# Civil Services Examination Mathematics Syllabus

## Civil Services Syllabus for Math

In the civil services main examination, the candidate is allowed to take two optional subjects out of 55 subjects. Every subject paper is divided into two parts that are part I and part II and each part consist of 300 marks questions.

**Syllabus for Mathematics**

**Paper I (Maximum Marks 300)**

**1. Linear Algebra:**This paper includes 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’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. Ordinary Differential Equations**: Equations of first order but not of first degree, Clairaut's equation, singular solution, formulation of differential equations, equations of first order and first degree, integrating factor and orthogonal trajectory. Paper I also encompass second and higher order linear equations with constant coefficients, particular integral, general solution and complementary function. It also covers Euler-Cauchy equation, determination of complete solution when one solution is known using method of variation of parameters and second order linear equations with variable coefficients. Along with this, the paper also includes Laplace transforms of elementary functions, Laplace and inverse Laplace transforms and their properties, application to initial value problems for 2nd order linear equations with constant coefficients.

**3. Calculus:**Functions of a real variable, real number, differentiability, limits, continuity, maxima and minima, Taylor’s theorem with remainders, indeterminate forms, mean-value theorem, asymptotes, curve tracing. The paper also includes Lagrange's method of multipliers, Jacobian, functions of two or three variables, limits, continuity, partial derivatives, maxima and minima, Riemann's definition of definite integrals, infinite and improper integrals, indefinite integrals, double and triple integrals (evaluation techniques only) and volumes, area and surface.

**4. Dynamics & Statics:**Kepler's laws, orbits under central forces, simple harmonic motion, rectilinear motion, projectiles, motion in a plane, constrained motion, conservation of energy and work & energy. Equilibrium of a system of particles, work and potential energy, friction, common catenary and principle of virtual work, stability of equilibrium of forces in three dimensions is also included in the syllabus of paper I.

**5.**

**Vector Analysis:**Differentiation of vector field of a scalar variable, scalar and vector fields, higher order derivatives, vector identities and vector equations, gradient, divergence and curl in Cartesian and cylindrical coordinates. Moreover, paper I syllabus also cover the application to geometry like curves in space, curvature and torsion, Serret-Frenet’s formulae and Green’s identities, Gauss and Stokes’ theorems.

**6. Analytic Geometry:**Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties, 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.

**Paper II (Maximum Marks 300)**

**1. Real Analysis:**The real number system as an ordered field with least upper bound property, sequences, limit of a sequence, completeness of real line, Cauchy sequence, Riemann integral, fundamental theorems of integral calculus, improper integrals, series and its convergence, rearrangement of series, absolute and conditional convergence of series of real and complex terms. Syllabus of Part II also covers topic like 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. Further, continuity and uniform continuity of functions, properties of continuous functions on compact sets are also included in the syllabus.

**2. Complex Analysis:**Cauchy-Riemann equations, Cauchy's integral formula, Cauchy's theorem, Cauchy’s residue theorem, contour integration, analytic functions, power series representation of an analytic function, Laurent’s series, Taylor’s series and singularities.

**3. Algebra:**Cayley’s theorem, Lagrange’s theorem, groups, subgroups, normal subgroups, homomorphism of groups, cyclic groups, quotient groups, permutation groups, cosets, basic isomorphism theorems. The syllabus of paper II also includes Euclidean domains, integral domains, unique factorization domains, principal ideal domains, fields, quotient fields and rings, subrings, ideals and homomorphisms of rings.

**4. Mechanics and Fluid Dynamics**: Hamilton equations; moment of inertia, motion of rigid bodies in two dimensions, generalized coordinates; D' Alembert's principle and Lagrange's equations. Besides this, Navier-Stokes equation for a viscous fluid, 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 and vortex motion is also included in the syllabus of part II.

**5. Partial Differential Equations:**Solution of quasilinear partial differential equations of the first order, family of surfaces in three dimensions and formulation of partial differential equations, Cauchy's method of characteristics, Laplace equation and their solutions, Linear partial differential equations of the second order with constant coefficients, canonical form, equation of a vibrating string and heat equation.

**6. Linear Programming:**Linear programming problems, basic solution, basic feasible solution and optimal solution, graphical method and simplex method of solutions, transportation and assignment problems and duality.

**7. Numerical Analysis and Computer Programming:**The part II paper includes numerical methods, solution of algebraic and transcendental equations of one variable by bisection, Newton's (forward and backward) interpolation, Lagrange's interpolation, Regula-Falsi and Newton-Raphson methods, solution of system of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss-Seidel(iterative) methods.

- Numerical integration: Trapezoidal rule, Simpson's rules, Gaussian quadrature formula.
- Numerical solution of ordinary differential equations: Euler 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.

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