully and write down what is given in the question. After this, solve the question step by step.

You should present your solutions line by line in steps.

Don't write haphazardly answers because it will make the examiners work harder and always write the solutions clearly.

Before writing the solution, always understand the questions correctly because once you start writing the incorrect answer you will lose precious space.

Keep a formula sheet handy with you. It will help you memorize certain formulas that would be needed to solve the problems.

Don't panic because it can cause confusion. Always solve mathematical questions with a relaxed mind otherwise you will end up badly in your UPSC math's optional paper.
"Mathematics is the queen of the sciences"
Johann Carl Friedrich
Conclusion
Mathematics is a subject that needs patience and a relaxed mind. Don't get stressed in the exam hall otherwise, there are higher chances of making mistakes. Practice is the most important aspect of math's preparation. Don't write unnecessary things as mathematics is a logical subject and check your answer after completing the question paper. Have faith in yourself and make your own goals to score as much as you can. You will definitely achieve success.
Mathematics is a beautiful subject and concerns everything around us. Treat this subject with the respect that it deserves and you will be rewarded in your UPSC exam.
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