The whole syllabus, one chapter at a time.

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1. Unit 1 · Chapter 17 min read

   The Real Line

   Elements of set theory: operations, De Morgan and difference

   Rewrite every set difference as $A \setminus B = A \cap B^{c}$, and the whole subject collapses into Boolean algebra where De Morgan does the rest.

   [⚠️ Where intuition breaks:

   Difference is not associative.

   ]

   6 PYQs inside Mark as readRead Chapter
2. Unit 1 · Chapter 27 min read

   The Real Line

   Completeness, sup/inf, Archimedean property

   Completeness is the single axiom that separates $\mathbb{R}$ from $\mathbb{Q}$, and almost every question in this area is really asking which property fails once you take it away.

   [⚠️ Where intuition breaks:

   "The supremum is the maximum."

   ]

   8 PYQs inside Mark as readRead Chapter
3. Unit 1 · Chapter 37 min read

   The Real Line

   Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

   Four properties — convergent, Cauchy, bounded, has a convergent subsequence — sit in a chain of one-way implications, and the exam tests whether you know which arrows reverse and which do not.

   [⚠️ Where intuition breaks:

   "Bounded, so it converges."

   ]

   12 PYQs inside Mark as readRead Chapter
4. Unit 1 · Chapter 47 min read

   The Real Line

   limsup, liminf and subsequential limits

   Every bounded sequence has a highest and a lowest place its terms keep returning to, and those two numbers exist even when the limit does not.

   [⚠️ Where intuition breaks:

   "$\limsup(a_n + b_n) = \limsup a_n + \limsup b_n$."

   ]

   8 PYQs inside Mark as readRead Chapter
5. Unit 1 · Chapter 57 min read

   The Real Line

   Series: comparison, ratio, root, Raabe, condensation, alternating, rearrangements

   Choosing the right test takes ten seconds if you recognise the shape of the terms, and the questions are built almost entirely around the handful of implications that look true and are not.

   [⚠️ Where intuition breaks:

   "$a_n \to 0$, so $\sum a_n$ converges."

   ]

   4 PYQs inside Mark as readRead Chapter
6. Unit 1 · Chapter 68 min read

   Continuity and Differentiation

   Continuity, uniform continuity, Lipschitz

   Continuity is a promise made separately at each point; uniform continuity is one promise for the whole domain; Lipschitz puts a number on it — and the exam lives in the two gaps.

   [⚠️ Where intuition breaks:

   "Continuous on a bounded interval, so bounded."

   ]

   14 PYQs inside Mark as readRead Chapter
7. Unit 1 · Chapter 77 min read

   Continuity and Differentiation

   Differentiability, mean value theorems, Taylor, L'Hôpital

   Every theorem in this family carries hypotheses that look like formalities, and the questions are built almost entirely from removing one and seeing whether you notice.

   [⚠️ Where intuition breaks:

   "Differentiable, so continuously differentiable."

   ]

   7 PYQs inside Mark as readRead Chapter
8. Unit 1 · Chapter 87 min read

   Integration

   Riemann integration and criteria

   A bounded function is Riemann integrable exactly when its discontinuities form a set of measure zero — and almost every question here is that criterion applied to a deliberately strange function.

   [⚠️ Where intuition breaks:

   "Bounded, so integrable."

   ]

   9 PYQs inside Mark as readRead Chapter
9. Unit 1 · Chapter 96 min read

   Integration

   Improper integrals and convergence tests

   An improper integral must be split at every point where it misbehaves, and each piece compared with $1/x^p$ — where the threshold runs one way near zero and the other way near infinity.

   [⚠️ Where intuition breaks:

   "$\int_0^\infty f$ converges, so $f(x) \to 0$."

   ]

   3 PYQs inside Mark as readRead Chapter
10. Unit 1 · Chapter 108 min read

    Sequences and Series of Functions

    Pointwise vs uniform convergence, M-test, Dini

    Pointwise convergence is a promise made separately to each point; uniform convergence is a single promise made to all of them at once — and the exam lives entirely in the gap between.

    [⚠️ Where intuition breaks:

    "The curves are getting flatter, so they must be converging uniformly."

    ]

    13 PYQs inside Mark as readRead Chapter
11. Unit 1 · Chapter 116 min read

    Sequences and Series of Functions

    Power series, radius of convergence, Abel's theorem

    A power series has a radius inside which everything works and outside which nothing does, and the whole subject is what happens on the circle in between.

    [⚠️ Where intuition breaks:

    "The radius is $\lim |a_n/a_{n+1}|$."

