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TNPSC Combined Statistical Service Examination Syllabus 2021 – Overview

TNPSC CSSE Exam Pattern 2021

The questions in Paper–I & Paper-II will be set both in Tamil and English.

TNPSC CSSE Exam Syllabus 2021

UNIT I – Algebra and Trigonometry

• Theory of Equations: Polynomial equations; Imaginary and irrational roots; Symmetric functions of roots in terms of coefficient; Sum of the powers of roots; Reciprocal equations; Transformations of equations.
• Descartes rule of signs: Approximate solutions of roots of polynomials by Newton – Raphson Method – Horner’s method; Cardan’s method of solution of a cubic polynomial.
• Summation of Series: Binomial, Exponential and Logarithmic series theorems; Summation of finite series using the method of differences – simple problems.
• Expansions of sin x, cos x, tan x in terms of x; sin nx, cos nx, tan nx, sin nx, cos nx , tan nx, hyperbolic and inverse hyperbolic functions – simple problems.

UNIT II – Calculus, Coordinate Geometry Of 2 Dimensions And Differential Geometry

• nth derivative; Leibnitz’s theorem and its applications; Partial differentiation. Total differentials; Jacobians; Maxima and Minima of functions of 2 and 3 independent variables – necessary and sufficient conditions; Lagrange’s method – simple problems on these concepts.
• Methods of integration; Properties of definite integrals; Reduction formulae – Simple problems.
• Conics – Parabola, ellipse, hyperbola and rectangular hyperbola – pole, polar, co-normal points, con-cyclic points, conjugate diameters, asymptotes and conjugate hyperbola.
• Curvature; the radius of curvature in Cartesian coordinates; polar coordinates; equation of a straight line, circle and conic; the radius of curvature in polar coordinates; p-r equations; evolutes; envelopes.
• Methods of finding asymptotes of rational algebraic curves with special cases. Beta and Gamma functions, properties and simple problems. Double Integrals; change of order of integration; triple integrals; applications to area, surface are volume.

UNIT III – Differential Equations and Laplace Transforms

• First-order but of higher degree equations – solvable for p, solvable for x, solvable for y, claimant’s form – simple problems.
• Second-order differential equations with constant coefficients with particular integrals for eax, xm , eax sin mx, eax cos mx
• Method of variation of parameters; Total differential equations, simple problems.

UNIT IV – Vector Calculus, Fourier Series and Fourier Transforms

• Vector Differentiation: Gradient, divergence, curl, directional derivative, unit normal to a surface.
• Vector integration: line, surface and volume integrals; theorems of Gauss, Stokes and Green – simple problems.
• Fourier Series: Expansions of a periodic function of period 2π; expansion of even and odd functions; half range series.
• Fourier Transform: Infinite Fourier transforms (Complex form, no derivation); sine and cosine transform; simple properties of Fourier Transforms; Convolution theorem; Parseval’s identity.

UNIT V – Algebraic Structures

• Groups: Subgroups, cyclic groups and properties of cyclic groups – simple problems; Lagrange’s Theorem; Normal subgroups; Homomorphism; Automorphism; Cayley’s Theorem, Permutation groups.
• Rings: Definition and examples, Integral domain, homomorphism of rings, Ideals and quotient Rings, Prime ideal and maximum ideal; the field and quotients of an integral domain, Euclidean Rings.
• Vector Spaces: Definition and examples, linear dependence and independence, dual spaces, inner product spaces.
• Linear Transformations: Algebra of linear transformations, characteristic roots, matrices, canonical forms, triangular forms.

UNIT VI – Real Analysis

• Sets and Functions: Sets and elements; Operations on sets; functions; real-valued functions; equivalence; countability; real numbers; least upper bounds.
• Sequences of Real Numbers: Definition of a sequence and subsequence; limit of a sequence; convergent sequences; divergent sequences; bounded sequences; monotone sequences; operations on convergent sequences; operations on divergent sequences; limit superior and limit inferior; Cauchy sequences.
• Series of Real Numbers: Convergence and divergence; series with non-negative numbers; alternating series; conditional convergence and absolute convergence; tests for absolute convergence; series whose terms form a non-increasing sequence; the class I 2
• Limits and metric spaces: Limit of a function on a real line; metric spaces; limits in metric spaces.

UNIT VII – Complex Analysis

• Complex numbers: Point at infinity, Stereographic projection
• Analytic functions: Functions of a complex variable, mappings, limits, theorems of limits, continuity, derivatives, differentiation formula, Cauchy-Riemann equations, sufficient conditions Cauchy-Riemann equations in polar form, analytic functions, harmonic functions.
• Mappings by elementary functions: linear functions, the function 1/z, linear fractional transformations, the functions w=zn, w=ez, special linear fractional transformations.
• Integrals: definite integrals, contours, line integrals, Cauchy-Goursat theorem, Cauchy integral formula, derivatives of analytic functions, maximum moduli of functions.

UNIT VIII – Dynamics and Statics

• DYNAMICS: kinematics of a particle, velocity, acceleration, relative velocity, angular velocity, Newton’s laws of motion, equation of motion, rectilinear motion under constant acceleration, simple harmonic motion.
• Projectiles: Time of flight, horizontal range, range in an inclined plane. Impulse and impulsive motion, collision of two smooth spheres, direct and oblique impact-simple problems.
• Central forces: Central orbit as a plane curve, p-r equation of a central orbit, finding law of force and speed for a given central orbit, finding the central orbit for a given law of force.
• Moment of inertia: Moment of inertia of simple bodies, theorems of parallel and perpendicular axes, a moment of inertia of triangular lamina, circular lamina, circular ring, right circular cone, sphere (hollow and solid).

UNIT IX – Operations Research

• Linear programming – formulation – graphical solution – simplex method
• Big-M method – Two-phase method-duality- primal-dual relation – dual simplex method – revised simplex method – Sensitivity analysis. Transportation problem – assignment problem.
• Sequencing problem – n jobs through 2 machines – n jobs through 3 machines – two jobs through m machines – n jobs through m machines.
• PERT and CPM: project network diagram – Critical path (crashing excluded) – PERT computations.

UNIT X – Mathematical Statistics

• Statistics – Definition – functions – applications – complete enumeration – sampling methods – measures of central tendency – measures of dispersion – skewness- kurtosis.
• Sample space – Events, Definition of probability (Classical, Statistical & Axiomatic ) – Addition and multiplication laws of probability – Independence – Conditional probability – Bayes theorem – simple problems.
• Random Variables (Discrete and continuous), Distribution function – Expected values & moments – Moment generating function – probability generating function – Examples. Characteristic function – Uniqueness and inversion theorems – Cumulants, Chebychev’s inequality – Simple problems.

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