**How to for prepare mathematics’ optional?**

- Practice

Practice as much as you can. Take a particular concept of the subject and practice different set of questions of the concept. This will make you concepts clearer and better. For every concept try out different possible solutions through which the question can be solved and rectify every kind of doubt in a particular question.

- Try to avoid small and silly mistakes.

Sometimes in exams students are so busy that they tend to commit many silly mistakes. Maths is all about numbers, so while solving questions many students note down wrong figures of the questions which lead to the wrong answer and hence students loose marks in the paper. Try to avoid doing such mistakes in the exam and read every question carefully before giving any kind of answer.

- Answer in a proper manner

Many of the students know the answer during the time of exam. But still are not able to score good marks in the exam. This is because they don’t answer the questions in a proper manner due to which marks are deducted. So to score good marks, it’s better to answer each question and present it in a proper manner.

- Don’t panic and be relaxed

The best thing to do during preparation is to be relaxed and prepare with peace in mind. Being panic and stressed will not help in scoring good marks. If you prepare with stress and panic then it will hinder your preparation and you may tend to lose marks in the exam.

So it’s better to stay calm and relaxed and preparing without any kind of stress. This will boost your score and result in better marks in the exam.

- Go through the syllabus

Before beginning the preparation go through the syllabus well. Understand the syllabus to know what actually is asked in the exams. If you the syllabus well then it gives a clear understanding to you as what could be asked in the exam. also it helps to know which all topics to be studied.

Marks distribution is another aspect about which a student should know well. Through this one can categorise topic which carries the highest weight age and the ones which doesn’t have high weight age. This helps in proper planning for the main exam.

- Choose right preparatory books

There are books available in abundance in the market. Theses bulk of books tends to confuse the candidates as which one to opt for better understanding. This creates chaos in the mind of the candidates and results in sheer wastage of time and energy.

So it’s always better to select some books out of many books available in the market and stick to them for the preparation. Clear your concepts and study in depth about a particular topic till the time you become well versed with it. Also don’t solely rely on one book try to explore other books as well if your topic is not cleared by one book. This doesn’t imply to refer 10 books just for one topic. Try to be selective in your choice when it comes to books and study material.

- Practice past year papers

Practicing past year paper is one of the most vital factors involved in the preparation. If you practice past year papers then you can an idea what exactly has been asked in previous exam. And what are the important topics through which most of the questions arises in the exam. This helps to plan your preparation in a better way and study properly. If you are well versed with previous exam papers then you get an insight what is most important from the exam point of view and accordingly you can solve and practice questions for the final examination

- Making notes

Most of the books available are too huge and thick that day before the exam a candidate cannot solely study from it. So it’s always better to make notes out of the available books and study materials.

Writing down 1000pages into a short summary of 100pages helps in a quick revision. This exercise of making notes also saves a lot of time and energy. At the end of the notes made by you is something which will help you a lot in exam preparation.

- Time management

Most of the students are unable to manage the time during the exam. This happens because many a times a candidate spends a lot of time in thinking about an answer of particular questions and this leads to a lot of wastage of time. Instead of this one needs to specify and allocate proper time to each and every question.

It will help to finish the paper on time. Also time management is important to avoid any last minute hassle. When less time is left and there are more number of questions left to be answer then many a times a candidate marks a wrong answer even if he/she knows the correct answer.

- Revision of important topics

It is very important to thoroughly study the whole syllabus twice and thrice. And then keep on revising it as many times as possible. Revision helps to absorb the things in a better way. It helps to keep whole of the syllabus in our brain for a longer duration.

If a student keeps on learning and doesn’t revise what he has studied then he tends to forget the thing on the day of the exam. This is because what he/studies didn’t ever got registered in his brain. Thus revision is the key factor to restore things in the brain

**About Mathematics Optional: **

Mathematics Optional is one of the Best Optional Subject in IAS, UPSC, IFS, IFoS, CSE-Civil Service Mains Examinations. It will give maximum scoring among rest of the Optionals in the IAS Mains Optional Exams.

**MATHEMATICS**

** PAPER I **

**(1) Linear Algebra: **Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions, Linear transformations, rank and nullity, a 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 a 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) 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, the 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 the first order and first degree, integrating factor; Orthogonal trajectory; Equations of first order but not of the first degree, Clairaut’s equation, singular solution. Second and higher-order linear equations with constant coefficients, complementary function, particular integral and general solution. Section order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using the 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, the equilibrium of forces in three dimensions.

**(6) VectorAnalysis: **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 identities.

**PAPER-II **

**(1) Algebra: **Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism 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 the least upper bound property; Sequences, the 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 a system of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss-Seidel (iterative) methods. Newton’s (forward and backwards) and interpolation, Lagrange’s interpolation. Numerical integration: Trapezoidal rule, Simpson’s rule, 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.

**(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, the path of a particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.