    ]

    1 PYQ inside Mark as readRead Chapter
12. Unit 1 · Chapter 126 min read

    Sequences and Series of Functions

    Arzelà–Ascoli and equicontinuity

    Boundedness gives you a convergent subsequence for numbers but not for functions; equicontinuity is the extra hypothesis that repairs it, and Arzelà–Ascoli is the repair.

    [⚠️ Where intuition breaks:

    "Uniformly bounded in $C[0,1]$, so a uniformly convergent subsequence exists."

    ]

    No PYQs filed yet Mark as readRead Chapter
13. Unit 1 · Chapter 1311 min read

    Functions of Several Variables

    Partial derivatives, differentiability, chain rule

    In one variable there is one notion of derivative; in two there are four, they are not equivalent, and the exam is built almost entirely on the gaps between them.

    [⚠️ Where intuition breaks:

    "Both partials exist at $a$, so $f$ is continuous at $a$."

    ]

    10 PYQs inside Mark as readRead Chapter
14. Unit 1 · Chapter 1411 min read

    Functions of Several Variables

    Inverse and implicit function theorems, extrema

    Both theorems are local, both need a derivative that is continuous and invertible, and nearly every wrong option is produced by upgrading "local" to "global" or by forgetting which of the two variables the hypothesis is about.

    [⚠️ Where intuition breaks:

    "$\det Df \ne 0$ everywhere, so $f$ is a bijection."

    ]

    3 PYQs inside Mark as readRead Chapter
15. Unit 1 · Chapter 159 min read

    Metric Spaces

    Open/closed sets, limit points, closure, interior

    Open and closed are not opposites, closure is not "the set plus its edge" in any naive sense, and almost every question in this topic is decided by producing one set in one metric where the option fails.

    [⚠️ Where intuition breaks:

    "A set is either open or closed."

    ]

    4 PYQs inside Mark as readRead Chapter
16. Unit 1 · Chapter 169 min read

    Metric Spaces

    Compactness: open covers, sequential, Heine–Borel

    In a metric space compactness, sequential compactness, and completeness-plus-total-boundedness are the same thing — and "closed and bounded" is not one of them outside $\mathbb{R}^n$.

    [⚠️ Where intuition breaks:

    "Closed and bounded, so compact."

    ]

    1 PYQ inside Mark as readRead Chapter
17. Unit 1 · Chapter 179 min read

    Metric Spaces

    Completeness and Baire category

    Completeness is a property of the metric, not of the topology, and both of the theorems built on it — Banach's fixed point theorem and Baire's category theorem — fail loudly when their hypotheses are trimmed.

    [⚠️ Where intuition breaks:

    "$(0,1)$ and $\mathbb{R}$ are homeomorphic, so both are complete or neither is."

    ]

    1 PYQ inside Mark as readRead Chapter
18. Unit 1 · Chapter 189 min read

    Metric Spaces

    Connectedness and path-connectedness

    Path-connected implies connected and not the other way round, the connected subsets of $\mathbb{R}$ are exactly the intervals, and one curve answers most of the questions in this topic.

    [⚠️ Where intuition breaks:

    "Connected, so path-connected."

    ]

    No PYQs filed yet Mark as readRead Chapter
19. Unit 1 · Chapter 199 min read

    Lebesgue Measure and Integration

    Measurable sets and functions

    Countable, measure zero and nowhere dense are three different kinds of small, no two of them imply each other, and this topic is mostly the business of keeping them apart.

    [⚠️ Where intuition breaks:

    "Measure zero, so countable."

    ]

    3 PYQs inside Mark as readRead Chapter
20. Unit 1 · Chapter 209 min read

    Lebesgue Measure and Integration

    Lebesgue integral, MCT, DCT, Fatou

    All three convergence theorems answer the same question — when does $\int \lim f_n = \lim \int f_n$ — and each one is examined by deleting the hypothesis that stops the mass escaping.

    [⚠️ Where intuition breaks:

    "$f_n \to f$ pointwise, so $\int f_n \to \int f$."

    ]

    1 PYQ inside Mark as readRead Chapter
21. Unit 1 · Chapter 218 min read

    Lebesgue Measure and Integration

    L^p spaces essentials

    Which $L^p$ contains which depends on the measure space and on nothing else — nested one way when the measure is finite, not nested at all on $\mathbb{R}$, and nested the other way for sequences.

    [⚠️ Where intuition breaks:

    "$L^2 \subseteq L^1$."

    ]

    3 PYQs inside Mark as readRead Chapter
22. Unit 1 · Chapter 229 min read

    Vector Spaces and Linear Maps

    Bases, dimension, rank–nullity

    Dimension is a statement about a set and a field, rank–nullity is the one theorem that constrains everything else, and rank behaves badly under every operation candidates expect it to respect.

