Mathematics Optional: answer writing, test series and mentorship
UPSC-style normalization across all 48 optionals26 tests (12 sectional + 14 full-length), each evaluated by experts within 2 hours, with model answers written around your own style. Every test gets your All-India rank and your Mathematics rank.
Papers set on UPSC's own pattern, a verified model answer for every question, every copy evaluated in about 2 hours, and a plan made for your hours.
Choose how you prepare for Mathematics
Three programmes, one standard: every paper set on UPSC's own pattern, every answer evaluated in about 2 hours, a verified model answer for every question.
Daily answer writing
A 50-mark paper every weekday, module by module through the whole syllabus. Write 35 minutes a day and never fall out of practice.
- Every module tested, the heavy ones again and again
- Real UPSC questions mixed with new ones of the same standard
- Evaluated in about 2 hours, with a model answer
- Your practice days; papers held when you fall behind
Mentorship
Everything you need to clear Mathematics: study material, a personal plan made for your hours, a test for every module, weekly and topic tests, full UPSC mocks and a mentor.
- Complete, comprehensive study material for every module, designed to score high in Mains: foundation, full notes, quick revision
- A personal plan for your hours and exam date, re-dated when you slip
- Read first, then the module's test; weekly, topic and mock tests in UPSC form
- Every copy reviewed; mentor calls and progress reports
Test series
Topic tests and full-length mocks, each a copy of a real UPSC paper's structure, with an All India Rank after every test.
- Mocks exactly in UPSC form: 8 questions, Q1 and Q5 compulsory, attempt 5
- Every question written for its probability of coming in 2027, printed beside it
- All India Rank and Mathematics rank after every test
- Change your test dates after joining
PYQ Topicwise · ₹750
Every Mathematics question UPSC has asked (2014–2026), arranged topic by topic, with how many marks each topic has carried year after year.
PYQ Solutions · ₹1,750
A model answer to every previous year question (2014–2026), module by module: worked step by step, with the figures a topper draws. Get one module free on WhatsApp.
GS + Essay + Mathematics: opening soon
The same three programmes for GS + Essay + Mathematics. Prices below; tell us on WhatsApp and we will message you the day it opens.
| Programme | 4 months | 8 months | 12 months |
|---|---|---|---|
| Daily answer writing | ₹17,999 | ₹27,999 | ₹34,999 |
| Mentorship | ₹49,999 | ₹79,999 | ₹99,999 |
| Test series | Lite ₹17,999 · Full ₹29,999 | ||
GS + Essay: opening soon
The same three programmes for GS + Essay. Prices below; tell us on WhatsApp and we will message you the day it opens.
| Programme | 4 months | 8 months | 12 months |
|---|---|---|---|
| Daily answer writing | ₹11,999 | ₹19,999 | ₹24,999 |
| Mentorship | ₹34,999 | ₹54,999 | ₹69,999 |
| Test series | Lite ₹12,999 · Full ₹21,999 | ||
Count the tests, not just the price
A test series is usually 8 to 16 tests. Here every rupee buys far more evaluated writing.
Up to 215 papers, each read, marked and returned with a model answer in about 2 hours.
26 tests in the full series: topic tests and mocks that copy real UPSC papers.
Up to 137 tests, with the study material, your personal plan and your mentor at no extra cost.
| What you get | Daily answer writing | Test series | Mentorship |
|---|---|---|---|
| Every copy evaluated in about 2 hours - marks, comments on each answer and a verified model answer | ✓ | ✓ | ✓ |
| A test on every module of the syllabus - nothing left untested | ✓ | — | ✓ |
| Weekly and topic tests that build the speed and stamina of the real exam | — | ✓ | ✓ |
| Full mocks set exactly like the UPSC paper, every question written for its probability of coming in 2027 | — | ✓ | ✓ |
| All India Rank after every test - know exactly where you stand | — | ✓ | ✓ |
| Complete, comprehensive study material, designed to score high in UPSC Mains - foundation, full notes and quick revision for every module | — | — | ✓ |
| Your own day-by-day study plan, built around your hours - it re-plans itself when you fall behind | — | — | ✓ |
| A mentor beside you - every copy reviewed, mentor calls and progress reports | — | — | ✓ |
Who sets the papers? How good are they?
The question every serious aspirant asks. Here is exactly how every Mathematics paper is made, and you can check one before you pay.
