Pi Craft › UPSC Maths PYQ Analysis
854Questions analysed
11,200Marks mapped to topics
14Years, 2013 to 2026
10%Biggest scorer: Ordinary Differential Equations
Where the 11,200 marks went
Share of all marks, by subject
Ordinary Differential Equations
9.6%
Calculus
8.8%
Mechanics and Fluid Dynamics
8.8%
Numerical Analysis and Computer Programming
8.5%
Linear Algebra
8.5%
Dynamics and Statics
8.4%
Partial Differential Equations
7.7%
Analytic Geometry
7.7%
Real Analysis
7.3%
Vector Analysis
7.0%
Algebra
6.1%
Complex Analysis
5.9%
Linear Programming
5.6%
How the marks are split
By question size
- 10-mark 31% of marks
- 15-mark 39% of marks
- 20-mark 20% of marks
- Other 9% of marks
Paper I · 50%Paper II · 50%
Year-by-year heat map
Marks per subject in each exam year. Darker means more marks.
1314151617181920212223242526Ordinary Differential Equations701007080871067070707070707070Calculus6070707560837075707070707070Mechanics and Fluid Dynamics7050757060658060757575757575Numerical Analysis and Computer Programming6095606070757570656565656565Linear Algebra7070687070457070707070707070Dynamics and Statics7070847560227070707070757070Partial Differential Equations7055657070604570606060606060Analytic Geometry7060625570726055606060606060Real Analysis5755556555558060556040606060Vector Analysis6030464553726060606060556060Algebra7355506045454050404050405050Complex Analysis2540454555404545655560554545Linear Programming4550503045603545404550454545
Trend of the six biggest subjects
Marks per year, each on the same scale (2013 to 2026). Big dot marks the peak year.
Ordinary Differential Equationsavg 77 marks a year
Peak 106 in 2018Latest 70
Calculusavg 70 marks a year
Peak 83 in 2018Latest 70
Mechanics and Fluid Dynamicsavg 70 marks a year
Peak 80 in 2019Latest 75
Numerical Analysis and Computer Programmingavg 68 marks a year
Peak 95 in 2014Latest 65
Linear Algebraavg 68 marks a year
Peak 70 in 2013Latest 70
Dynamics and Staticsavg 68 marks a year
Peak 84 in 2015Latest 70
Top topics by marks
The eight topics that carried the most marks
| Topic | Questions | Marks |
|---|---|---|
| Linear Transformations, Rank and Nullity, Matrix of a Linear TransformationLinear Algebra | 21 | 254 |
| Double and Triple Integrals; Areas, Surfaces and VolumesCalculus | 18 | 235 |
| The SphereAnalytic Geometry | 18 | 225 |
| Second Order Variable Coefficients: One Solution Known, Variation of ParametersOrdinary Differential Equations | 17 | 215 |
| Simplex Method (including Big-M and Two-Phase)Linear Programming | 13 | 215 |
| Equations of First Order and First Degree; Integrating FactorsOrdinary Differential Equations | 19 | 206 |
| Functions of Several Variables: Partial Derivatives, Maxima and MinimaReal Analysis | 13 | 205 |
| Two-Dimensional and Axisymmetric Motion; Sources, Sinks and ImagesMechanics and Fluid Dynamics | 13 | 190 |
We tagged every Mathematics optional question from 2013 to 2026 to its syllabus topic. The tables below show how many questions and how many marks each topic has carried, and how many of them were long (20-mark) questions. Use it to decide where your first revision hours should go.