    [⚠️ Where intuition breaks:

    "The dimension of $\mathbb{C}^n$ is $n$."

    ]

    15 PYQs inside Mark as readRead Chapter
23. Unit 1 · Chapter 239 min read

    Vector Spaces and Linear Maps

    Linear transformations, matrix representation, change of basis

    A matrix is a linear map plus a choice of basis, similarity is what changing that choice does, and the invariants of similarity are exactly the things a question can legitimately ask about a map.

    [⚠️ Where intuition breaks:

    "Same characteristic polynomial, so similar."

    ]

    11 PYQs inside Mark as readRead Chapter
24. Unit 1 · Chapter 248 min read

    Eigenvalues and Canonical Forms

    Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

    The characteristic polynomial tells you which eigenvalues there are, the minimal polynomial tells you how badly the matrix fails to be diagonal, and almost every question here turns on the second rather than the first.

    [⚠️ Where intuition breaks:

    "Cayley–Hamilton says the minimal polynomial is the characteristic polynomial."

    ]

    26 PYQs inside Mark as readRead Chapter
25. Unit 1 · Chapter 257 min read

    Eigenvalues and Canonical Forms

    Diagonalisability criteria

    Diagonalisable means enough eigenvectors, which is the same as the minimal polynomial having no repeated factor, which is the same as geometric equalling algebraic multiplicity everywhere — and distinct eigenvalues are sufficient but never necessary.

    [⚠️ Where intuition breaks:

    "Diagonalisable, so $n$ distinct eigenvalues."

    ]

    5 PYQs inside Mark as readRead Chapter
26. Unit 1 · Chapter 2610 min read

    Eigenvalues and Canonical Forms

    Jordan canonical form

    The Jordan form is a picture of block sizes, every feature of that picture is a rank you can compute, and the questions are almost all "how many structures are possible" or "which structure is this".

    [⚠️ Where intuition breaks:

    "Same characteristic and minimal polynomial, so similar."

    ]

    6 PYQs inside Mark as readRead Chapter
27. Unit 1 · Chapter 278 min read

    Eigenvalues and Canonical Forms

    Rational canonical form

    The rational canonical form does what the Jordan form cannot — it exists over every field — and its main consequence is that similarity never depends on which field you are working in.

    [⚠️ Where intuition breaks:

    "Similar over $\mathbb{C}$ but not over $\mathbb{R}$."

    ]

    No PYQs filed yet Mark as readRead Chapter
28. Unit 1 · Chapter 287 min read

    Inner Product Spaces and Forms

    Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

    Normal is exactly the class the spectral theorem applies to, Hermitian and unitary and skew-Hermitian are the three special cases whose eigenvalues sit on the real axis, the unit circle and the imaginary axis, and every question here is asking which of those you are in.

    [⚠️ Where intuition breaks:

    "Symmetric, so diagonalisable."

    ]

    9 PYQs inside Mark as readRead Chapter
29. Unit 1 · Chapter 297 min read

    Inner Product Spaces and Forms

    Quadratic forms, positive definiteness, Sylvester's law

    A quadratic form is classified by its signature and by nothing else, congruence is the relation that preserves it, and congruence is not similarity — it changes eigenvalues freely.

    [⚠️ Where intuition breaks:

    "Leading principal minors $\ge 0$, so positive semidefinite."

    ]

    14 PYQs inside Mark as readRead Chapter
30. Unit 1 · Chapter 307 min read

    Determinants and Matrix Tricks

    Determinants, trace, block matrices, rank inequalities

    The determinant is multiplicative and violently non-additive, block structure is the fastest route through any large matrix, and the adjugate's rank takes only three values.

    [⚠️ Where intuition breaks:

    "$\det(cA) = c\det A$."

    ]

    3 PYQs inside Mark as readRead Chapter
31. Unit 2 · Chapter 319 min read

    Analytic Functions

    Cauchy–Riemann equations, harmonic functions

    Complex differentiability asks for a single complex number where the real derivative offers four, and the Cauchy–Riemann equations are exactly the two conditions that force the four to collapse into one.

    [⚠️ Where intuition breaks:

    "Cauchy–Riemann holds at $z_0$, so $f$ is differentiable at $z_0$."

    ]

    7 PYQs inside Mark as readRead Chapter
32. Unit 2 · Chapter 328 min read

    Analytic Functions

    Power series and analyticity

    Holomorphic and analytic are the same thing in one complex variable, the radius of convergence is the distance to the nearest singularity, and the identity theorem makes holomorphic functions so rigid that agreeing on a small set forces agreeing everywhere.

    [⚠️ Where intuition breaks:

    "Smooth but not analytic."

