## UPSC CSE Mathematics Optional TOPIC WISE PYQ

## UPSC CSE Mathematics Optional TOPIC WISE PYQ

## UPSC CSE Mathematics Optional TOPIC WISE PYQ

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## BPSC Mathematics Optional Syllabus

SECTION- I

Linear Algebra :

Vector space bases, dimension of finitely generated space. Linear transformations, Rank and nulity of a linear transformation, Cayley Hamilton theorem. Eigenvalues and Eigenvectors. Matrix of a linear transformation. Row and Column reduction. Echelon form. Equivalence. Congruence and similarity. Reduction to canonical forms. Orthogonal, symmetrical, skew-symmetrical, unitary, Hermitian and Skew-Hermitian matrices – their eigenvalues, orthogonal and unitary reduction of quadric and Hermitian forms, positive definite quadratic forms. Simultaneous reduction.

Calculus :

Real numbers, limits, continuity, differentiability, Mean-Value theorem, Taylor's theorem, indeterminate forms, Maxima and Minima, Curve Tracing, Asymptotes, Functions of several variables, partial derivatives. Maxima and Minima, Jacobian. Definite and indefinite integrals, double and triple integrals (techniques only). Application to Beta and Gamma functions. Areas, Volumes, Centre of gravity. Analytic Geometry of two and three dimensions: First and second-degree equations in two dimensions in Cartesian and polar coordinates. Plane, Sphere, Paraboloid, Ellipsoid. Hyperboloid of one and two sheets and their elementary properties. Curves in space, curvature, and torsion. Frenet's formula.

Differential Equations :

Order and Degree and a differential equation, differential equation of first order and degree, variables separable. Homogeneous, Linear, and exact differential equations. Differential equations with constant coefficients. The complementary function and the particular integral of e ax, cosax , sinax,xm , eax , cosbx, eax ,sin bx.

Vector Analysis :

Vector Algebra, Differential of Vector function of a scalar variable, Gradient, Divergence and Curl in Cartesian Cylindrical and spherical coordinates and their physical interpretation. Higher-order derivatives. Vector identities and Vector equations, Gauss and Stocks theorems.

Tensor Analysis :

Definition of Tensor, transformation of coordinates, contravariant and covariant tensor. Addition and multiplication of tensors, contraction of tensors, Inner product, fundamental tensor, Christoffel symbols, covariant differentiation. gradient, Curl, and divergence in tensor notation.

Statics :

Equilibrium of a system of particles, work, and potential energy. Friction, Common catenary. Principles of Virtual Work stability of equilibrium, equilibrium of forces in three dimensions.

Dynamics :

Degree of freedom and constraints. Rectilinear motion. Simple harmonic motion. Motion in a plane. Projectiles. Constrained motion. Work and energy motion under impulsive forces. Kepler's laws. Orbits under central forces. The motion of varying mass. Motion under resistance.

Hydrostatics :

Pressure of heavy fluids. Equilibrium of fluids under a given system of forces Centre of pressure. Thrust on curved surfaces, Equilibrium, and pressure of gases, problems relating to the atmosphere.

SECTION- II

Algebra :

Groups, sub-groups, normal subgroups, homomorphism of groups, quotient groups. Basic isomorphism theorems. Sylow theorems. Permutation Groups, Cayley's theorem. Rings and Ideals, Principal Ideal Domains, unique factorçation domains, and Euclidean domains. Field Extensions. Finite fields.

Real Analysis :

Metric spaces, their topology with special reference to Rn sequence in a metric space, Cauchy sequence Completeness, Completion Continuous functions, Uniform Continuity, Properties of Continous function on Compact sets. Riemanh Stieltjes intergral, Improper intergrals and their conditions of existence. Differentiation of functions of several variables, Implicit function theorem, maxima and minima, Absolute and conditional convergence series of real and complex terms, Re-arrangement of series. Uniform convergence, Infinite products, Continuity, differentiability, and integrability for series, Multiple integrals.

Complex Analysis :

Analytic functions, Cauchy's theorem, Cauchy's integral formula, Power series, Taylor's, Singularities, Cauchy's Residue theorem and Contour integration.

Partial Differential Equations :

Formations of partial differential equations. Types of integrals of partial differential equations of first order Charpits method. Partial differential equation with constant coefficient.

Mechanics :

Generalized coordinates, Constraints, Holonomic and Non-holonomic systems, D' Alembert's principle, and Lagranges equations. Moment of Inertia, Motion of rigid bodies in two dimensions.

Hydrodynamics :

Equation of continuity, Momentum, and energy. Inviscid Flow Theory – Two-dimensional motion, streaming motion, Sources, and Sinks.

Numerical Analysis :

Transcendental and Polynomial Equations – Methods of tabulation, bisection, regula-falsi, secant, and Newton-Raphson and order of its convergence.

Interpolation and Numerical differentiation – Polynomial interpolation with equal or unequal step sçe. Spline interpolation – Cubic splines. Numerical differentiation formulae with error terms.

Numerical integration – Problems of approximate quadrative quadrature formulae with equispaced arguments. Caussina quardrature convergence.

Ordinary differential equations – Euler's method, Multistep-predictor corrector methods – Adam's and Milne's method, convergence and stability, Runge – Kutta methods.

Probability and Statistics :

1. Statistical methods – Concept of statistical population and random sample. Collection and presentation of data. Measure of location and dispersion. Moments and shepherd's correction cumulants. Measures of Skewness and Kurtosis.

Curve fitting by least-squares regression, correlation, and correlation ratio. Rank correlation, Partial correlation co-efficient, and Multiple correlation co-efficient.

2. Probability – Discrete sample space, Events, their union, and intersection, etc., Probability – Classical relative frequency and axiomatic approaches. Probability in continuum probability space conditional probability and independence, Basic laws of probability, Probability of combination of events, Bayes theorem, Random variable probability function, Probability density function. Distributions function, Mathematical expectation. Marginal and conditional distributions, Conditional expectation.

3. Probability distributions – Dinomial, Poisson Normal Gamma, Beata. Cauchy, Multinomial, Hypergeometirc, Negative Binomial, Chebychey's Lemma. (Weak) law of large numbers, Central limit theorem for independent and identical varieties, standerd errors, Sampling distribution of T.F and Chi-square and their uses in tests of significance large sample tests for mean and proportion.

Operational Research :

Mathematical Programming – Definition and some elementary properties of convex sets, simplex methods, degeneracy, duality, sensitivity analysis rectangular games and their solutions. Transportation and assignment problems. Kuha Tucker condition for non-linear programming Bellman's optimality principle and some elementary applications of dynamic programming.

Theory of Queues – Analysis of steady-state and transient solution for queueing system with poission arrivals and exponential service time.

Deterministic replacement models, sequencing problems with two machines, n jobs, 3 machines, n jobs (special case) and n machines, 2 jobs.