June 2025
180 questions, booklet 2025 JUN PII.
Section I · General (compulsory) · Q1–20
Practise this section →- 1Metric spaces — connectednessAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 2Metric spaces — the agreement setAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 3Metric spaces — what can never happenAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 4Series — divergence by comparisonAnalysisSequences and SeriesMedium
- 5Sequences — when the liminf/limsup chain is not strictAnalysisSequences and SeriesHard
- 6Uniform convergence on [0,1] versus [0,tfrac12]AnalysisSequences and SeriesHard
- 7Differentiation — evaluating a derivative expressionAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 8Two variables — partials can exist without continuityAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsHard
- 9Continuity — removing a 0/0 indeterminacyAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 10Counting subsetsAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 11Vector spaces — dimension statementsAnalysisElementary Set Theory, Real Number System, Supremum, InfimumHard
- 12Linear maps — reconstructing T from its valuesLinear AlgebraAlgebra of Linear TransformationsHard
- 13Matrix geometric seriesLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 14Determinants — multiplicativity and scalingLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 15Linear systems — consistencyLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsHard
- 16Eigenvalues of a polynomial in ALinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 17Nilpotent matricesLinear AlgebraMatrix Representation, Change of Basis, Canonical and Jordan FormsHard
- 18The all-ones matrixLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremHard
- 19The adjoint (adjugate) matrixLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsHard
- 20Quadratic form — definiteness by leading minorsLinear AlgebraQuadratic Forms, Reduction and ClassificationMedium
Section II · Mathematics · Q21–100
Practise this section →- 21Complex analysis — modulus of an exponentialComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 22Complex analysis — stereographic projectionComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 23Complex analysis — harmonic parts and reflectionComplex AnalysisAnalytic Functions, Cauchy-Riemann EquationsMedium
- 24Complex analysis — a removable singularityComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 25Complex analysis — Schwarz's lemmaComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 26Complex analysis — Cauchy's integral formulaComplex AnalysisContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaEasy
- 27Complex analysis — disk to right half-planeComplex AnalysisConformal Mappings, Mobius TransformationsEasy
- 28Complex analysis — radii of convergenceComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 29Complex analysis — when an analytic function is constantComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 30Complex analysis — conformality on the unit diskComplex AnalysisConformal Mappings, Mobius TransformationsMedium
- 31Complex analysis — zeros of entire and analytic functionsComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 32Complex analysis — logarithms of non-vanishing functionsComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsHard
- 33Complex analysis — Casorati–WeierstrassComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesMedium
- 34Complex analysis — a square root of 1+iComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 35Algebra — minimal polynomial of sqrt2+sqrt3AlgebraPolynomial Rings and Irreducibility CriteriaEasy
- 36Algebra — Galois group of an irreducible cubicAlgebraFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 37Algebra — subfields of mathbb F64AlgebraFields, Finite Fields, Field Extensions, Galois TheoryEasy
- 38Algebra — finite fields are never algebraically closedAlgebraFields, Finite Fields, Field Extensions, Galois TheoryEasy
- 39Algebra — irreducibility of x^3-9 modulo nAlgebraPolynomial Rings and Irreducibility CriteriaMedium
- 40Algebra — subrings of UFDs and gcdsAlgebraRings, Ideals, UFD, PID, Euclidean DomainMedium
- 41Algebra — small non-commutative ringsAlgebraRings, Ideals, UFD, PID, Euclidean DomainMedium
- 42Group theory — SL2(Z/2Z)cong S3AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 43Group theory — normality from cyclic subgroups; index 2AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 44Group theory — the structure of A4AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 45Group theory — groups of order p^2: no option always holdsAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 46Number theory — the Frobenius number of 3 and 8AlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremMedium
- 47Number theory — units of Z12AlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 48Group theory — conjugacy in S4 is cycle typeAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 49Analysis — subsets of ell^infty: every option failsAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 50Topology — (-1,1)congRcong a parabolaTopologyBasis, Dense Sets, Subspace and Product TopologyEasy
- 51Functional analysis — an evaluation condition defines an open setAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 52Topology — a two-valued map and connectedness: only one direction holdsTopologySeparation Axioms, Connectedness and CompactnessEasy
- 53Topology — compactness in a discrete spaceTopologySeparation Axioms, Connectedness and CompactnessEasy
- 54Analysis — a closed dense subset of RAnalysisElementary Set Theory, Real Number System, Supremum, InfimumEasy
- 55Topology — products and punctured circlesTopologySeparation Axioms, Connectedness and CompactnessMedium
- 56Topology — normality via shrinking neighbourhoodsTopologySeparation Axioms, Connectedness and CompactnessMedium
- 57Topology — the curve x^2y^2=1TopologySeparation Axioms, Connectedness and CompactnessEasy
- 58Topology — comparing metric topologiesTopologyBasis, Dense Sets, Subspace and Product TopologyMedium
- 59Topology — the upper-limit topologyTopologyBasis, Dense Sets, Subspace and Product TopologyMedium
- 60Analysis — Cantor's nested interval theoremAnalysisElementary Set Theory, Real Number System, Supremum, InfimumEasy