    ]

    8 PYQs inside Mark as readRead Chapter
33. Unit 2 · Chapter 338 min read

    Cauchy Theory

    Cauchy's theorem and integral formula

    On a simply connected domain every closed contour integral of a holomorphic function vanishes, the single exception $\oint dz/z = 2\pi i$ is where all the content lives, and the integral formula turns that exception into a machine for evaluating functions and their derivatives.

    [⚠️ Where intuition breaks:

    "$f$ is holomorphic on the domain, so $\oint f = 0$."

    ]

    3 PYQs inside Mark as readRead Chapter
34. Unit 2 · Chapter 349 min read

    Cauchy Theory

    Liouville, Morera, maximum modulus principle

    Holomorphic functions are rigid: bounded on all of $\mathbb{C}$ forces constant, a maximum of $|f|$ inside forces constant, and the converse of Cauchy's theorem holds under a hypothesis weak enough to be useful.

    [⚠️ Where intuition breaks:

    "$f$ is entire and $|f| \le 1$ on $\mathbb{R}$, so $f$ is constant."

    ]

    12 PYQs inside Mark as readRead Chapter
35. Unit 2 · Chapter 359 min read

    Singularities and Residues

    Laurent series, classification of singularities, Casorati–Weierstrass

    Count the negative powers in the Laurent series: none means removable, finitely many means a pole, infinitely many means essential — and each of those is equivalent to a statement about $|f|$ that you can check without expanding anything.

    [⚠️ Where intuition breaks:

    "Essential means $|f| \to \infty$ faster."

    ]

    5 PYQs inside Mark as readRead Chapter
36. Unit 2 · Chapter 368 min read

    Singularities and Residues

    Residue theorem and standard contour integrals

    The residue is the single Laurent coefficient $a_{-1}$, the residue theorem says a contour integral is $2\pi i$ times the residues enclosed, and the exam's real content is the handful of standard contours that turn a real integral into that sum.

    [⚠️ Where intuition breaks:

    "Apply the semicircle to any rational function."

    ]

    8 PYQs inside Mark as readRead Chapter
37. Unit 2 · Chapter 379 min read

    Zeros and Mappings

    Argument principle, Rouché's theorem, open mapping

    The argument principle counts zeros minus poles by watching how far the image winds, Rouché turns that count into a comparison with a simpler function, and the open mapping theorem is the structural fact underneath both.

    [⚠️ Where intuition breaks:

    "$|g| \le |f|$ on $\gamma$, so Rouché applies."

    ]

    3 PYQs inside Mark as readRead Chapter
38. Unit 2 · Chapter 388 min read

    Zeros and Mappings

    Conformal maps, Möbius transformations, Schwarz lemma

    Conformal means angle-preserving, which for holomorphic maps is exactly $f' \ne 0$; Möbius transformations are the maps of the sphere to itself and send circles-or-lines to circles-or-lines; and Schwarz's lemma pins down every map of the disc to itself.

    [⚠️ Where intuition breaks:

    "Möbius maps send circles to circles and lines to lines."

    ]

    6 PYQs inside Mark as readRead Chapter
39. Unit 2 · Chapter 397 min read

    Groups

    Subgroups, cosets, Lagrange, cyclic groups

    Lagrange says the order of a subgroup divides the order of the group, the converse is false, and $A_4$ having no subgroup of order $6$ is the fact the whole topic is built to test.

    [⚠️ Where intuition breaks:

    "$d$ divides $|G|$, so $G$ has a subgroup of order $d$."

    ]

    9 PYQs inside Mark as readRead Chapter
40. Unit 2 · Chapter 407 min read

    Groups

    Normal subgroups, quotients, isomorphism theorems

    Normal subgroups are exactly the kernels, quotients exist exactly for them, and the property that catches people is that normality is a relation between two groups rather than something a subgroup possesses.

    [⚠️ Where intuition breaks:

    "$N \trianglelefteq H$ and $H \trianglelefteq G$, so $N \trianglelefteq G$."

    ]

    4 PYQs inside Mark as readRead Chapter
41. Unit 2 · Chapter 417 min read

    Groups

    Permutation groups: cycles, sign, conjugacy in S_n and A_n

    Cycle type is the complete conjugacy invariant in $S_n$, so counting classes is counting partitions — and the two facts that catch people are that a cycle of even length is an odd permutation, and that classes can split on passing to $A_n$.

    [⚠️ Where intuition breaks:

    "A $4$-cycle is even because $4$ is even."

    ]

    3 PYQs inside Mark as readRead Chapter
42. Unit 2 · Chapter 427 min read

    Groups

    Group actions, class equation, p-groups

    Orbit-stabiliser converts group theory into counting, the class equation is that count applied to conjugation, and Sylow's theorems are what let you decide the structure of a group from its order alone.

