Madhavi Gaur
August 16, 2023 03:00
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UPSC Maths Optional Syllabus covers linear algebra, geometry, differential equations, algebra, and real analysis. For the UPSC Math Optional exam, candidates need to attempt two papers based on this syllabus. Check the latest IAS Maths Optional Syllabus here.
UPSC Maths Optional Syllabus: The mathematics optional syllabus consists of two papers, Paper I and Paper II, each carrying a weightage of 250 marks. Both papers are divided into various sections, and candidates are required to choose topics from each section. Let’s delve into the details of each paper.
Selecting an optional subject can be a significant decision. The combined score of both optional papers contributes 500 out of 1750 marks in the UPSC Main Examination. Therefore, it’s crucial to choose wisely. As a general guideline, opt for a subject that genuinely interests you. Mathematics stands as a well-liked optional subject for the UPSC Main Exam. However, it’s recommended mainly for candidates who pursued Mathematics in their graduation.
UPSC Maths Syllabus: The UPSC Civil Services Examination is one of the most prestigious and competitive exams in India. Among the various optional subjects offered, mathematics has gained popularity among aspirants with a strong aptitude for numbers and problem-solving. The UPSC Maths optional syllabus is designed to test the analytical and logical thinking abilities of aspirants while providing them with an opportunity to showcase their mathematical prowess. In this article, we will explore the mathematics optional syllabus for the UPSC Civil Services Examination, its structure, and the key topics covered.
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Check the Math Optional Syllabus Paper 1 in given below table:
| UPSC Maths Optional Syllabus For Paper I | |
|---|---|
| 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, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues. |
| 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. |
| 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. |
| 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 equation; Determination 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. |
| 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. |
| 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 identities. |
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Below is the UPSC Math Optional Syllabus for Paper II:
| UPSC Maths Optional Syllabus For Paper 2 | |
|---|---|
| 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. |
| 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. |
| 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. |
| 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. |
| 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. |
| 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. Algorithms and flow charts for solving numerical analysis problems. |
| 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. |
The UPSC Mathematics optional syllabus is available in PDF format. This syllabus outlines the topics and content covered in the Mathematics optional subject for the UPSC examination. Aspirants preparing for UPSC Exam can check the UPSC Mathematics Optional Syllabus PDF given below:
There are several books that are widely recommended for UPSC Mathematics optional preparation. Some popular ones include:
To effectively prepare for the mathematics optional syllabus, aspirants should consider the following tips:
1. Understand the Syllabus: Familiarize yourself with the entire syllabus and identify the areas where you need more focus. This will help you plan your preparation strategy accordingly.
2. Study from Standard Books: Refer to standard textbooks recommended by experts and experienced candidates. Some popular books for mathematics optional include “Higher Algebra” by Hall and Knight, “Introduction to the Theory of Numbers” by Niven and Zuckerman, and “Real Analysis” by Royden.
3. Solve Previous Year Question Papers: Practice solving previous years’ question papers to get an idea of the exam pattern and the type of questions asked. This will also help in time management during the actual examination.
4. Seek Guidance: If needed, enroll in coaching institutes or seek guidance from experienced mentors who can provide valuable insights and strategies for tackling the UPSC Maths optional syllabus.
5. Practice Regularly: Dedicate ample time to practice solving mathematical problems. This will not only improve your speed and accuracy but also enhance your understanding of various concepts.
6. Develop Short Notes: Prepare concise notes on important theorems, formulas, and concepts. These notes will serve as quick revision material during the last days before the examination.
7. Test Yourself: Take regular mock tests to assess your preparation level and identify areas that need improvement. Analyze your performance and work on your weak points.
Regular practice of mathematical problems and equations will help in building a strong foundation and enhance problem-solving skills.
Choosing Mathematics as an optional subject in the UPSC (Union Public Service Commission) civil services examination requires careful consideration. Here are some important factors to keep in mind before selecting Mathematics as your optional subject:
The most crucial aspect is your interest and aptitude for mathematics. Do you enjoy studying and solving complex mathematical problems? Do you have a strong foundation in mathematics? Assessing your passion and proficiency in the subject is essential.
Mathematics can be a high-scoring subject in the UPSC examination if you have a solid understanding and problem-solving skills. It is important to evaluate your ability to consistently perform well in mathematics and your comfort level in handling the subject’s intricacies.
If you have a background in mathematics, such as a degree in mathematics or engineering, it may give you an advantage in understanding and tackling advanced mathematical concepts. Additionally, consider the time required for preparation as mathematics might demand rigorous practice and continuous learning.
Analyze the UPSC Mathematics syllabus thoroughly. Check whether it overlaps with other subjects in the UPSC syllabus or with your chosen optional subject. Overlapping can be beneficial as it reduces the overall workload and helps in integrated preparation.
Availability of good study material and guidance plays a crucial role in your preparation. Ensure that you have access to quality books, previous years’ question papers, and guidance from mentors, coaching institutes, or online platforms.
Research the success rate of candidates who have chosen mathematics as an optional subject in the UPSC examination. Analyze the performance trends, success stories, and testimonials of candidates to understand the subject’s viability as an optional.
Review the past trends and marks distribution of the mathematics optional subject. Examine the scoring pattern, average marks obtained, and the probability of scoring well in comparison to other optional subjects.
Assess your own strengths and weaknesses. Consider whether mathematics aligns with your strengths and how it complements your overall preparation strategy. Evaluate whether it balances out any potential weaknesses you may have in other areas.
Evaluate your time management skills and your ability to handle the subject within the time constraints of the UPSC examination. Consider how well you can incorporate mathematics preparation into your overall exam strategy.
While the primary consideration should be your interest and aptitude, it is also worth reflecting on the potential utility of mathematics in your chosen career path beyond the UPSC examination. Reflect on how mathematics may contribute to your future goals and aspirations.
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