There are several topics and chapters in Mathematics that are frequently asked in UPSC exams. These topics are essential for candidates who have chosen Mathematics as their optional subject. Here are some of the important topics:


Linear Algebra is a fundamental topic in Mathematics. Candidates should have a thorough understanding of concepts such as vector spaces, linear transformations, eigenvalues, eigenvectors, and matrices. Questions related to determinants, rank, and systems of linear equations are commonly asked in the exams.


Calculus is another crucial topic that is frequently tested in exams. Candidates should be well-versed in both differential and integral calculus. Key concepts include limits, continuity, differentiation, integration, and their applications. Questions related to maxima and minima, definite and indefinite integrals, and differential equations are commonly asked.


Real Analysis is an important branch of Mathematics that deals with the study of real numbers and real-valued functions. Candidates should have a strong understanding of topics such as sequences, series, limits, continuity, differentiability, and Riemann integration. Questions related to theorems, proofs, and convergence are often asked.


Complex Analysis deals with the study of functions of complex variables. Candidates should be familiar with topics such as complex numbers, analytic functions, contour integration, and residue theorem. Questions related to complex mappings, Cauchy-Riemann equations, and Taylor series expansions are commonly asked.


Differential Equations involve the study of mathematical equations involving derivatives. Candidates should have a good understanding of first-order and higher-order differential equations. Questions related to linear and nonlinear differential equations, boundary value problems, and initial value problems are frequently asked.


Linear Programming focuses on optimization problems where the objective is to maximize or minimize a linear function. Candidates should be familiar with topics such as linear inequalities, simplex method, duality, and transportation problems. Questions related to formulation and solving of linear programming problems are commonly asked.


Numerical Analysis deals with the study of algorithms and methods for solving mathematical problems using numerical approximation techniques. Candidates should have a good understanding of numerical methods for solving equations, interpolation, numerical integration, and numerical solutions of differential equations. Questions related to error analysis and convergence of numerical methods are frequently asked.

It is important for candidates to thoroughly understand and practice these topics to score well in the Mathematics optional subject for UPSC exams.