Mathematics Optional Test Series for UPSC Mains

Choose your plan

Everything is delivered and submitted on WhatsApp. Evaluated reports and analytics appear on your dashboard.

Test series

Mathematics test series

12 sectional + 14 full-length tests

₹9,999

Expert evaluation within 2 hours
Model answers built on your writing style
Your score vs UPSC weightage graphs
12 years of topic and subtopic analytics
Personalise your test dates after joining

Daily practice

One test a day (3 questions), on the days you choose

₹6,000 6 months

Plan your topic priority and test dates
Same-day evaluation and feedback
Model solutions for every test
Delivered and submitted on WhatsApp

Topic-wise PYQs

Last 10 years, organised by topic

₹250

Average yearly weightage per topic
UPSC focus areas highlighted
Quick reference for revision

PYQ solutions

2014-2024 model solutions, topic-wise

₹1,250

Detailed, easy-to-follow solutions
Learn how strong answers are structured
Built to lift your score toward 330+

Test schedule

12 sectional tests followed by 14 full-length mocks.

Test	Type	Topic	Date
1	Sectional – Paper I	Linear Algebra	10-Oct
2	Sectional – Paper I	Calculus	04-Oct
3	Sectional – Paper I	Analytic Geometry	17-Oct
4	Sectional – Paper I	Ordinary Differential Equations	11-Oct
5	Sectional – Paper I	Dynamics and Statics	24-Oct
6	Sectional – Paper I	Vector Analysis	18-Oct
7	Sectional – Paper II	Modern Algebra	31-Oct
8	Sectional – Paper II	Real Analysis	25-Oct
9	Sectional – Paper II	Complex Analysis & Linear Programming	07-Nov
10	Sectional – Paper II	Partial Differential Equations	01-Nov
11	Sectional – Paper II	Numerical Analysis and Computer Programming	14-Nov
12	Sectional – Paper II	Mechanics and Fluid Dynamics	08-Nov
13	Full-Length Test	Paper I – Full Syllabus	21-Nov
14	Full-Length Test	Paper II – Full Syllabus	15-Nov
15	Full-Length Test	Paper I – Full Syllabus	28-Nov
16	Full-Length Test	Paper II – Full Syllabus	22-Nov
17	Full-Length Test	Paper I – Full Syllabus	05-Dec
18	Full-Length Test	Paper II – Full Syllabus	29-Nov
19	Full-Length Test	Paper I – Full Syllabus	12-Dec
20	Full-Length Test	Paper II – Full Syllabus	06-Dec
21	Full-Length Test	Paper I – Full Syllabus	19-Dec
22	Full-Length Test	Paper II – Full Syllabus	13-Dec
23	Full-Length Test	Paper I – Full Syllabus	26-Dec
24	Full-Length Test	Paper II – Full Syllabus	20-Dec
25	Full-Length Test	Paper I – Full Syllabus	02-Jan
26	Full-Length Test	Paper II – Full Syllabus	27-Dec

Dates don't suit you? You can personalise your test schedule after joining.

Mathematics Optional syllabus

The UPSC syllabus this test series covers, topic by topic.

Paper-I

Linear Algebra: Vector spaces over R and C linear dependence and independence subspaces bases dimension Linear transformations rank and nullity Algebra of matrices including row or column reduction echelon form congruences similarity Rank and inverse of a matrix Solving linear systems Eigenvalues and eigenvectors characteristic polynomial Cayley–Hamilton theorem Properties of symmetric skew symmetric Hermitian skew Hermitian orthogonal and unitary matrices and their eigenvalues
Calculus: Real functions limits continuity differentiability Mean value theorem Taylor theorem with remainder Indeterminate forms Maxima and minima Asymptotes Curve tracing Multivariable functions limits continuity partial derivatives maxima and minima including Lagrange multipliers Jacobian Integration Riemann definite integrals indefinite integrals Improper integrals Double and triple integrals evaluation techniques only Area surface and volume calculations
Analytic Geometry: Cartesian and polar coordinates in three dimensions Quadratic surfaces in three variables reduction to canonical forms Straight lines Shortest distance between two skew lines Planes spheres cones cylinders paraboloids ellipsoids hyperboloids one or two sheets and their properties
Ordinary Differential Equations: Formulation of differential equations First order equations integrating factors orthogonal trajectories Clairaut equation singular solutions Second and higher order linear ODEs with constant coefficients complementary function particular integrals Variable coefficient cases Euler–Cauchy Variation of parameters Laplace and inverse Laplace transforms and applications to initial value problems for second 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 laws Kepler laws central force orbits Equilibrium of particle systems work potential energy friction common catenary virtual work principle stability of equilibrium three dimensional force equilibrium
Vector Analysis: Scalar and vector fields differentiation of vector fields with respect to a scalar variable Gradient divergence curl in Cartesian and cylindrical coordinates Higher order derivatives Vector identities and equations Geometry applications curves in space curvature and torsion Serret–Frenet formulae Integral theorems Gauss theorem Stokes theorem Green identities

Paper-II

Modern Algebra: Groups subgroups normal subgroups homomorphisms quotient groups First Second Third isomorphism theorems Sylow theorems and applications Rings ideals integral domains fields Ring homomorphisms Field extensions Finite fields
Real Analysis: The real number system as a complete ordered field Sequences limits Cauchy sequences completeness Series convergence tests absolute or conditional convergence Continuity and uniform continuity Properties of continuous functions on compact sets Riemann integration improper integrals Fundamental theorems of calculus Uniform convergence of sequences and series of functions continuity differentiability integrability Multivariable partial derivatives maxima and minima
Complex Analysis: Analytic functions and the Cauchy–Riemann equations Cauchy integral theorem and formula Taylor and Laurent series Residue theorem and applications
Linear Programming: Linear programming problems basic concepts Graphical method Simplex method Duality and its interpretation
Partial Differential Equations: Geometry of curves and surfaces in three dimensions Formulation of PDEs First order PDEs Second order linear PDEs with constant coefficients Standard PDEs vibrating string heat equation Laplace equation
Numerical Analysis and Computer Programming: Numerical methods root finding bisection regula falsi Newton Raphson Linear systems by Gaussian elimination or Gauss Jordan methods Numerical interpolation Newton forward and backward Lagrange Numerical integration trapezoidal Simpson rules Numerical ODE solution Euler Runge Kutta Programming in C data types operators input output control structures functions
Mechanics and Fluid Dynamics: Generalized coordinates and Lagrange equations with simple applications Central force motion Rigid body motion in two dimensions Mechanics of particle systems equations of motion for mass moments Fluid mechanics continuity equation Euler equation for perfect fluids streamlines particle paths potential flow two dimensional and axisymmetric flows sources sinks vortex motion flow past a circular cylinder method of images

Free 2014 PYQ solutions

Expert-solved previous year questions. Read them free to see our answer quality.

Linear AlgebraRead free Real AnalysisRead free

Want every year? Get topic-wise solutions for 2014-2024.

UPSC weightage analytics

Where UPSC has awarded marks, topic by topic. Focus on what scores, and compare your test results once you join.

Mathematics Optional test series