- 61ODE — eliminating a constantOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 62ODE — Abel's formula gives W(1)=e, not an optionOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 63ODE — a Lipschitz constant on a rectangleOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 64ODE — stability of constant-coefficient equationsOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionMedium
- 65ODE — the logistic equation tends to its stable equilibriumOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 66ODE — integrating factor e^x^2Ordinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 67ODE — a linear combination that decouples the systemOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsMedium
- 68PDE — surfaces orthogonal to xyz=cPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 69PDE — data on a characteristic curvePartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 70PDE — uniqueness of general and complete integralsPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 71PDE — characteristic curves and envelopesPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 72PDE — integrability of Pfaffian equationsPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 73PDE — where yuxx+2xyuxy+xuyy is hyperbolicPartial Differential EquationsClassification of Second Order PDEsEasy
- 74PDE — envelope of planes through the originPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 75PDE — types of the classical equationsPartial Differential EquationsClassification of Second Order PDEsEasy
- 76Numerical analysis — nablaDelta=Delta-nabla=delta^2Numerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 77Numerical analysis — nth difference of a degree-n polynomialNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 78Numerical analysis — divided differences are linearNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 79Numerical analysis — composite trapezoidal error is O(h^2)Numerical AnalysisNumerical Differentiation and IntegrationEasy
- 80Numerical analysis — differences of an exponentialNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 81Numerical analysis — composite Simpson's one-third ruleNumerical AnalysisNumerical Differentiation and IntegrationEasy
- 82Calculus of variations — polar arc lengthCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 83Calculus of variations — geodesics in R^3Calculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 84Calculus of variations — speed v=x gives circles centred on the y-axisCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 85Calculus of variations — Euler's equation when F=F(x,y')Calculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 86Calculus of variations — a free end-point adds a natural boundary conditionCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 87Optimisation — coercive functions grow at infinityCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 88Integral equations — a separable kernel with no real eigenvaluesLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 89Integral equations — Abel's equationLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 90Integral equations — BVP to Fredholm equationLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 91Integral equations — eigenvalues of cos(t+s) on [0,pi]Linear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 92Integral equations — resolvent of a^t-sLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 93Integral equations — a variable lower limit makes it Volterra, second kindLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 94Mechanics — with air resistance, H is the energy but is not conservedClassical MechanicsHamilton's Canonical Equations, Hamilton's PrincipleMedium
- 95Mechanics — canonical momentum on a parabolaClassical MechanicsHamilton's Canonical Equations, Hamilton's PrincipleMedium
- 96Mechanics — Euler's theorem on homogeneous parts of TClassical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 97Mechanics — the energy relation is a consequence, not a restatementClassical MechanicsGeneralized Coordinates, Lagrange's EquationsMedium
- 98Mechanics — velocity perpendicular to gravity at the topClassical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 99Mechanics — adding a total time derivative to LClassical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 100Mechanics — the moment of inertia about the x-axisClassical MechanicsMotion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsEasy
Section III · Statistics · Q101–180
Practise this section →- 101Descriptive statistics — AM and GPDescriptive StatisticsDescriptive Statistics, Exploratory Data AnalysisMedium
- 102Scales of measurementDescriptive StatisticsDescriptive Statistics, Exploratory Data AnalysisHard
- 103Quartile deviationDescriptive StatisticsDescriptive Statistics, Exploratory Data AnalysisMedium
- 104Bayes' theoremProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremMedium
- 105Conditional distribution of a binomial given the sumProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 106Jump points of a distribution functionStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 107Independence of eventsProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 108Bayes' theorem — the truthful witnessProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremMedium
- 109Total probability and Bayes' theoremProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 110Which transformations of a c.d.f. are c.d.f.sProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 111Symmetry about the medianProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 112Jensen's inequality — direction of the inequalityProbabilityCharacteristic Functions, Probability InequalitiesHard
- 113Recognising a characteristic functionProbabilityCharacteristic Functions, Probability InequalitiesHard
- 114Borel–Cantelli lemmasProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 115Continuous mapping theoremProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 116Indicators of vanishing eventsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 117Slutsky's theoremProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 118What convergence in distribution does and does not giveProbabilityCharacteristic Functions, Probability InequalitiesHard
