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Mathematics as an Optional for UPSC?

Success rate for Mathematics optional is approximately around 7% per annum like FD. Which means 7 in 100 students choosing this optional get selected for the service.

The Union Public Service Commission (UPSC) conducts one of the most challenging and prestigious exams in the country, the Civil Services Examination (CSE).


The CSE consists of three stages, namely, the Preliminary exam, the Mains exam, and the Personality Test.


For the Mains exam, the UPSC offers a list of optional subjects, and one of them is Mathematics.
Optional will have 2 papers each carrying 250 marks each. However, many students are confused about whether to choose Maths as an optional subject or not. Let’s explore the benefits and drawbacks of opting for Maths as an optional subject.


Benefits of Choosing Maths as an Optional Subject for UPSC:

  • Scoring Subject: Mathematics is a scoring subject. If you have a good understanding of the concepts, you can score well in the exam.
  • Short Syllabus: The Maths syllabus for the UPSC exam is relatively shorter than other optional subjects like History or Public Administration. Therefore, it is easier to complete the syllabus in less time.
  • Objective Evaluation: The Maths paper is evaluated objectively, which means that the answers are either right or wrong. There is no scope for ambiguity or subjectivity in the evaluation.
  • Utilization of Common Syllabus: Many topics of the Maths syllabus overlap with the General Studies paper, such as Statistics, Data Interpretation, and Logical Reasoning. Therefore, studying Maths can help you in the General Studies paper as well.

Drawbacks of Choosing Maths as an Optional Subject for UPSC:

  • Difficult Concepts: Mathematics can be a challenging subject for some students as it involves complex concepts and problem-solving.
  • Time-consuming: Solving Maths problems can be time-consuming. Therefore, it may take longer to complete the Maths paper, leaving you with less time for other papers.
  • No Connection with Job Profile: Mathematics is not directly related to the job profile of a civil servant. Therefore, choosing Maths as an optional subject may not help you in your career as a civil servant.

High-Scoring Strategies for UPSC Maths Optional

If your teachers emphasized the importance of practicing maths problems frequently and avoiding careless mistakes to achieve high scores, they were right.

These are fundamental strategies for success in Mathematics. Here, we have compiled some tips to help you score high in the Maths Optional subject for the UPSC Mains Exam.

  • Practice Diligently: Regular practice improves problem-solving speed and accuracy, which can help you achieve a high score in the exam.
  • Avoid Careless Mistakes: One incorrect symbol or number can be costly, so maintain focus while solving questions to avoid such errors. Practicing frequently will also help you avoid mistakes. Additionally, reserve a few minutes for review just before the exam ends to double-check your answers and make corrections if necessary.
  • Be Organized: Presentation is crucial, along with the correct answer. Therefore, write your solution neatly and systematically in the proper steps. Don’t skip any essential steps in a rush or write the solution carelessly.
  • Avoid Memorization: Maths is logical, so avoid memorizing solutions or theorems. Instead, understand the logical flow and build a strong foundation of concepts. This will enable you to solve all types of questions, as you will be able to apply logic in the exam and not skip any crucial steps of the solution.
  • Formula Sheet: Every aspirant who has chosen Maths as an optional subject should keep a formula sheet. Formulae are the backbone of maths. Prepare a formula sheet with all the necessary formulae, so you can easily revise it anytime, anywhere.

The syllabus of both the papers of Maths optional as per UPSC Official Notification is as follows:

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.

Mathematics 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.

(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.

Choosing Maths as an optional subject for UPSC can have both benefits and drawbacks. Therefore, it is essential to assess your strengths and weaknesses before deciding on the subject.

If you have a good grasp of Maths concepts and are willing to put in the effort, it can be a scoring subject for you. However, if you find Maths challenging or time-consuming, it may not be the right choice for you.

July 2024