    [⚠️ Where intuition breaks:

    "$n_p \equiv 1 \pmod p$, so $n_p = 1$."

    ]

    5 PYQs inside Mark as readRead Chapter
43. Unit 2 · Chapter 438 min read

    Groups

    Sylow theorems and groups of small order

    Sylow's counting conditions usually pin down enough structure to classify a group from its order alone, and when they do not, counting elements of prime order finishes the job.

    [⚠️ Where intuition breaks:

    "$n_p \equiv 1 \pmod p$, so $n_p = 1$."

    ]

    3 PYQs inside Mark as readRead Chapter
44. Unit 2 · Chapter 447 min read

    Groups

    Finite abelian groups

    Every finite abelian group is a direct sum of cyclic groups of prime power order, that decomposition is unique, and counting abelian groups of a given order is counting partitions of the prime exponents.

    [⚠️ Where intuition breaks:

    "$C_4\times C_2 \cong C_8$."

    ]

    2 PYQs inside Mark as readRead Chapter
45. Unit 2 · Chapter 456 min read

    Rings and Fields

    Ideals, quotient rings, prime & maximal ideals, CRT

    An ideal is prime when the quotient is a domain and maximal when the quotient is a field, so maximal implies prime and the converse fails exactly where a domain falls short of being a field.

    [⚠️ Where intuition breaks:

    "Prime implies maximal."

    ]

    14 PYQs inside Mark as readRead Chapter
46. Unit 2 · Chapter 466 min read

    Rings and Fields

    Euclidean, PID, UFD hierarchy

    Euclidean $\subsetneq$ PID $\subsetneq$ UFD $\subsetneq$ integral domain, every inclusion is strict, and the examinable content is the witness in each gap plus the fact that irreducible and prime coincide only from UFD onwards.

    [⚠️ Where intuition breaks:

    "Irreducible means prime."

    ]

    2 PYQs inside Mark as readRead Chapter
47. Unit 2 · Chapter 477 min read

    Rings and Fields

    Polynomial rings and irreducibility tests

    $K[x]$ over a field is Euclidean and so as well behaved as $\mathbb{Z}$, and every irreducibility test is a one-way implication whose failure proves nothing.

    [⚠️ Where intuition breaks:

    "No roots, so irreducible."

    ]

    8 PYQs inside Mark as readRead Chapter
48. Unit 2 · Chapter 487 min read

    Rings and Fields

    Field extensions, splitting fields, finite fields

    The degree of an extension is a dimension, the tower law multiplies degrees, and a splitting field is one that contains all the roots — which $\mathbb{Q}(\sqrt[3]{2})$ conspicuously does not.

    [⚠️ Where intuition breaks:

    "$L$ contains a root of $f$, so $f$ splits in $L$."

    ]

    4 PYQs inside Mark as readRead Chapter
49. Unit 2 · Chapter 498 min read

    Rings and Fields

    Galois theory essentials

    For a normal and separable extension the subgroups of the Galois group correspond exactly to the intermediate fields, inclusion-reversing — and the whole subject is the consequences of that dictionary.

    [⚠️ Where intuition breaks:

    "Every finite extension is Galois."

    ]

    3 PYQs inside Mark as readRead Chapter
50. Unit 2 · Chapter 507 min read

    Topology

    Topological spaces, bases, subspace/product/quotient

    A topology is a choice of which sets are open, everything else is defined from that choice, and the one construction that behaves unexpectedly is the infinite product — where the useful topology is the coarser one.

    [⚠️ Where intuition breaks:

    "The product topology has all products of open sets as basic opens."

    ]

    No PYQs filed yet Mark as readRead Chapter
51. Unit 2 · Chapter 517 min read

    Topology

    Continuity, homeomorphism, separation axioms

    Continuity is preimages of open sets being open, a homeomorphism is a continuous bijection with continuous inverse, and the separation axioms measure how far a space is from being metric.

    [⚠️ Where intuition breaks:

    "A continuous bijection is a homeomorphism."

    ]

    5 PYQs inside Mark as readRead Chapter
52. Unit 2 · Chapter 526 min read

    Topology

    Compactness and Tychonoff

    Compactness is still every open cover having a finite subcover, but outside metric spaces it no longer coincides with sequential compactness, and Tychonoff — arbitrary products of compact spaces are compact — is the theorem that makes the product topology worth having.

    [⚠️ Where intuition breaks:

    "Compact means sequentially compact."

    ]

    2 PYQs inside Mark as readRead Chapter
53. Unit 2 · Chapter 536 min read

    Topology

    Connectedness and components

    Connectedness in a general space is still the absence of a separation, components are always closed and need not be open, and the gap between components and path components is where the questions live.

