HCS mains syllabus mathematics optional paper

# HCS mains syllabus mathematics optional paper

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## HCS Syllabus:  Mathematics HCS Mains Optional Syllabus

HPSC HCS mains optional subject syllabus

Part - I

1. Linear Algebra
Vector space, Linear dependance and independance, Sub spaces, Bases, Dimensions, Finite dimensional vector spaces. Matrics, Cayley-Hamilton theorem, Eigenvalues and Eigenvectors, Matrix of linear transformation, Row and column reduction, Echelon form, Equivalence, Congruence and Similarity, Reduction to Canonical form, Rank, Orthogonal, symmertrical, Skew Symmetrical, Unitary, Hermitian Skew-Hermitian forms–their given values. Orthogonal and Unitary reduction of quardratic and Hermition forms, Positive definite quadratic forms, Simultaneous reduction, Sylvester‘s law of inertia.

2. Calculus
Real numbers, Limits, Continuity, Differentiability, Mean-value Theorems, Taylor‘s theorem with remainders, Indeterminate forms, Maxima and Minima, Asymptotes, Functions of several variables, Continuity, Differentiability, Partial derivatives,Maxima and Minima, Lagrange‘s method of Multipliers, Jacobian, Riemann‘s definition of Definite integrals; Indefinite integrals, infinite and improper integral, Double and triple integrals (techniques only). Repeated integrals, Beta and Gamma functions. Areas, Surface and Volumes, Centre of Gravity.

3. Geometry
Cartesian and Polar coordinates in two and three dimension, Second degree equations in two and three dimensions, Reduction to Cannonical forms, Straight lines, Plane, Sphere, Cone, Cylinder, Paraboloid, Ellipsoid, Hyperboloid of one and two sheets and their properties. Shortest distance between two skew lines, Curves in space, Curvature and torsion. Serret-Frenet‘s formulae.

4. Ordinary Differential Equations
Formation of differential equations, Order and Degree, Equations of first order and first degree, Integrating factor, Equations of first order but not of first degree, lairaut‘s equation, singular solution. Higher order linear equations with constant coefficients. Complementary function and particular integral. General solution. Euler-Cauchy equation. Second order linear equations with variable coefficients. Determination of complete solution when one solution is known. Method of variation of parameters.

5. Statics
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. ?Lemi‘s theorem‘.

6. Dynamics
Degree of freedom and constraints, Rectilinear motion, Simple Harmonic motion. Motion in a plane, Projectiles. Constrained Motion, Work and energy, Conservation of energy, Motion under Impulsive forces, Kepler‘s laws, Orbits under Central forces, Motion of varying mass, Motion under resistance.

7. Hydrostatics
Pressure of heavy fluids. Equilibrium of fluids under given system of forces, Centre of pressure, Thrust on curved surfaces, Equilibrium of floating bodies. Stability of equilibrium. Metacentre, Pressure of gases, problems relating to atmosphere.

Part-II

1. Vector Analysis
Scalar and vector fields, triple products. Differentiation 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, Application to Geometry, Gauss and Stoke‘s Theorems, Green‘s identities.

2. Real Analysis
Real number system, Ordered sets. Bounds, Ordered Field, Real number systems as an Ordered Field with least Upper Bound, Cauchy Sequence, Completeness. Completion Continuous Functions, Uniform Continuity. Properties of continuous functions on compact sets. Riemann Integral, Improper integrals. Differentiation of functions of several variables, Maxima and Minima, Absolute and conditional Convergence of series of real and Complex terms, Rearrangement of series, Uniform convergence, Infinite Products. Continuity, differentiability and integrability for series, Multiple integrals. Infinite and alternating series.

3. Numerical Analysis
Numerical Methods : Solution of algebraic and transcendental equations of one variable by bisection, Regula-falsi, and Newton-Raphsons methods. Solution of system of linear equations by Gaussian elimination and Gauss-Jordan (direct) methods. Gauss Seidel (iterative) method. Interpolation : Nedwton‘s (forward and backword) and Lagrange‘s method.

4. Mechanics
Concepts of particles, Lamina, Rigid Body, Displacement, Force, Mass, Weight, Motion, Velocity, Speed, Acceleration. Parallelogram of forces. Parallelogram of velocity, acceleration, resultant, equilibrium of coplanar forces. Moments, Couple, Friction, Centre of mass, Gravity. Laws of motion. Motion under conservative forces. Motion under gravity. Projectile, Escape velocity; Motion of artificial satellites.

5. Probability
Sample space, Events, Algebra of events, Probability–Classical, Statistical and Axiomatic Approaches. Conditional Probability and Baye‘s Theorem Random Variables and Probability. Distributions–Discrete and Continuous. Mathematical Expectations. Binomial, Poisson and Normal Distributions.

6. Statistical Methods
Collection, Classification, tabulation and presentation of data. Measures of central value. Measures of dispersion. Skewness, moments and Kurtosis. Correlation and regressiion.