UPSC Maths Optional – Syllabus, Previous Year Papers & Strategy by Dr. Gajendra Purohit
There are two papers called Paper I and Paper II in the UPSC mains for the optional subject. Each paper is out of 250 marks with a total of 500 marks. Get detailed syllabus by downloading the IAS Notification, PDF of which is available in the linked article.
UPSC Mathematics Optional 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.
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(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.
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(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.
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(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.
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(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.
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 GATE EXAM MATHEMATICS PREVIOUS YEAR QUESTIONS – DOWNLOAD NOW
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UPSC Mathematics Optional Paper II
(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.
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(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.
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(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.
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(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.
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(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.
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(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
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UPSC Mathematics Optional – Preparation Strategy
Preparing for the UPSC (Union Public Service Commission) examination in mathematics requires a well-structured and disciplined approach. Here is a step-by-step strategy to help you in your preparation:
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Understand the Syllabus: Begin by thoroughly understanding the UPSC mathematics syllabus. Familiarize yourself with the topics and sub-topics that need to be covered. This will help you create a clear roadmap for your preparation.
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Collect Study Material: Gather the necessary study material, including textbooks, reference books, previous years’ question papers, and online resources. Ensure that the material you choose covers the entire syllabus comprehensively.
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Create a Study Plan: Devise a study plan that suits your schedule and allows you to cover all the topics within a specified time frame. Break down the syllabus into smaller sections and allocate sufficient time to each topic. Remember to include regular revisions in your plan.
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Strengthen Your Fundamentals: Mathematics is a subject that builds upon fundamental concepts. Start by strengthening your foundational knowledge of various mathematical concepts, including algebra, calculus, geometry, and statistics. Ensure that you have a clear understanding of the basic principles and formulas.
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Solve Previous Years’ Question Papers: Solve as many previous years’ question papers as possible to familiarize yourself with the exam pattern and types of questions asked. Analyze your performance and identify areas that require improvement.
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Take Mock Tests: Mock tests are an essential part of your preparation. Take regular mock tests to simulate the actual exam environment. This will help you gauge your speed, accuracy, and time management skills. Analyze your performance in these tests and work on your weaknesses.
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Revise & Practice Regularly: Make regular revision & practice a part of your study routine. Set aside dedicated time for revisiting important topics, formulas, and concepts. Revision helps reinforce your learning and ensures better retention of information.
UPSC Mathematics Optional – Previous Year Papers
| Subject | Paper I | Paper II |
| IAS – Mathematics Optional – 2022 Question Paper | Download | Download |
| IAS – Mathematics Optional – 2021 Question Paper | Download | Download |
| IAS – Mathematics Optional – 2020 Question Paper | Download | Download |
| IAS – Mathematics Optional – 2019 Question Paper | Download | Download |
| IAS – Mathematics Optional – 2018 Question Paper | Download | Download |
| IAS – Mathematics Optional – 2017 Question Paper | Download | Download |
| IAS – Mathematics Optional – 2016 Question Paper | Download | Download |
| IAS – Mathematics Optional – 2015 Question Paper | Download | Download |
| IAS – Mathematics Optional – 2014 Question Paper | Download | Download |
Thank you for reading, to gain more information about CSIR NET and learn mathematics and general aptitude from Dr. Gajendra Purohit Sir visit – https://www.youtube.com/c/DrGajendraPurohitMathematics
And for university courses, blogs, quizzes and ask doubts from sir visit – https://www.mathscare.com/
