Study Guide
One chapter per syllabus topic: the recurring question patterns, the minimum theory that cracks them, and verified worked examples from thirteen years of papers. We're publishing chapters in exam-priority order — sphere is the flagship sample.
Paper 1
Dynamics & Statics
- Rectilinear motion under variable force — T3 · Rectilinear motion under variable force
- Simple harmonic motion (free, damped, forced) — T1 · Simple harmonic motion (free, damped, forced)
- Motion in a Plane (Resolved Components / Polar) — T4 · Motion in a plane (resolved components / polar)
- Projectile motion — T2 · Projectile motion
- Constrained motion — T2 · Constrained motion; circular motion
- Work-energy theorem — T3 · Work-energy theorem; conservation of energy
- Central force motion and Kepler's laws — T2 · Central force motion; Kepler's laws
- Orbits under inverse-square central force — T3 · Orbits under inverse-square central force
- Equilibrium of a system of particles — T2 · Equilibrium of a system of particles
- Work and Potential Energy; Conservation — T4 · Work and potential energy; conservation
- Friction (limiting friction) — T3 · Friction (limiting friction; equilibrium with friction)
- Common catenary — T3 · Common catenary
- Principle of virtual work — T2 · Principle of virtual work
- Stability of equilibrium (energy criterion) — T2 · Stability of equilibrium (energy criterion)
- Equilibrium of Forces in Three Dimensions — T4 · Equilibrium of forces in three dimensions
ODEs
- Formulation of differential equations — T3 · Formulation of differential equations
- Variables separable — T3 · Variables separable
- Homogeneous Equations and Reduction — T4 · Homogeneous equations and reduction
- Linear first-order — T3 · Linear first-order; integrating factor
- Exact equations — T2 · Exact equations; integrating factors for non-exact
- Orthogonal trajectories (cartesian and polar) — T2 · Orthogonal trajectories (cartesian and polar)
- First-order higher-degree ODEs — T3 · First-order higher-degree: solvable for p, x, y
- Clairaut's equation — T2 · Clairaut's equation; singular solution
- Linear ODE with constant coefficients — T3 · Linear ODE with constant coefficients: complementary function
- Particular integral via operator method — T2 · Particular integral via operator method
- Euler-Cauchy equation — T1 · Euler-Cauchy equation
- Method of variation of parameters — T1 · Method of variation of parameters
- Reduction of order with one solution known — T3 · Reduction of order with one solution known
- Laplace transform — T3 · Laplace transform: definition; transforms of elementary functions
- Properties of Laplace transform (linearity, shift, derivative, convolution) — T3 · Properties of Laplace transform (linearity, shift, derivative, convolution)
- Inverse Laplace transform — T3 · Inverse Laplace transform; partial fractions
- Laplace transform applied to IVP for second-order linear ODE with constant coefficients — T1 · Laplace transform applied to IVP for second-order linear ODE with constant coefficients
- Picard's Existence/Uniqueness Theorem; Lipschitz Condition — T4 · Picard's existence/uniqueness theorem; Lipschitz condition
Vector Analysis
- Scalar and Vector Fields — T4 · Scalar and vector fields
- Differentiation of a vector function of a scalar variable — T3 · Differentiation of a vector function of a scalar variable
- Gradient: definition, geometric meaning, computation — T3 · Gradient: definition, geometric meaning, computation
- Curl: definition, physical meaning, computation — T3 · Curl: definition, physical meaning, computation
- Higher order derivatives; Laplacian — T3 · Higher order derivatives; Laplacian
- Vector identities (curl of grad, div of curl, product rules) — T3 · Vector identities (curl of grad, div of curl, product rules)
- Curves in space: tangent, normal, binormal — T3 · Curves in space: tangent, normal, binormal
- Curvature and torsion — T2 · Curvature and torsion
- Serret-Frenet formulae — T3 · Serret-Frenet formulae
- Line integrals — T1 · Line integrals; circulation
- Gauss divergence theorem — T1 · Gauss divergence theorem