Built on UPSC's own paper
Each mock copies the structure of a recent UPSC paper: its sections, its questions and parts, its marks. Q1 and Q5 with five 10-mark parts, attempt five, at least one from each section, exactly as in the exam hall.
Weighted by 12 years of UPSC marks
Every topic and module gets questions in proportion to the marks UPSC has given it. You practise most what UPSC asks most.
Real PYQs and new questions of the same standard
Previous-year questions sit beside new questions written to the UPSC standard, each checked against the syllabus and given its chance of coming in 2027.
A verified model answer for every question
Worked step by step, with the figures a topper draws, checked before it is printed. You get it with your evaluated copy.
Reviewed and approved by our faculty
No paper reaches you before it is reviewed and approved. The approval is printed on every page of the paper.
Evaluated in about 2 hours
Your copy comes back marked question by question, with what you missed and how to write it better, while the paper is still fresh in your mind.
See it before you join
Judge the papers, the model answers and the plan yourself.
Write a free test
Get a real Mathematics paper on WhatsApp, write it, send it back and get it evaluated free. We also send you a solved paper and the sample plans.
- Open WhatsApp
- Tap the camera
- Scan this code
- Tap Send
A solved paper
One of our daily papers with its full model answers: see the standard of the questions and how every answer is written.
- Open WhatsApp
- Tap the camera
- Scan this code
- Tap Send
A sample personal plan
Day by day from the coming Monday: what to read, which test comes when, the revision weeks and the mocks, made for 6 hours a day. Preview the first weeks here; get the whole plan on WhatsApp.
- Open WhatsApp
- Tap the camera
- Scan this code
- Tap Send
Questions aspirants ask us
Who sets the question papers?
Every paper is built on the structure of a real UPSC paper and on 12 years of UPSC's marks for each topic, mixes real PYQs with new questions of the same standard, and is reviewed and approved by our faculty before it reaches you. Download the solved sample above and judge it yourself.
Who writes the model answers?
Every question has a model answer worked out step by step, with figures where a topper would draw one, checked before it is printed. You receive it with your evaluated copy.
How fast is the evaluation, and what do I get back?
About 2 hours after you submit. Your copy comes back marked question by question with comments, your marks, the model answer and, in the test series and mocks, your All India Rank.
How do I write and submit?
Print the paper or write on plain A4 sheets with margins, scan the pages into one PDF named with the test code, and send it on WhatsApp or upload it in the IndianCivils app.
Daily answer writing, test series or mentorship: which one is for me?
Daily answer writing keeps you writing every day through the syllabus. The test series checks you in exam conditions with topic tests and full mocks. Mentorship does both and adds the material, a personal plan and a mentor: the complete preparation for the optional.
What if I fall behind my plan?
In the mentorship your plan is re-dated around your real pace. In daily answer writing new papers wait while you have several unwritten ones, so nothing piles up.
Test series timetable
12 topic tests, then 14 full UPSC mocks, at the weekend, from your batch's first week. A free weekend follows every mock, to work on its evaluation.
| Test | Type | What it covers | When |
|---|---|---|---|
| 1 | Topic test | Paper I - Topic test 1: Linear Algebra | Week 1 · Sat (FN) |
| 2 | Topic test | Paper II - Topic test 2: Modern Algebra | Week 2 · Sat (FN) |
| 3 | Topic test | Paper I - Topic test 3: Calculus | Week 3 · Sat (FN) |
| 4 | Topic test | Paper II - Topic test 4: Real Analysis | Week 4 · Sat (FN) |
| 5 | Topic test | Paper I - Topic test 5: Analytic Geometry | Week 5 · Sat (FN) |
| 6 | Topic test | Paper II - Topic test 6: Complex Analysis; Linear Programming | Week 6 · Sat (FN) |
| 7 | Topic test | Paper I - Topic test 7: Ordinary Differential Equations | Week 7 · Sat (FN) |
| 8 | Topic test | Paper II - Topic test 8: Partial Differential Equations | Week 8 · Sat (FN) |
| 9 | Topic test | Paper I - Topic test 9: Dynamics and Statics | Week 9 · Sat (FN) |
| 10 | Topic test | Paper II - Topic test 10: Numerical Analysis and Computer Programming | Week 10 · Sat (FN) |
| 11 | Topic test | Paper I - Topic test 11: Vector Analysis | Week 11 · Sat (FN) |