Linear Algebra
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Linear Transformations, Rank and Nullity, Matrix of a Linear Transformation | 21 | 254 | 8 |
| Vector Spaces, Subspaces, Linear Dependence and Independence, Bases and Dimension | 17 | 186 | 4 |
| Eigenvalues, Eigenvectors, Characteristic Polynomial, Similarity and Diagonalisation | 13 | 150 | 2 |
| Algebra of Matrices, Row Reduction, Echelon Form, Rank and Inverse | 11 | 121 | 3 |
| Solution of System of Linear Equations | 8 | 95 | 3 |
| Symmetric, Skew-symmetric, Hermitian, Skew-Hermitian, Orthogonal and Unitary Matrices and their Eigenvalues | 7 | 76 | 2 |
| Cayley-Hamilton Theorem | 6 | 71 | 2 |
Calculus
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Double and Triple Integrals; Areas, Surfaces and Volumes | 18 | 235 | 9 |
| Functions of Two or Three Variables: Limits, Continuity, Partial Derivatives and Jacobians | 14 | 171 | 6 |
| Maxima and Minima (One Variable) and Applied Optimisation | 9 | 134 | 5 |
| Maxima and Minima of Several Variables and Lagrange's Method of Multipliers | 6 | 110 | 6 |
| Riemann Integral, Definite, Indefinite and Improper Integrals | 11 | 108 | 1 |
| Mean Value Theorems, Taylor's Theorem and Indeterminate Forms | 10 | 105 | 1 |
| Asymptotes and Curve Tracing | 4 | 65 | 3 |
| Functions of a Real Variable: Limits, Continuity and Differentiability | 5 | 55 | 1 |
Analytic Geometry
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| The Sphere | 18 | 225 | 8 |
| The Cone | 14 | 181 | 6 |
| Straight Lines and Shortest Distance between Skew Lines | 12 | 144 | 4 |
| Central Conicoids: Ellipsoid and Hyperboloids of One and Two Sheets | 9 | 130 | 6 |
| Paraboloids | 5 | 66 | 2 |
| The Cylinder | 5 | 55 | 1 |
| Coordinates in Three Dimensions and the Plane | 4 | 38 | 0 |
| Second Degree Equations and Reduction to Canonical Forms | 2 | 25 | 1 |
Ordinary Differential Equations
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Second Order Variable Coefficients: One Solution Known, Variation of Parameters | 17 | 215 | 8 |
| Equations of First Order and First Degree; Integrating Factors | 19 | 206 | 2 |
| Application of Laplace Transforms to Initial Value Problems | 14 | 181 | 6 |
| Euler-Cauchy (Homogeneous Linear) Equations | 9 | 121 | 4 |
| First Order Higher Degree Equations, Clairaut's Equation and Singular Solutions | 10 | 116 | 2 |
| Higher Order Linear Equations with Constant Coefficients | 8 | 80 | 1 |
| Orthogonal Trajectories | 7 | 70 | 0 |
| Laplace and Inverse Laplace Transforms and their Properties | 6 | 46 | 0 |
| Formulation of Differential Equations and Simple Applications | 4 | 38 | 0 |
Dynamics and Statics
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Rectilinear Motion and Simple Harmonic Motion | 12 | 160 | 6 |
| Equilibrium of Particles and Rigid Bodies; Forces in Three Dimensions | 9 | 115 | 5 |
| Constrained Motion | 7 | 112 | 5 |
| Kepler's Laws and Orbits under Central Forces | 8 | 108 | 3 |
| Principle of Virtual Work | 7 | 100 | 4 |
| Motion in a Plane and Projectiles | 7 | 88 | 3 |
| Stability of Equilibrium | 6 | 87 | 4 |
| Common Catenary | 5 | 72 | 2 |
| Friction | 5 | 58 | 1 |
| Work, Energy and Conservation of Energy | 4 | 46 | 0 |
Vector Analysis
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Stokes' Theorem | 11 | 163 | 7 |
| Gauss Divergence Theorem | 10 | 151 | 8 |
| Curves in Space: Curvature, Torsion and Serret-Frenet Formulae | 12 | 138 | 4 |
| Line Integrals and Green's Theorem in the Plane | 8 | 108 | 5 |
| Gradient, Divergence and Curl; Directional Derivatives; Irrotational Fields | 10 | 104 | 0 |
| Higher Order Derivatives and Vector Identities | 6 | 67 | 1 |
| Vector Algebra and Differentiation of Vector Functions of a Scalar Variable | 5 | 50 | 0 |
Algebra
