Maths Optional for UPSC Preparation Strategy

Choosing the right optional subject for UPSC CSE Mains is a crucial decision every UPSC aspirant must make. A general rule is that you should take the issue in which you have some academic background and a genuine interest. Along with this, other factors like – past performance of the subject in UPSC, the time frame required to prepare the issue, contribution of the subject towards the General Studies syllabus and coaching available in the subject should also be considered in deciding the Optional Subject. The same criteria apply to opting for Mathematics as an Optional Subject for UPSC CSE Mains.

An applicant must have studied maths during their undergraduate degree, first and foremost. Studying maths only until the 12th grade is insufficient for the UPSC Mains, regardless of how well one did.

Second, a candidate must genuinely enjoy the subject, as someone who doesn't have a passion for it will find the curriculum overwhelming. But a student passionate about maths will find it easy to develop and implement concepts. 

Everything you need to know about the mathematics option for UPSC Civil Services is covered in detail in this post.

Advantages of selecting mathematics as an optional subject for the UPSC Main Exams

1. High Scoring

Whether it is the UPSC exam or another exam, maths has always been an area that can help you score well. Students who choose maths as an optional subject have regularly done well and earned excellent grades over the past few years. In the UPSC, presentation style is almost as important as substance, particularly in maths. One may estimate the grade on such a question quite easily if a student learns how to give a solution in steps while emphasizing the key points. Since the questions are factual, not subjective or opinion-based, it is not the examiner's responsibility to determine whether the substance and presentation are appropriate.

2. little emphasis on memory

Of course, basic equations and essential steps in theorems must be remembered, but nothing compared to what students must memorize for prelims/mains/other optional topics. This makes it appealing to individuals who dislike cramming. Furthermore, the items to remember in maths are only about 10% of the complete syllabus, which one retains due to practice and utilization in most questions.

3. Static Syllabus

The Maths Optional for UPSC syllabus is the curriculum most universities and institutes in India follow. Most concepts are familiar to anyone with Maths as an undergraduate course, whether engineering students or B.Sc/B.Students. All that remains is concept development and careful revision. This differs from other optional disciplines, particularly for engineering students, who must learn the subject from the ground up. Also, because the curriculum is not tied to current events, once you finish it, you do not need to refresh your knowledge constantly; you only need to revise.

4. Less time to prepare the subject

Suppose a student devotes enough time and effort to developing strong mathematical basics and learning how to categorize/recognize problem types. In that case, they can effortlessly address most possible questions from that field. It is a fallacy that maths requires more time to cover. While the syllabus is extensive, due to the static structure of the subject, if one topic is entirely studied, one must make an effort to revise that area and not learn other stuff simply frequently. So, if carefully organized and managed, it can be completed in six months.

Tips to score high in Maths Optional

If your teachers in school and college frequently encouraged you to practise maths problems and prevent dumb mistakes to get good grades, they were perfectly correct. These are the fundamental keys to excelling in Mathematics. Here are some pointers to help you do well in Math Optional in the UPSC Mains Exam.

Practice more: Practicing makes you faster at problem-solving and improves your speed and accuracy, which will help you perform well in exams.

Avoid Making Silly Mistakes: Even one incorrect sign or number can cost money. Stay calm and concentrated when answering questions to prevent making such foolish mistakes. Again, a lot of practice will help you avoid such errors. Also, set aside a few minutes for revision before the exam ends so you can cross-check your answers and make corrections as needed.

Be Systematic: Presentation, coupled with the proper solution, is essential. So, put the solution cleanly and in sequential order. Take your time with any critical steps, or write the answer clumsily.

Refrain from cramming maths: Because maths is a logical subject, never try to fill in solutions or theorems. Comprehend the logical flow and strengthen your concepts; this will help you tackle all types of problems since you can apply logic in the exam and will not neglect any critical steps of the solution.

Formula Sheet: This is something that every applicant who has Maths as an optional subject should have on hand. Formulae are the foundation of mathematics. Prepare a formula sheet with all the formulae on it so you can easily update it anytime and from any location.

Syllabus

The syllabus of both the papers of Maths optional as per UPSC Official Notification is as follows:

Mathematics Paper 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 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 integrals; 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 functions, particular integral and general solutions. Section order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using a 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 identities.

Mathematics 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 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 the system of linear equations by Gaussian Elimination and Gauss-Jorden (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.

Bottomline

If you have a strong interest in and aptitude for maths and are prepared to put in the time and effort necessary to master it, maths can be an excellent alternative for UPSC. If maths is not your most vital subject, there are other optional subjects you can consider as it is not required for UPSC.