| 12 | Topic test | Paper II - Topic test 12: Mechanics and Fluid Dynamics | Week 12 · Sat (FN) |
| 13 | Full mock | Paper I - Full mock 1 | Week 13 · Sat (FN) |
| 14 | Full mock | Paper II - Full mock 1 | Week 13 · Sat (AN) |
| 15 | Full mock | Paper I - Full mock 2 | Week 15 · Sat (FN) |
| 16 | Full mock | Paper II - Full mock 2 | Week 15 · Sat (AN) |
| 17 | Full mock | Paper I - Full mock 3 | Week 17 · Sat (FN) |
| 18 | Full mock | Paper II - Full mock 3 | Week 17 · Sat (AN) |
| 19 | Full mock | Paper I - Full mock 4 | Week 19 · Sat (FN) |
| 20 | Full mock | Paper II - Full mock 4 | Week 19 · Sat (AN) |
| 21 | Full mock | Paper I - Full mock 5 | Week 21 · Sat (FN) |
| 22 | Full mock | Paper II - Full mock 5 | Week 21 · Sat (AN) |
| 23 | Full mock | Paper I - Full mock 6 | Week 23 · Sat (FN) |
| 24 | Full mock | Paper II - Full mock 6 | Week 23 · Sat (AN) |
| 25 | Full mock | Paper I - Full mock 7 | Week 25 · Sat (FN) |
| 26 | Full mock | Paper II - Full mock 7 | Week 25 · Sat (AN) |
The Lite series takes 6 of these topic tests and 3 mock rounds. Dates don't suit you? Change your test dates after joining.
Mathematics Optional syllabus
The UPSC syllabus this test series covers, topic by topic.
Mathematics - PAPER-I · 6 topics, 37 modules
- Linear Algebra
- Vector Spaces: Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions
Linear Transformations: Linear transformations, rank and nullity, matrix of a linear transformation
Matrix Algebra: Algebra of Matrices; Inverse of a matrix
Matrix Reduction: Row and column reduction, Echelon form, congruence’s and similarity; Rank of a matrix
Linear Equations: Solution of system of linear equations
Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem
Special Matrices: Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues - Calculus
- Limits and Continuity: Real numbers, functions of a real variable, limits, continuity
Differential Calculus: differentiability, mean-value theorem, Taylor’s theorem with remainders, indeterminate forms
Extrema and Curve Tracing: maxima and minima, asymptotes; Curve tracing
Multivariable Calculus: Functions of two or three variables; Limits, continuity, partial derivatives, maxima and minima, Lagrange’s method of multipliers, Jacobian
Integral Calculus: Riemann’s definition of definite integrals; Indefinite integrals
Improper Integrals: Infinite and improper integral
Multiple Integrals: Double and triple integrals (evaluation techniques only); Areas, surface and volumes - Analytic Geometry
- Coordinates and Canonical Forms: Cartesian and polar coordinates in three dimensions, second degree equations in three variables, reduction to Canonical forms
Lines and Planes: straight lines, shortest distance between two skew lines, Plane
Spheres: sphere
Cones and Cylinders: cone, cylinder
Quadric Surfaces: paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties - Ordinary Differential Equations
- First-Order Equations: Formulation of differential equations; Equations of first order and first degree, integrating factor; Orthogonal trajectory
First Order Higher Degree: Equations of first order but not of first degree, Clairaut’s equation, singular solution
Constant-Coefficient Equations: Second and higher order liner equations with constant coefficients, complementary function, particular integral and general solution
Variable-Coefficient Equations: Section order linear equations with variable coefficients, Euler-Cauchy equation
Variation of Parameters: Determination of complete solution when one solution is known using method of variation of parameters
Laplace Transforms: 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
- Particle Motion: Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; Constrained motion
Work and Energy: Work and energy, conservation of energy
Central Force Motion: Kepler’s laws, orbits under central forces
Static Equilibrium: Equilibrium of a system of particles; Stability of equilibrium, equilibrium of forces in three dimensions
Friction: friction
Catenary: Common catenary
Virtual Work: Work and potential energy; Principle of virtual work - Vector Analysis
- Vector Fields: Scalar and vector fields, differentiation of vector field of a scalar variable
Vector Differential Operators: Gradient, divergence and curl in cartesian and cylindrical coordinates; Higher order derivatives
Vector Identities: Vector identities and vector equation
Space Curves: Application to geometry : Curves in space, curvature and torsion; Serret-Furenet's formulae
Integral Theorems: Gauss and Stokes’ theorems, Green's identities
