UPSC Maths Optional Syllabus

The UPSC Civil Services Mains Exam includes Mathematics as an Optional Subject with two papers (Paper I and Paper II).

For CSE 2023, the Maths Optional remains the same as it was in 2021. Keep up-to-date with the current IAS Maths Syllabus by checking the UPSC Notification 2023.

The IAS Mathematics Optional papers carry 250 marks each, with a total of 500 marks. There are a total of nine papers in the IAS Exam Mains.

Syllabus For UPSC Maths Optional-2023

UPSC Maths Optional Syllabus – Paper 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.
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.
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 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.
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 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.

UPSC Maths Optional 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 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.
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-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.
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.

UPSC Maths Optional – Syllabus & Strategy

Deciding on an optional subject is important since both its papers make up a significant portion (500/1750 marks) of the UPSC Main Examination. It’s crucial to select an optional subject carefully. A good rule of thumb is to choose a subject that interests you. Maths is a popular optional subject for UPSC, but it’s only suitable for those who have completed their graduation in Mathematics.

Some of the benefits of Maths optional include:

Static Syllabus: The syllabus for UPSC Maths Optional remains the same and doesn’t change over time. If you have already studied this subject during your graduation, you will only need to review and refresh your understanding of the concepts. The syllabus is not related to current events, so you don’t have to keep updating your notes for review.

High scoring: Maths is an objective subject and is known for being highly scoring. This is because there is only one correct answer, so there is less room for comparison. The questions are based on facts and not opinions, which means that as long as your content and presentation are good, the examiner will not have to judge your answer and award marks

Important of memorization: You don’t have to worry about memorizing a lot while studying Maths. Although you need to remember some theorems and formulas, the subject mainly relies on logical reasoning. So, you don’t need to cram too much information.

UPSC Maths Optional preparation Tips For Main Exam?

Engineering students often choose UPSC Maths Optional as their preferred subject because its syllabus is objective. Here are some tips to help you prepare for UPSC Maths Optional:

Engineering students often choose UPSC Maths Optional as their preferred subject because its syllabus is objective. Here are some tips to help you prepare for UPSC Maths Optional:

1. Understand the concepts clearly: To do well in Maths, it’s important to have a clear understanding of the concepts. So, make sure you understand each topic that is part of the UPSC Maths Optional.
2. Revise regularly: To retain the information you’re studying, set aside time for regular revision. Practice with previous year papers and mock tests to prepare well.
3. Be organized: Presentation is important when writing answers for UPSC, so learn from toppers’ answer scripts or ask mentors to evaluate your own answers. It’s essential to write in a systematic way.
4. Don’t cram: Instead of trying to memorize everything, focus on building logical reasoning skills. This will help you to solve different types of questions asked in the exam.
5. Create a formula sheet: Maths has many formulas and theorems, so it’s important to learn them. Keep a separate notebook for formulas and review them frequently to avoid forgetting anything important.
6. Avoid careless errors: Practice enough to avoid making silly mistakes while solving problems.

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FAQs

CONCLUSION

The UPSC Maths Optional Syllabus for 2023 is the same as the 2022 syllabus and consists of two papers, each of which is of 250 marks. The syllabus covers topics such as Algebra, Real Analysis, Complex Analysis, Linear Algebra, Ordinary Differential Equations, Partial Differential Equations, Numerical Analysis, and Calculus of Variations. While Maths is considered a difficult optional subject for UPSC, candidates who have a background in mathematics or a strong interest in the subject may find it easier to prepare for this optional. To prepare for the UPSC Maths Optional exam, candidates should start by understanding the syllabus and exam pattern, gather study materials, practice solving problems regularly, and consider joining coaching classes or taking online courses to supplement their preparation. The duration of each paper in the UPSC Maths Optional exam is three hours, and the marking scheme includes both objective and subjective questions. Additionally, candidates can take the exam in Hindi or English.