IFoS Maths Optional Syllabus

IFoS Maths Optional Syllabus in English is one of the most scoring subjects in the UPSC exam. This subject is well-suited for candidates who have a background in science and possess strong problem-solving skills. It is also a good choice for those who want to broaden their analytical skills. The subject requires a lot of time and effort, but it can be very rewarding. If you are considering taking this subject, it is important to prepare thoroughly and focus on the topic-wise preparation tips.In order to pass the UPSC Maths optional, you should take advantage of all available resources. This includes study notes, online videos, and previous year exam papers. It is also a good idea to attend a coaching class. This will help you understand the course material better and improve your score. Moreover, you should keep a notebook that contains all the formulas and theorems that you might need to solve questions. This will help you avoid silly mistakes and save a lot of time.

IFoS maths optional syllabus

UPSC Maths is a very challenging subject, but you can improve your chances of passing by preparing well. The best way to prepare is by systematically going through all the topics and practicing as many questions as possible. You should also try to write answers in a logical manner. This will help you score more marks and secure a high rank in the exam.Besides practicing and studying, you should also try to take some mock tests and full-length exams from previous years. It is also important to make a schedule and stick to it. By following this plan, you will be able to cover the entire syllabus and improve your score in the exam. In addition, you should try to find out the best books for UPSC Maths optional to study from.

The IFoS Mathematics optional syllabus

The IFoS Mathematics optional syllabus consists of two papers with 250 marks each. This is a great choice for science graduates, as it offers an opportunity to increase their score and get a top rank in the exam. It is also a useful subject to learn, as it develops analytical and problem-solving skills.The IFoS syllabus covers topics like calculus, linear algebra with matrices, ordinary differential equations, and analytic geometry. This subject is an excellent choice for those who wish to become a civil servant and develop their analytical and problem-solving skills. It is also one of the most popular subjects in the IAS and IFoS examinations. It is also a good choice for candidates who are interested in working as an engineer or B.Sc Maths.This subject is also considered one of the most scoring options in the IAS, IFoS, and CSE-Civil Services Mains exams. Candidates should remember that this subject is not linked to current affairs, so it is not as difficult to pass as other subjects. Moreover, it is not as time-consuming as other subjects. In addition, it is easy to master if you are a science graduate and have a good understanding of the theory behind it.

IFS(IFoS) Maths Optional Syllabus Paper-1

Section-A

(1) Linear Algebra:

Vector, space, linear dependence and independence, subspaces, bases, dimensions. Finite dimensional vector spaces. Matrices, Cayley-Hamilton theorem, Eigen values and Eigenvectors, matrix of linear transformation, row and column reduction, Echelon form, equivalence, congruence and similarity, reduction to canonical form, rank, orthogonal, symmetrical, skew symmetrical, unitary, Hermitian, skew-Hermitian forms their Eigen values. Orthogonal and unitary reduction of quadratic and Hermitian forms, positive definite quadratic forms.

(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 integrals, beta and gamma functions. Double and triple integrals (evaluation techniques only). Areas, surface and volumes, center of gravity.

(3) Analytic Geometry:

Cartesian and polar coordinates in two and three dimensions, second degree equations in two and three dimensions, 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.

Section-B

(4) Ordinary Differential Equations:

Formulation of differential equations, order and degree, equations of first order and first degree, integrating factor, equations of first order but not of first degree, Clairaut’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) Dynamics, Statics and Hydrostatics:

(i) 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.

(ii) 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.

(iii) Hydro Statics: Pressure of heavy fluids, equilibrium of fluids under given system of forces Bernoulli’s equation, centre of pressure, thrust on curved surfaces, equilibrium of floating bodies, stability of equilibrium, metacentre, pressure of gases.

(6) 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 interpretations. 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.

IFS(IFoS) Maths Optional Syllabus Paper-2

Section-A

(1) Algebra:

Groups, subgroups, normal subgroups, homomorphism of groups quotient groups basic isomorphism theorems, Sylow’s group, permutation groups, Cayley theorem. Rings and ideals, principal ideal domains, unique factorization domains and Euclidean domains. Field extensions, finite fields.

(2) Real Analysis:

Real number system, ordered sets, bounds, ordered field, real number system as an ordered field with least upper bound property, Cauchy sequence, completeness, Continuity and uniform continuity of functions, properties of continuous functions on compact sets. Riemann integral, improper integrals, absolute and conditional convergence of series of real and complex terms, rearrangement of series. Uniform convergence, continuity, differentiability and integrability for sequences and series of functions. Differentiation of functions of several variables, change in the order of partial derivatives, implicit function theorem, maxima and minima. Multiple integrals.

(3) Complex Analysis:

Analytic function, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, power series, Taylor’s series, Laurent’s Series, Singularities, Cauchy’s residue theorem, contour integration. Conformal mapping, bilinear transformations.

(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. Travelling salesman problems.

Section-B

(5)Partial differential equations:

Curves and surfaces in three dimensions, formulation of partial differential equations, solutions of equations of type dx/p=dy/q=dz/r; orthogonal trajectories, Pfaffian differential equations; partial differential equations of the first order, solution by Cauchy’s method of characteristics; Charpit’s method of solutions, linear partial differential equations of the second order with constant coefficients, equations of vibrating string, heat equation, Laplace equation.

(6) Numerical Analysis and Computer programming:

(i) 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) methods, Gauss-Seidel(iterative) method. Newton’s (Forward and backward) and Lagrange’s method of interpolation.

(ii) Numerical integration: Simpson’s one-third rule, trapezoidal rule, Gaussian quadrature formula.

(iii) Numerical solution of ordinary differential equations: Euler and Runge Kutta-methods.

(iv) Computer Programming: Storage of numbers in Computers, bits, bytes and words, binary system. arithmetic and logical operations on numbers. Bitwise operations. AND, OR, XOR, NOT, and shift/rotate operators. Octal and Hexadecimal Systems. Conversion to and from decimal Systems. Representation of unsigned integers, signed integers and reals, double precision reals and long integers. Algorithms and flow charts for solving numerical analysis problems. Developing simple programs in Basic for problems involving techniques covered in the numerical analysis.

(7) Mechanics and Fluid Dynamics:

(i) Mechanics: Generalized coordinates, constraints, holonomic and non-holonomic, systems. D’Alembert’s principle and Lagrange’ equations, Hamilton equations, moment of inertia, motion of rigid bodies in two dimensions.

(ii) Fluid Dynamics: Equation of continuity, Euler’s equation of motion for in viscid flow, stream-lines, path of a particle, potential flow, two-dimensional and axisymmetric motion, sources and sinks, vortex motion, flow past a cylinder and a sphere, method of images. Navier-Stokes equation for a viscous fluid.