    [⚠️ Where intuition breaks:

    "Connected implies path-connected."

    ]

    1 PYQ inside Mark as readRead Chapter
54. Unit 2 · Chapter 547 min read

    Topology

    Standard spaces: cofinite, cocountable, Sorgenfrey, Cantor set

    Five spaces refute most of what can plausibly be claimed in general topology, and knowing which property each one breaks is faster than reasoning from definitions.

    [⚠️ Where intuition breaks:

    "$T_{1}$ implies Hausdorff."

    ]

    3 PYQs inside Mark as readRead Chapter
55. Unit 3 · Chapter 557 min read

    Ordinary Differential Equations

    Existence–uniqueness, Picard, Lipschitz

    Continuity of $f$ gives a solution, a Lipschitz condition in $y$ makes it unique, and neither gives a solution that lasts forever.

    [⚠️ Where intuition breaks:

    "$f$ continuous, so the solution is unique."

    ]

    8 PYQs inside Mark as readRead Chapter
56. Unit 3 · Chapter 567 min read

    Ordinary Differential Equations

    Linear ODE, Wronskian, variation of parameters, systems

    The solution set of a linear $n$-th order equation is an $n$-dimensional vector space, the Wronskian detects a basis — but only for solutions of a common equation, and that qualifier is the whole trap.

    [⚠️ Where intuition breaks:

    "$W \equiv 0$, so the functions are dependent."

    ]

    15 PYQs inside Mark as readRead Chapter
57. Unit 3 · Chapter 577 min read

    Ordinary Differential Equations

    Sturm–Liouville problems and Green's functions

    A Sturm–Liouville problem has real, simple eigenvalues increasing to infinity with eigenfunctions orthogonal in a weighted inner product, and the Green's function inverts the operator exactly when zero is not among those eigenvalues.

    [⚠️ Where intuition breaks:

    "Eigenfunctions are orthogonal in the usual $L^2$ inner product."

    ]

    7 PYQs inside Mark as readRead Chapter
58. Unit 3 · Chapter 587 min read

    Ordinary Differential Equations

    Stability and phase portraits

    Near a hyperbolic equilibrium the linearisation tells you everything, the trace and determinant classify every planar linear system between them, and the centre is the one case where linearisation decides nothing.

    [⚠️ Where intuition breaks:

    "The linearisation always determines stability."

    ]

    2 PYQs inside Mark as readRead Chapter
59. Unit 3 · Chapter 597 min read

    Partial Differential Equations

    First-order PDE: Lagrange, Charpit, characteristics

    A first-order PDE is an ODE in disguise: find the curves along which it becomes one, carry the initial data along them, and the only way it fails is when those curves do not reach where you need them.

    [⚠️ Where intuition breaks:

    "The general solution has arbitrary constants."

    ]

    10 PYQs inside Mark as readRead Chapter
60. Unit 3 · Chapter 606 min read

    Partial Differential Equations

    Classification and canonical forms

    For $Au_{xx} + Bu_{xy} + Cu_{yy} + \cdots = 0$ the sign of $B^2 - 4AC$ decides the type, the type decides the behaviour, and the type can change from point to point.

    [⚠️ Where intuition breaks:

    "The type is a property of the equation."

    ]

    2 PYQs inside Mark as readRead Chapter
61. Unit 3 · Chapter 617 min read

    Partial Differential Equations

    Laplace, heat and wave equations: separation of variables

    Separation of variables turns a PDE into a Sturm–Liouville problem plus an ODE in time, and the difference between heat and wave is entirely in that time factor — decaying exponential against oscillation.

    [⚠️ Where intuition breaks:

    "Heat and wave have different eigenfunctions."

    ]

    15 PYQs inside Mark as readRead Chapter
62. Unit 3 · Chapter 626 min read

    Numerical Analysis

    Root finding: bisection, Newton–Raphson, fixed point, order of convergence

    Bisection is slow and certain, Newton is fast and unguaranteed, and Newton's famous quadratic rate holds only at a simple root.

    [⚠️ Where intuition breaks:

    "Newton always converges."

    ]

    5 PYQs inside Mark as readRead Chapter
63. Unit 3 · Chapter 636 min read

    Numerical Analysis

    Interpolation and numerical integration with error terms

    There is exactly one polynomial of degree $\le n$ through $n+1$ points, adding equispaced points can make the approximation worse, and Simpson's rule is exact for cubics rather than for quadratics.

    [⚠️ Where intuition breaks:

    "More interpolation nodes give a better approximation."