- 119Stationary distributions of two-state chainsStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 120First-passage probabilities and closed classesStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 121Transience via the return seriesStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 122First-return decomposition for a periodic chainStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 123Poisson process — waiting for the first arrivalStochastic ProcessesPoisson and Birth-and-Death ProcessesHard
- 124The Yule–Furry pure birth processProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 125Normalising a geometric-type p.m.f. on even integersProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 126Probability generating function of a linear transformProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 127Mean and variance of a mixture distributionDistribution TheoryStandard Discrete and Continuous Univariate DistributionsHard
- 128An m.g.f. identity forcing symmetry about 5ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 129Expectation from a c.d.f. — discrete and continuousProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 130Ratios of exponentials — F and BetaProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 131Sufficiency for the Bernoulli parameterProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 132UMP tests are unbiasedStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 133Existence of UMP tests: one-sided versus two-sidedStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 134Minimal sufficiency for N(theta,theta)Statistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 135MLE at the boundary of a theta-dependent supportStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 136Rao–Blackwell versus Lehmann–Scheff\'eStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 137Consistency of mean and median for U(theta-1,theta+1)Statistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 138Which families are one-parameter exponential familiesDistribution TheoryStandard Discrete and Continuous Univariate DistributionsHard
- 139Consistency in the N(mu,mu^2) modelStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 140Fisher's variance-stabilising z-transformationDistribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsHard
- 141Asymptotic confidence interval for the exponential meanStatistical InferenceConfidence IntervalsHard
- 142Definition of an unbiased testStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 143Pearson's chi-square for a fully specified multinomialStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 144Non-parametric analogue of the goodness-of-fit testStatistical InferenceNonparametric Tests, Rank Correlation, Test for IndependenceHard
- 145The Bayes factorProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 146Poisson likelihood with an exponential priorStatistical InferenceElementary Bayesian InferenceHard
- 147Non-estimable linear parametric functionsStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 148Identifying estimable functions from the designProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 149Completing a two-way ANOVA tableLinear Models & RegressionAnalysis of Variance and Covariance, Fixed/Random/Mixed EffectsHard
- 150What ANOVA actually testsLinear Models & RegressionAnalysis of Variance and Covariance, Fixed/Random/Mixed EffectsHard
- 151Variance of a prediction in simple linear regressionLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionHard
- 152Which mean squares are unbiased for sigma^2Linear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionHard
- 153What the adjusted R^2 adjusts forLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionHard
- 154Degrees of freedom for the deviance goodness-of-fit testLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionHard
- 155A conditional variance that is really an independence checkProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 156Conditional distribution in a trivariate normalProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 157Likelihood ratio test for a multivariate normal meanMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 158Definiteness of a quadratic formLinear AlgebraQuadratic Forms, Reduction and ClassificationHard
- 159When a quadratic form is not chi-squareStatistical InferenceAnalysis of Discrete Data and Chi-Square Test of Goodness of FitHard
- 160Feasible range of a third correlationProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 161Mahalanobis distance between two normal populationsMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 162Choosing the number of principal componentsMultivariate AnalysisData Reduction: PCA, Discriminant, Cluster AnalysisHard
- 163Sample size for a specified margin of errorSampling TheorySimple Random, Stratified and Systematic SamplingHard
- 164PPSWOR — probability at the third drawSampling TheoryProbability Proportional to Size SamplingHard
- 165The Horvitz–Thompson estimatorSampling TheorySimple Random, Stratified and Systematic SamplingHard
- 166Post-stratification versus proportional allocationSampling TheorySimple Random, Stratified and Systematic SamplingHard
- 167When a stratum deserves a larger sampleProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 168Balanced incomplete block designsDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDHard
- 169The C matrix of an incomplete block designDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDHard
- 170Checking BIBD parametersDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDHard
- 171Completing a Latin square ANOVADesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsHard
- 172Orthogonality in two-way data and orthogonal contrastsDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDHard
- 173Characterising the hazard rateDemography / Vital StatisticsLife Tables, Mortality RateHard
- 174Failure rate of a series systemSurvival AnalysisHazard Function and Failure Rates, Censoring and Life TestingHard
- 175Likelihood under Type I censoringSurvival AnalysisHazard Function and Failure Rates, Censoring and Life TestingHard
- 176Forming the dual of a linear programmeOperations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 177Solving a small LPP by substitutionOperations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 178An LPP with unrestricted variablesOperations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 179M/M/1 waiting probabilitiesOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Hard
- 180Stationary distribution of M/M/1Operations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Hard