UPPSC Maths Optional Syllabus

Mathematics is widely considered as one of the most scoring subjects in the UPPSC State Civil Services Examinations. The reason behind this is that unlike other subjective subjects like English or History, a maths score has a definitive answer that can be scored based on your understanding and application of the concepts.

Moreover, the questions asked in Maths are logic-based and don’t require too much memorization apart from formulas and theorems. Hence, the subject is easy to master for candidates with a strong mathematical background and academic inclination. However, it is important to understand the marking scheme of this paper before preparing for it. The correct and logical approach can help you maximize your marks.

To ensure a high score in the exam, one should start with revision of NCERT Books of class 11 and 12. After that, it is advisable to solve previous year full-length papers. It will help you get familiar with the type of questions that are asked in the exam and also improve your time management skills.

Another thing that should be kept in mind is to practice step-by-step method while solving problems. This will not only help you solve the problems quickly but also avoid silly mistakes that can cost you marks in the exam. Additionally, it is a good idea to read some of the best books for IAS Maths Optional as they will provide you with a detailed analysis of the syllabus and exam pattern.

Lastly, candidates should try to avoid any distractions while preparing for this subject. They should not listen to music or watch movies as it will hamper their concentration level. In addition, they should keep a track of their progress by writing down the topics that need more focus.

Choosing the right optional subject for the UPSC exam is crucial as it will decide the fate of your entire civil services journey. Therefore, it is necessary to select a subject that suits your interests and academic profile. The best option for IAS aspirants is to opt for Maths. It is a scoring subject and the competition is low, which means you can easily get a top score in this subject. The subject static syllabus also makes it easy for applicants to revise and retain the material. Moreover, since the subject is unrelated to current affairs, candidates can prepare at a much faster pace.

UPPSC Maths Optional Syllabus Paper-1

1. Linear Algebra and Matrix:

Vector spaces, Sub Spaces, basis and dimensions, Quotient. space, co-ordinates, linear transformation, rank and nullity of a linear transformation, matrix representation of linear transformation, linear functionals, dual space, transpose of a linear transformation, characteristic values, annihilating polynomials, Cayley-Hamilton theorem, Inner product spaces, Cauchy-Schwarz inequality, Orthogonal vectors, orthogonal complements, orthonormal sets and bases, Bessel’s inequality of finite dimensional spaces, Gram-Schmidt orthogonalisation process. Rank of Matrix, Echelon form, Equivalence, congruence and similarity, Reduction to canonical form, orthogonal, symmetrical, skew-symmetrical, Hermitian and skew-Hermitian matrices, their eigen values, orthogonal and unitary reduction of quadratic and Hermitian form, Positive definite quadratic forms, simultaneous reduction.

2. Calculus:

Limits, continuity, differentiability, mean value theorems, Taylor’s theorem, indeterminate forms, maxima and minima, tangent and normal, Asymptotes, curvature, envelope and evolute, curve tracing, continuity and differentiability of function of several variables Interchangeability of partial derivatives, Implicit functions theorem, double and triple integrals. (techniques only), application of Beta and Gamma functions, areas, surface and volumes, centre of gravity.

3. Analytical Geometry of two and three dimensions:

General equation of second degree, system of conics, confocal conics, polar equation of conics and its properties. Three dimensional co-ordinates, plane, straight line, sphere, cone and cylinder. Central conicoids, paraboloids, plane section of conicoids, generating lines, confocal conicoids.

4. Ordinary differential equations:

Order and Degree of a differential equation, linear, and exact differential equations of first order and first degree, equations of first order but not of first degree, Singular solutions, Orthogonal trajectories, Higher order linear equations with constant coefficients, Complementary functions and particular integrals.

Second order linear differential equations with variable coefficients: use of known solution to find another, normal form, method of undetermined coefficients method of variation of parameters.

5. Vector and Tensor Analysis:

Vector Algebra, Differentiation and integration of vector function of a scalar variable gradient, divergence and curl in cartesian, cylindrical and spherical coordinates and their physical interpretation, Higher order derivates, vector identities and, vector equations, Green’s, Gauss and Stoke’s theorems

UPPSC Maths Optional Syllabus Paper-2

1. Modern Algebra: Groups, Cyclic groups, subgroups, Cosets of a subgroup, Lagrange’s theorem, Normal subgroups, Homomorphism of groups, Factor groups, basic Isomorphism theorems, Permutation groups, Cayley’s theorem.

Rings, Subrings, Ideals, Integral domains, Fields of quotients of an integral domain, Euclidean domains, Principal ideal domains, Polynomial rings over a field, Unique factorization domains.

2. Real Analysis: Metric spaces and their topology with special reference to sequence, Convergent sequence, Cauchy sequences, Cauchy’s criterion of convergence, infinite series and their convergence, nth term test, series of positive terms, Ratio and root tests, limit comparison tests, logarithmic ratio test, condensation test, Absolute and conditional convergence of general series in R, Abel’s Dirichlet’s theorems. Uniform convergence of sequences and series of functions over an interval, Weierstrass M-test, Abel’s and Dirichlet’s tests, continuity of limit function. Term by term integrability and differentiability.

Riemann’s theory of integration for bounded functions, integrability of continuous functions. Fundamental theorem of calculus. Improper integrals and conditions for their existence, tests.

3. Complex Analysis: Analytic functions, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, Power series representation of an analytic function. Taylor’s series. Laurent’s series, Classification of singularities, Cauchy’s Residue theorem, Contour integration.

4. Partial Differential Equations: Formation of partial differential equations. Integrals of partial differential equations of first order, Solutions of quasi linear partial differential equations of first order, Charpit’s method for non-linear partial differential equations of first order, Linear Partial differential equations of the second order with constant coefficients and their canonical forms, Equation of vibrating string. Heat equation. Laplace equation and their solutions.

5. Mechanics: Generalized co-ordinates, generalized velocities, Holonomic and nonholonomic systems, D’Alembert’s principle and Lagrange’s equations of motion for holonomic systems in a conservative field, generalized momenta, Hamilton’s equations. Moments and products of inertia, Principal axes, Moment of inertia about a line with direction cosines (l,m,n), Momental ellipsoid, Motion of rigid bodies in two dimensions.

6. Hydrodynamics: Equation of continuity, Velocity Potential, Stream lines, Path Lines, Momentum and energy. Inviscid flow theory: Euler’s and Bernoulli’s equations of motion. Two-dimensional fluid motion, Complex potential, Momentum and energy, Sources and Sinks, Doublets and their images with respect line and circle.

7. Numerical Analysis: Solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi and Newton-Raphson methods and order of their convergence. Interpolation (Newton’s and Lagrange’s) and Numerical differentiation formula with error terms. Numerical Integration: Trapezoidal and Simpson’s rules. Numerical solutions of Ordinary differential Equations: Euler’s method, Runge-Kutta method.