- Stokes' theorem — T1 · Stokes' theorem
Analytic Geometry
- Straight lines in 3D — T2 · Straight lines in 3D: parametric/symmetric/vector forms
- Shortest distance between two skew lines — T3 · Shortest distance between two skew lines
- Plane — T2 · Plane: equations, distance, angle
- Second-degree equations in three variables — T3 · Second-degree equations in three variables: general form
- Reduction of Second-Degree Equation to Canonical Form — T4 · Reduction of second-degree equation to canonical form
- Sphere — T1 · Sphere: equation, tangent plane, intersection
- Cone — T1 · Cone: equation; right circular cone
- Cylinder — T3 · Cylinder: equation; right circular cylinder
- Paraboloid (elliptic and hyperbolic) — T2 · Paraboloid (elliptic and hyperbolic)
- Ellipsoid — T3 · Ellipsoid
- Hyperboloid of one sheet — T3 · Hyperboloid of one sheet; ruled surface
Linear Algebra
- Linear dependence and independence — T3 · Linear dependence and independence
- Subspaces — T2 · Subspaces: definition, intersection, sum, direct sum
- Bases and dimension; coordinates in a basis — T2 · Bases and dimension; coordinates with respect to a basis
- Linear transformations — T3 · Linear transformations: definition, kernel, image
- Rank and nullity; rank-nullity theorem — T2 · Rank and nullity; rank-nullity theorem
- Matrix of a linear transformation — T1 · Matrix of a linear transformation; change of basis
- Algebra of matrices — T3 · Algebra of matrices: addition, multiplication, transpose
- Row and Column Reduction; Echelon Form — T4 · Row and column reduction; Echelon form
- Congruence and similarity of matrices — T3 · Congruence and similarity of matrices
- Rank of a matrix — T2 · Rank of a matrix
- Inverse of a matrix (adjoint and row reduction) — T3 · Inverse of a matrix (adjoint and row reduction)
- Solution of system of linear equations — T2 · Solution of system of linear equations; Gauss elimination
- Eigenvalues and eigenvectors — T1 · Eigenvalues and eigenvectors: computation
- Cayley-Hamilton theorem — T3 · Cayley-Hamilton theorem
- Diagonalization via Eigenvectors — T4 · Diagonalization via eigenvectors
- Symmetric and Skew-Symmetric Matrices — T4 · Symmetric and skew-symmetric matrices
- Hermitian and skew-Hermitian matrices — T3 · Hermitian and skew-Hermitian matrices
- Orthogonal and unitary matrices — T3 · Orthogonal and unitary matrices
- Quotient Vector Space V/W — T4 · Quotient vector space V/W: cosets; dimension formula dim(V/W)=dim V - dim W
Calculus
- Continuity of real functions — T3 · Continuity of real functions
- Differentiability — T3 · Differentiability; rules of differentiation
- Mean-value theorems (Rolle, Lagrange, Cauchy) — T3 · Mean-value theorems (Rolle, Lagrange, Cauchy)
- Taylor's theorem with remainders — T3 · Taylor's theorem with remainders
- Indeterminate forms — T3 · Indeterminate forms; L'Hopital's rule
- Maxima and minima of single-variable functions — T2 · Maxima and minima of single-variable functions
- Asymptotes — T4 · Asymptotes
- Curve tracing (cartesian and polar) — T3 · Curve tracing (cartesian and polar)
- Functions of two/three variables: limits, continuity — T3 · Functions of two/three variables: limits, continuity
- Partial derivatives — T2 · Partial derivatives; equality of mixed partials
- Maxima and Minima of Multi-Variable Functions (Unconstrained) — T4 · Maxima and minima of multi-variable functions (unconstrained)
- Lagrange's method of multipliers (constrained extrema) — T1 · Lagrange's method of multipliers (constrained extrema)
- Jacobian — T3 · Jacobian; change of variables
- Riemann Definite Integral; Integrability — T4 · Riemann definite integral; integrability
- Indefinite integrals — T2 · Indefinite integrals: substitution, parts, partial fractions
- Improper integrals (unbounded interval/integrand) — T3 · Improper integrals (unbounded interval/integrand)
- Double integrals — T1 · Double integrals; change of order/variables
- Triple Integrals; Cylindrical and Spherical Coordinates — T4 · Triple integrals; cylindrical, spherical coordinates
