Mathematics Optional Syllabus for Civil Service Examination

The Mathematics optional subject consist of two papers and each paper carries 250 marks each. This makes a total of 500 marks for this paper.

Mathematics Syllabus for History Optional – Paper I

(1) Linear Algebra

•  Vector spaces over R and C, 
• linear dependence and independence, 
• subspaces, bases, dimension; 
• linear transformations, 
• rank and nullity, 
• matrix of a linear transformation.
• Algebra of Matrices; 
• row and column reduction, 
• echelon form, congruence 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; f
• unctions 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 integrals;
•  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 linear equations with constant coefficients, complementary function, particular integral and general solution. 
• Second 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 & 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 equations. 
• Application to geometry: 
  • curves in space, curvature and torsion; 
• Serret-Frenet’s formulae. 
• Gauss and Stokes’ theorems, 
• Green’s identities.

Mathematics Syllabus for History 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.

(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 functions, 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-Jordan (direct), Gauss-Seidel(iterative) methods. 
• Newton’s (forward and backward) interpolation, Lagrange’s interpolation. 
• Numerical integration: Trapezoidal rule, Simpson’s rules, Gaussian quadrature formula. 
• Numerical solution of ordinary differential equations: Euler and Runge 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.

Practicing previous year question papers and attending mock test can help you in conquering top score in the Mathematics civil service optional paper.