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IAS/UPSC Mathematics Optional Classes

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Mathematics is the most beautiful and most powerful creation of the human spirit.
Stefan Banach, Polish mathematician

A UPSC aspirant is often faced with the dilemma of choosing the right optional. Both the papers contain 500 marks and have a massive impact in your selection in the final list. So one should chose their optional subject very carefully after proper due diligence.


Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.


Candidates who have interest in Maths and have studied it in their graduation can opt to take Maths as their optional subject. Only candidates that have some academic background in mathematics should consider choosing Maths as it is a technical subject. Most of the syllabus is static in nature so you don’t have to link it with current affairs. Also, since it is objective in nature, it is extremely scoring.

Some of the Toppers with Mathematics as their optional subject

Name of the candidate Year of passing Rank secured Total marks (500) MATHEMATICS
Kanishk Kataria CSE- 2018 1 361
Ganesh Kumar Bhaskar CSE- 2019 7 310
Gss Pravenchand CSE- 2018 64 342

Optional subject marks play an important role in improving your UPSC-CSE all India ranking. Also, if you have not scored well in GS papers, you can still find your name in UPSC-CSE final list if you score well in optional papers.

What Mathematics Has To Offer?
  • Static Syllabus– The syllabus for this subject is static. It is not linked to current affairs. So, once you are done with the syllabus you don’t have to constantly update your knowledge, you just need to revise
  • High Scoring Potential- Maths has a high scoring potential compared with other subjects. If you have prepared well and have written the answers correctly, you will most definitely score high marks.
  • Objective in Nature- Since the questions are not subjective or opinion based but factual, the examiner has to give you full marks if your presentation and content is up to the mark.

Features of Chahal Academy MATHEMATICS optional subject Online Course:-

  • Detailed coverage of both MATHEMATICS paper I & II for UPSC mains exam by Delhi based faculties.
  • Access to the best lecturers anytime and anywhere.
  • 100+ hours of online lectures by Delhi based faculty members.
  • The faculty has a year of experience in guiding students of MATHEMATICS optional. The faculty is passionate about the subject and is the only teacher for MATHEMATICS optional who teaches the subject in both Hindi and English medium.
  • Videos can be played both on Web & Application
  • Flexibility to watch a lecture 3 times
Super Affordable fees:

Usually, the cost of MATHEMATICS optional coaching fee is approx. Rs.50,000/- to Rs.75,000/- whereas online coaching costs very less.

Your time, Your Place:

While offline IAS coaching has a fixed time schedule, online coaching is flexible. Aspirants can save time from travelling between home, college and other places.


One of the biggest advantages of online MATHEMATICS coaching is that no lecture or topic will be missed as you can watch it anytime anywhere on your laptop or mobile.


Mathematics Paper I

  1. 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’s 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.
  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 integral; 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.
  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 liner equations with constant coefficients, complementary function, particular integral and general solution. Section 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.
  5. 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.
  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 equation. Application to geometry: Curves in space, curvature and torsion; Serret-Furenet's formulae. Gauss and Stokes’ theorems, Green's identities.

TMathematics Paper I

  1. 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.
  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 function, 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.
  5. 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.
  6. 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-Jorden (direct), Gauss-Seidel (iterative) methods. Newton’s (forward and backward) and interpolation, Lagrange’s interpolation. Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula. Numerical solution of ordinary differential equations: Eular 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.
  7. Mechanics and Fluid Dynamics:Generalised 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.
Book list for UPSC mathematics optional
  • Linear Algebra: A.R.Vasista, Schaum Series
  • Calculus and Real Analysis: S.C Malik and Savita Arora, Shanti Narayana
  • 3-D Geometry: P.N. Chatterjee
  • Ordinary Differential Equations: M.D. Raisinghania, Ian Sneddon
  • Vector Analysis: A.R.Vasista, Schaum Series
  • Algebra: Joseph A. Gallian, Shramik Sen Upadhayay
  • Complex Analysis: Schaum Series, J.N. Sharma, Ponnu Swami, G K Ranganath
  • Linear Programming: Shanti Swarup, Kanti Swarup, S D Sharma
  • Numerical Analysis: Jain and Iyengar, K. Shankar Rao, S. S. Sastry
  • Computer Programming: Raja Raman
  • Dynamics & Statics: A.R.Vasista, M. Ray
  • Mechanics and Fluid Dynamics: M.D. Raisinghania, R.K. Gupta, J.K. Goyal and K.P. Gupta, Azaroff Leonid

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