- Areas, surface areas, volumes via integration — T3 · Areas, surface areas, volumes via integration
Paper 2
Algebra
- Groups: definition, axioms, examples — T3 · Groups: definition, axioms, examples
- Subgroups; Subgroup Criterion — T4 · Subgroups; subgroup criterion
- Cyclic groups — T1 · Cyclic groups: structure, generators, order of element
- Cosets and Lagrange's theorem — T3 · Cosets and Lagrange's theorem
- Normal subgroups; quotient groups — T3 · Normal subgroups; quotient groups
- Group homomorphisms: kernel, image — T3 · Group homomorphisms: kernel, image
- Isomorphism theorems (First, Second, Third) — T3 · Isomorphism theorems (First, Second, Third)
- Permutation Groups (S_n): Cycle Decomposition, Sign, A_n — T4 · Permutation groups (S_n): cycle decomposition, sign, A_n
- Cayley's Theorem — T4 · Cayley's theorem
- Rings: Definition, Axioms, Examples — T4 · Rings: definition, axioms, examples
- Subrings and ideals — T3 · Subrings and ideals
- Ring homomorphisms; quotient rings — T3 · Ring homomorphisms; quotient rings
- Integral domains; characteristic — T3 · Integral domains; characteristic
- Principal Ideal Domains (PID) — T4 · Principal ideal domains (PID)
- Euclidean domains — T2 · Euclidean domains
- Fields and finite fields — T2 · Fields: definition; finite fields F_p; characteristic
- Field Extensions; Tower Law; Algebraic Closure — T4 · Field/algebraic extensions; tower law; algebraic closure
Complex Analysis
- Analytic Functions: Complex Differentiability — T4 · Analytic functions: complex differentiability
- Cauchy-Riemann equations (necessary and sufficient) — T2 · Cauchy-Riemann equations (necessary and sufficient)
- Harmonic functions and harmonic conjugate — T2 · Harmonic functions; harmonic conjugate
- Cauchy's theorem (Cauchy-Goursat) — T3 · Cauchy's theorem (Cauchy-Goursat)
- Power Series of Analytic Functions; Radius of Convergence — T4 · Power series of analytic functions; radius of convergence
- Taylor's Series for Analytic Functions — T4 · Taylor's series for analytic functions
- Singularities: removable, pole, essential — T3 · Singularities: removable, pole, essential
- Laurent's series in an annulus — T2 · Laurent's series in an annulus
- Residues: computation at poles of various orders — T3 · Residues: computation at poles of various orders
- Cauchy's residue theorem — T2 · Cauchy's residue theorem
- Contour integration of real integrals using residues — T1 · Contour integration of real integrals using residues
Linear Programming
- LPP: standard form; basic, basic feasible, optimal solutions — T3 · LPP: standard form; basic, basic feasible, optimal solutions
- Graphical method — T2 · Graphical method (two-variable LPPs)
- Simplex method (basic) — T1 · Simplex method (basic)
- Big-M / two-phase method (artificial variables) — T2 · Big-M / two-phase method (artificial variables)
- Duality — T3 · Duality: dual formulation, weak/strong duality, complementary slackness
- Transportation problem — T2 · Transportation problem: BFS (NWCR/LCM/VAM), MODI optimality
- Assignment problem (Hungarian method) — T2 · Assignment problem (Hungarian method)
Numerical Analysis
- Bisection Method (Convergence, Error) — T4 · Bisection method (convergence, error)
- Regula Falsi (False Position) — T4 · Regula Falsi (false position)
- Newton-Raphson method (convergence, geometric meaning) — T2 · Newton-Raphson method (convergence, geometric meaning)
- Gaussian Elimination — T4 · Gaussian elimination
- Gauss-Jordan method — T3 · Gauss-Jordan method
- Gauss-Seidel iteration — T2 · Gauss-Seidel iteration; convergence criteria
- Newton's forward difference interpolation — T3 · Newton's forward difference interpolation
- Newton's Backward Difference Interpolation — T4 · Newton's backward difference interpolation
- Lagrange's interpolation — T2 · Lagrange's interpolation
- Trapezoidal rule (composite; error) — T3 · Trapezoidal rule (composite; error)
- Simpson's 1/3 and 3/8 rules — T2 · Simpson's 1/3 and 3/8 rules
- Gaussian quadrature — T3 · Gaussian quadrature
- Euler's method (and modified Euler) — T3 · Euler's method (and modified Euler)
- Runge-Kutta methods (RK2/RK4) — T3 · Runge-Kutta methods (RK2/RK4)