    ]

    8 PYQs inside Mark as readRead Chapter
64. Unit 3 · Chapter 646 min read

    Numerical Analysis

    Numerical ODE: Euler, Runge–Kutta

    Euler is first order, Runge–Kutta 4 is fourth, and being higher order says nothing about being stable — an explicit method with too large a step produces growth where the true solution decays.

    [⚠️ Where intuition breaks:

    "Higher order means more stable."

    ]

    5 PYQs inside Mark as readRead Chapter
65. Unit 3 · Chapter 656 min read

    Calculus of Variations

    Euler–Lagrange equation and standard functionals

    Extremising $\int F(x,y,y')\,dx$ gives the Euler–Lagrange equation, two special forms of $F$ collapse it to a first-order equation, and a solution of it is stationary rather than minimal.

    [⚠️ Where intuition breaks:

    "An Euler–Lagrange solution is a minimum."

    ]

    16 PYQs inside Mark as readRead Chapter
66. Unit 3 · Chapter 666 min read

    Calculus of Variations

    Isoperimetric problems

    An isoperimetric problem extremises one functional subject to another being fixed, and the method is a Lagrange multiplier applied to functionals rather than to functions.

    [⚠️ Where intuition breaks:

    "Choose $\lambda$ to make the algebra convenient."

    ]

    4 PYQs inside Mark as readRead Chapter
67. Unit 3 · Chapter 677 min read

    Linear Integral Equations

    Fredholm and Volterra equations

    Fredholm equations integrate over a fixed interval and have eigenvalues; Volterra equations integrate up to the variable point and have none, so a Volterra equation of the second kind is always uniquely solvable.

    [⚠️ Where intuition breaks:

    "A Volterra equation may have no solution."

    ]

    14 PYQs inside Mark as readRead Chapter
68. Unit 3 · Chapter 686 min read

    Linear Integral Equations

    Separable kernels and resolvent kernels

    A separable kernel turns an integral equation into a finite linear system, so the whole apparatus of eigenvalues and the Fredholm alternative becomes ordinary linear algebra.

    [⚠️ Where intuition breaks:

    "A separable kernel can have infinitely many eigenvalues."

    ]

    3 PYQs inside Mark as readRead Chapter
69. Unit 3 · Chapter 697 min read

    Classical Mechanics

    Lagrangian formalism and generalised coordinates

    Write $L = T - V$ in whatever coordinates suit the constraints, apply the Euler–Lagrange equations, and the constraint forces never appear — which is the entire reason the formalism exists.

    [⚠️ Where intuition breaks:

    "$L = T + V$."

    ]

    8 PYQs inside Mark as readRead Chapter
70. Unit 3 · Chapter 707 min read

    Classical Mechanics

    Hamiltonian formalism and conservation laws

    The Legendre transform replaces $n$ second-order equations with $2n$ first-order ones on phase space, where the flow preserves volume — and $H$ equals the energy only under conditions that must be checked rather than assumed.

    [⚠️ Where intuition breaks:

    "$H = T+V$."

    ]

    3 PYQs inside Mark as readRead Chapter
71. Unit 4 · Chapter 715 min read

    Probability

    Axioms, conditional probability, independence, Bayes

    Probability is a measure of total mass one, conditioning renormalises it, and the two things that catch people are that pairwise independence is weaker than mutual, and that a positive test on a rare condition usually means nothing.

    [⚠️ Where intuition breaks:

    "A 99% accurate positive means 99% chance of disease."

    ]

    9 PYQs inside Mark as readRead Chapter
72. Unit 4 · Chapter 725 min read

    Probability

    Random variables, distributions, moments, MGF

    A distribution is determined by its CDF, the MGF determines it too when it exists near zero, and neither the mean nor the MGF is guaranteed to exist at all.

    [⚠️ Where intuition breaks:

    "Every random variable has a mean."

    ]

    15 PYQs inside Mark as readRead Chapter
73. Unit 4 · Chapter 736 min read

    Probability

    Standard discrete and continuous distributions

    Almost every named distribution is a sum, a limit, a square or a ratio of the others, and knowing which relationship connects two of them answers most questions without any integration.

    [⚠️ Where intuition breaks:

    "The $t$ distribution always has a mean and variance."

    ]

    11 PYQs inside Mark as readRead Chapter
74. Unit 4 · Chapter 745 min read

    Probability

    Joint distributions, transformations, order statistics

    Independence means the joint density factors, uncorrelated is strictly weaker except for jointly normal variables, and order statistics come from one observation — the maximum is below $x$ exactly when all of them are.

    [⚠️ Where intuition breaks:

    "Uncorrelated implies independent."

    ]

    12 PYQs inside Mark as readRead Chapter
75. Unit 4 · Chapter 757 min read

    Limit Theorems and Markov Chains

    Modes of convergence, WLLN, SLLN, CLT

    Four modes of convergence sit in a strict hierarchy, the laws of large numbers give two of them under an integrability hypothesis, and the central limit theorem needs finite variance rather than normality.

