# HPSC/HCS Maths Optional Syllabus

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**HPSC/HCS Maths/Mathematics Optional Syllabus**

**PART-I**

**(1) Linear Algebra:** Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions, 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, skewHermitian, 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; Functions 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 integral; 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 liner equations with constant coefficients, complementary function, particular integral and general solution. Section order linear equations with variable coefficients, Euler-Cauchy equationDetermination 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 and 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 equation. Application to geometry: Curves in space, curvature and torsion; Serret-Furenet's formulae. Gauss and Stokes‘ theorems, Green's indentities

**PART II HPSC/HCS Maths/Mathematics Optional Syllabus**

**(1) Algebra:** Groups, subgroups, cyclic groups, cosets, Lagrange‘s Theorem, normal subgroups, quotient groups, hcsomomorphism 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 function, 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-Jorden (direct), Gauss-Seidel (iterative) methods. Newton‘s (forward and backward) and interpolation, Lagrange‘s interpolation. Numerical integration: Trapezoidal rule, Simpson‘s rule, Gaussian quadrature formula. Numerical solution of ordinary differential equations: Eular 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. 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.

About HPSC/HCS Maths/Mathematics Optional Syllabus

The government of Haryana conducts the exam of HPSC and is one of the most prestigious public service commissions in India. The HPSC maths optional syllabus is available on HPSC's official website and here. In addition, all the mathematics optional related state-level Public Service Commission classes are offered by Ramanasri IAS Institute.

The HPSC HCS exam has five conventional type question papers. The HPSC/HCS maths optional paper is the only paper that tests the general knowledge of the subject. The main exam, which is the most important part of the examination, consists of 5 questions, one essay, and two numerical papers. The HPSC/HCS exams are divided into two parts. The first part consists of a multi-choice examination to test general awareness and quantitative skills.

The second part of the exam is the essay. The essay is a requirement of the HPSC/HCS exam. The question paper consists of four parts and must be completed within three hours. The questions are of the conventional type, and if you have any problems, you must approach the instructor as soon as possible. The answer key is provided with each answer. The HPSC HCS Mathematics Optional Syllabus is updated frequently, and it is the best way to study.

The second section of the HPSC HCS exam is the maths optional. This subject has many requirements for applicants to succeed in this examination. For example, mechanics and fluid dynamics cover subjects like D'Alembert's principle, a moment of inertia, and generalized coordinates. Among other topics, the two-dimensional motion of rigid bodies and the vortex motion of a fluid is included in the syllabus.

The HCS/HPSC exam is divided into three parts - the prelims, the main exam, and the interview. In addition to the maths optional paper, the HPSC/HCS also requires you to write an essay in the third stage. During the exam, you must remember that there are five sections. The HPSC Maths Optional Syllabus contains the questions from the first stage.

The HPSC HCS examination is a three-tiered exam. The HPSC prelims exam is the first part of the HPSC examination. It is a qualifying examination, and your score in this paper is not considered in the final merit. The second stage of the HPSC maths optional syllabus consists of two papers, the prelims paper, and the interview.

The HPSC HCS mains exam comprises four papers. Each paper has four sections, which are known as sub-papers. For example, the maths sub-paper contains questions related to basic mathematical operations. The test pattern of the exam has changed from previous years to this one. Therefore, it would be best to study the new syllabus to prepare for the HPSC HCS. Once you pass the prelims, you will be able to ace the other two papers.