Mathematics - PAPER-II · 7 topics, 41 modules
- Modern Algebra
- Groups and Subgroups: Groups, subgroups, cyclic groups
Cosets and Quotient Groups: cosets, Lagrange’s Theorem, normal subgroups, quotient groups
Group Homomorphisms: homomorphism of groups, basic isomorphism theorems
Permutation Groups: permutation groups, Cayley’s theorem
Rings and Ideals: Rings, subrings and ideals, homomorphisms of rings
Integral Domains: Integral domains, principal ideal domains, Euclidean domains and unique factorization domains
Fields: Fields, quotient fields - Real Analysis
- Real Number System: Real number system as an ordered field with least upper bound property
Sequences and Limits: Sequences, limit of a sequence, Cauchy sequence, completeness of real line
Infinite Series: Series and its convergence, absolute and conditional convergence of series of real and complex terms, rearrangement of series
Continuous Functions: Continuity and uniform continuity of functions, properties of continuous functions on compact sets
Riemann Integration: Riemann integral, improper integrals; Fundamental theorems of integral calculus
Uniform Convergence: Uniform convergence, continuity, differentiability and integrability for sequences and series of functions
Multivariable Analysis: Partial derivatives of functions of several (two or three) variables, maxima and minima - Complex Analysis
- Analytic Functions: Analytic function, Cauchy-Riemann equations
Cauchy Theorem and Formula: Cauchy's theorem, Cauchy's integral formula
Power and Taylor Series: power series, representation of an analytic function, Taylor’s series
Singularities and Laurent Series: Singularities; Laurent’s series
Residues and Contours: Cauchy’s residue theorem; Contour integration - Linear Programming
- Basic and Feasible Solutions: Linear programming problems, basic solution, basic feasible solution and optimal solution
Graphical and Simplex Methods: Graphical method and simplex method of solutions
Duality: Duality
Transportation and Assignment: Transportation and assignment problems - Partial Differential Equations
- Formulation of PDEs: Family of surfaces in three dimensions and formulation of partial differential equations
First-Order PDEs: Solution of quasilinear partial differential equations of the first order, Cauchy’s method of characteristics
Second-Order Linear PDEs: Linear partial differential equations of the second order with constant coefficients, canonical form
Wave Heat Laplace Equations: Equation of a vibrating string, heat equation, Laplace equation and their solutions - Numerical Analysis and Computer Programming
- Equations and Linear Systems: 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
Interpolation: Newton’s (forward and backward) and interpolation, Lagrange’s interpolation
Numerical Integration: Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula
Numerical ODEs: Numerical solution of ordinary differential equations : Eular and Runga Kutta methods
Number Systems and Arithmetic: Computer Programming : Binary system; Arithmetic and logical operations on numbers; Octal and Hexadecimal Systems; Conversion to and from decimal Systems; Algebra of binary numbers; Representation of unsigned integers, signed integers and reals, double precision reals and long integers
Boolean Logic and Gates: Elements of computer systems and concept of memory; Basic logic gates and truth tables, Boolean algebra, normal forms
Algorithms and Flowcharts: Algorithms and flow charts for solving numerical analysis problems - Mechanics and Fluid Dynamics
- Lagrangian Mechanics: Generalised coordinates; D’Alembert’s principle and Lagrange’s equations
Hamiltonian Mechanics: Hamilton equations
Rigid Body Dynamics: Moment of inertia; Motion of rigid bodies in two dimensions
Fluid Kinematics: Equation of continuity; Stream-lines, path of a particle
Inviscid Flow: Euler’s equation of motion for inviscid flow
Potential Flow: Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion
Viscous Flow: Navier-Stokes equation for a viscous fluid
Free PYQ solutions
One module's previous year questions with full model answers, worked step by step with the figures a topper draws. We send it to you on WhatsApp.
- Open WhatsApp
- Tap the camera
- Scan this code
- Tap Send
Every module, UPSC 2014–2026, topic by topic:
UPSC weightage analytics
Where UPSC has awarded marks, topic by topic. Focus on what scores, and compare your test results once you join.
Questions about Mathematics?
Message us on WhatsApp or Telegram at +91 90000 18827. We're always live.
Chat on WhatsApp
Telegram