    [⚠️ Where intuition breaks:

    "Convergence in probability implies almost sure convergence."

    ]

    12 PYQs inside Mark as readRead Chapter
76. Unit 4 · Chapter 766 min read

    Limit Theorems and Markov Chains

    Markov chains: classification of states, stationary distributions

    Irreducibility makes the stationary distribution unique, positive recurrence makes it exist, and aperiodicity makes it the limit — three separate conditions doing three separate jobs.

    [⚠️ Where intuition breaks:

    "Every chain has a stationary distribution."

    ]

    17 PYQs inside Mark as readRead Chapter
77. Unit 4 · Chapter 776 min read

    Estimation

    Sufficiency, completeness, UMVUE, Cramér–Rao

    A complete sufficient statistic plus unbiasedness gives the UMVUE outright, while the Cramér–Rao bound is a separate route that needs regularity conditions and fails exactly where the support depends on the parameter.

    [⚠️ Where intuition breaks:

    "No unbiased estimator can beat the Cramér–Rao bound."

    ]

    13 PYQs inside Mark as readRead Chapter
78. Unit 4 · Chapter 786 min read

    Estimation

    MLE and method of moments

    The MLE maximises the likelihood, is invariant under reparametrisation, is asymptotically efficient — and is very often biased, sometimes on the boundary, and occasionally not unique.

    [⚠️ Where intuition breaks:

    "The MLE is unbiased."

    ]

    19 PYQs inside Mark as readRead Chapter
79. Unit 4 · Chapter 796 min read

    Hypothesis Testing

    Neyman–Pearson lemma and UMP tests

    Fix the type I error rate, then maximise power — and the Neyman–Pearson lemma says the likelihood ratio does it for a simple alternative, while for two-sided alternatives no uniformly best test exists at all.

    [⚠️ Where intuition breaks:

    "Failing to reject means $H_0$ is true."

    ]

    11 PYQs inside Mark as readRead Chapter
80. Unit 4 · Chapter 806 min read

    Hypothesis Testing

    Likelihood ratio and standard tests

    The likelihood ratio test compares the best fit under the null with the best fit overall, $-2\log\Lambda$ is asymptotically chi-square with degrees of freedom equal to the number of constraints, and every standard test is a special case.

    [⚠️ Where intuition breaks:

    "Degrees of freedom is the sample size."

    ]

    14 PYQs inside Mark as readRead Chapter
81. Unit 4 · Chapter 816 min read

    Linear Models and Multivariate

    Gauss–Markov, regression, ANOVA basics

    Gauss–Markov says least squares is best among linear unbiased estimators and needs no normality at all — normality is what buys the $t$ and $F$ tests, and the two claims are constantly merged.

    [⚠️ Where intuition breaks:

    "Gauss–Markov requires normal errors."

    ]

    19 PYQs inside Mark as readRead Chapter
82. Unit 4 · Chapter 825 min read

    Linear Models and Multivariate

    Multivariate normal distribution

    The multivariate normal is closed under every linear operation, its conditionals are normal with a linear mean, and it is the one family where zero correlation really does mean independence.

    [⚠️ Where intuition breaks:

    "Normal marginals means jointly normal."

    ]

    16 PYQs inside Mark as readRead Chapter
83. Unit 4 · Chapter 836 min read

    Sampling and Design of Experiments

    SRS, stratified and systematic sampling

    Simple random sampling is the baseline, stratification beats it when the strata differ, and systematic sampling is the only one of the three that can be worse — catastrophically so if the list has a period.

    [⚠️ Where intuition breaks:

    "Sampling with replacement is more precise."

    ]

    8 PYQs inside Mark as readRead Chapter
84. Unit 4 · Chapter 847 min read

    Sampling and Design of Experiments

    CRD, RBD, LSD essentials

    Each design removes one more source of variation from the error term, paying for it in degrees of freedom — and the degrees of freedom are where the arithmetic is tested.

    [⚠️ Where intuition breaks:

    "Blocking always improves the design."

    ]

    6 PYQs inside Mark as readRead Chapter
85. Unit 4 · Chapter 859 min read

    Linear Programming

    Linear programming, simplex and duality

    A linear programme optimises a linear objective over a polyhedron, so if an optimum exists it is attained at a vertex — and the dual programme tells you whether one exists at all.

    [⚠️ Where intuition breaks:

    An unbounded region does not mean an unbounded objective.

    ]

    5 PYQs inside Mark as readRead Chapter

Every hook and poison object above is quoted from the chapter itself.