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Fields and Quotient Fields | 9 | 135 | 8 |
| Integral Domains, PID, Euclidean Domains and UFD | 10 | 125 | 4 |
| Groups, Subgroups and Cyclic Groups | 8 | 110 | 5 |
| Rings, Subrings, Ideals and Ring Homomorphisms | 8 | 110 | 6 |
| Homomorphisms of Groups and Isomorphism Theorems | 8 | 100 | 4 |
| Cosets and Lagrange's Theorem | 4 | 45 | 1 |
| Permutation Groups and Cayley's Theorem | 3 | 33 | 0 |
| Normal Subgroups and Quotient Groups | 3 | 30 | 0 |
Real Analysis
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Functions of Several Variables: Partial Derivatives, Maxima and Minima | 13 | 205 | 11 |
| Riemann Integral and Fundamental Theorems of Integral Calculus | 13 | 175 | 8 |
| Continuity, Differentiability and Uniform Continuity | 8 | 115 | 6 |
| Real Number System, Sequences and Cauchy Sequences | 8 | 99 | 3 |
| Series: Convergence, Absolute and Conditional Convergence, Rearrangement | 7 | 90 | 3 |
| Uniform Convergence of Sequences and Series of Functions | 5 | 73 | 3 |
| Improper Integrals | 5 | 60 | 2 |
Complex Analysis
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Contour Integration: Evaluation of Real Integrals | 10 | 185 | 10 |
| Analytic Functions, Cauchy-Riemann Equations and Harmonic Functions | 13 | 145 | 3 |
| Singularities, Poles and Residues | 10 | 140 | 8 |
| Power Series, Taylor's and Laurent's Series | 9 | 130 | 5 |
| Complex Integration: Cauchy's Theorem and Integral Formula | 5 | 65 | 2 |
Linear Programming
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Simplex Method (including Big-M and Two-Phase) | 13 | 215 | 10 |
| Assignment Problems | 7 | 105 | 5 |
| Duality | 7 | 100 | 5 |
| Transportation Problems | 5 | 95 | 5 |
| Formulation; Basic, Basic Feasible and Optimal Solutions | 5 | 65 | 2 |
| Graphical Method | 5 | 50 | 0 |
Partial Differential Equations
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Linear PDEs with Constant Coefficients | 12 | 155 | 7 |
| Classification and Reduction to Canonical Form | 9 | 140 | 8 |
| First Order Quasilinear PDEs (Lagrange's Method); Integral Surfaces; Orthogonal Surfaces | 10 | 130 | 6 |
| Wave Equation (Vibrating String) | 7 | 130 | 7 |
| Non-linear First Order PDEs: Characteristics, Charpit and Complete Integrals | 6 | 90 | 6 |
| Laplace Equation | 5 | 85 | 4 |
| Family of Surfaces and Formulation of PDEs | 8 | 80 | 0 |
| Heat Equation | 3 | 55 | 3 |
Numerical Analysis and Computer Programming
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Boolean Algebra, Logic Gates, Truth Tables and Normal Forms | 14 | 185 | 9 |
| Numerical Integration: Trapezoidal, Simpson's and Gaussian Quadrature | 11 | 165 | 9 |
| Interpolation: Newton's Forward/Backward and Lagrange | 9 | 135 | 7 |
| Linear Systems: Gauss Elimination, Gauss-Jordan and Gauss-Seidel | 10 | 130 | 6 |
| Number Systems and Data Representation | 10 | 105 | 2 |
| Numerical Solution of ODEs: Euler and Runge-Kutta | 7 | 100 | 5 |
| Algorithms and Flow Charts | 5 | 70 | 3 |
| Roots of Equations: Bisection, Regula-Falsi and Newton-Raphson | 6 | 65 | 1 |
Mechanics and Fluid Dynamics
| Topic | Questions | Total marks | Long (20 marks) |
|---|---|---|---|
| Two-Dimensional and Axisymmetric Motion; Sources, Sinks and Images | 13 | 190 | 8 |
| Generalized Coordinates, D'Alembert's Principle and Lagrange's Equations | 10 | 155 | 7 |
| Hamilton's Equations | 10 | 140 | 5 |
| Kinematics: Continuity, Streamlines, Potential Flow, Vorticity | 9 | 115 | 4 |
| Moment of Inertia | 7 | 85 | 3 |
| Navier-Stokes Equation for Viscous Flow | 4 | 80 | 4 |
| Vortex Motion | 4 | 80 | 4 |
| Euler's Equation of Motion and Pressure | 4 | 70 | 4 |
| Motion of Rigid Bodies in Two Dimensions | 5 | 65 | 2 |
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