- Number systems — T2 · Number systems: binary, octal, hex; conversions
- Algebra of Binary Numbers — T4 · Algebra of binary numbers
- Logic gates and truth tables — T3 · Logic gates and truth tables
- Boolean algebra — T1 · Boolean algebra; normal forms (DNF, CNF)
- Representation of Integers, Signed Integers, and Reals (incl. Double Precision) — T4 · Representation of integers, signed integers, reals (incl. double precision)
- Algorithms and flowcharts for numerical analysis problems — T3 · Algorithms and flowcharts for numerical analysis problems
PDEs
- Family of surfaces — T2 · Family of surfaces; PDE formulation
- Quasilinear first-order PDEs (Lagrange's method) — T2 · Quasilinear first-order PDEs (Lagrange's method)
- Cauchy's method of characteristics — T3 · Cauchy's method of characteristics
- Classification and reduction to canonical form — T1 · Classification and reduction to canonical form
- Second-order linear PDEs with constant coefficients (CF, PI) — T1 · Second-order linear PDEs with constant coefficients (CF, PI)
- Wave equation — T2 · Wave equation: D'Alembert and separation of variables
- Heat equation — T3 · Heat equation: separation of variables; Fourier solutions
- Laplace equation: Dirichlet/Neumann, separation of variables — T3 · Laplace equation: Dirichlet/Neumann, separation of variables
- Charpit's method — T3 · Charpit's method; complete integrals of nonlinear first-order PDEs
Real Analysis
- Real number system as ordered field with LUB property — T3 · Real number system as ordered field with LUB property
- Sequences — T3 · Sequences: limit, bounded, monotone convergence
- Cauchy sequences; completeness of R — T3 · Cauchy sequences; completeness of R
- Series of real terms: convergence, standard tests — T3 · Series of real terms: convergence, standard tests
- Absolute and conditional convergence — T3 · Absolute and conditional convergence
- Rearrangement of Series; Riemann's Theorem — T4 · Rearrangement of series; Riemann's theorem (statement)
- Continuity of Functions on R; Epsilon-Delta — T4 · Continuity of functions on R; epsilon-delta
- Uniform continuity — T3 · Uniform continuity
- Properties of Continuous Functions on Compact Sets — T4 · Properties of continuous functions on compact sets
- Riemann integral — T1 · Riemann integral: Darboux sums, integrability
- Improper integrals (analysis perspective) — T3 · Improper integrals (analysis perspective)
- Fundamental theorems of integral calculus — T3 · Fundamental theorems of integral calculus
- Pointwise vs. Uniform Convergence of Sequences of Functions — T4 · Pointwise vs. uniform convergence of sequences of functions
- Uniform Convergence: Term-by-Term Differentiation — T4 · Uniform convergence: term-by-term differentiation
- Uniform convergence of series — T3 · Uniform convergence of series; Weierstrass M-test
- Partial derivatives; equality of mixed partials (Schwarz) — T3 · Partial derivatives; equality of mixed partials (Schwarz)
- Maxima and minima of multi-variable functions (analytic criteria) — T3 · Maxima and minima of multi-variable functions (analytic criteria)
- Maxima and minima of single-variable functions — T3 · Maxima and minima of single-variable functions
Mechanics & Fluid Dynamics
- D'Alembert's Principle — T4 · D'Alembert's principle
- Lagrange's equations — T1 · Lagrange's equations
- Hamilton's equations — T1 · Hamilton's equations
- Moment of inertia — T2 · Moment of inertia; perpendicular/parallel axis theorems
- Motion of rigid bodies in two dimensions — T3 · Motion of rigid bodies in two dimensions
- Equation of Continuity — T4 · Equation of continuity
- Euler's equation of motion for inviscid flow — T3 · Euler's equation of motion for inviscid flow
- Potential flow — T1 · Potential flow; velocity potential
- Two-Dimensional and Axisymmetric Flow — T4 · Two-dimensional and axisymmetric flow
- Sources, sinks, doublets — T1 · Sources, sinks, doublets
- Vortex motion; circulation — T3 · Vortex motion; circulation
- Navier-Stokes equation for a viscous fluid — T3 · Navier-Stokes equation for a viscous fluid