December 2025
120 questions, booklet 9167102.
Section A · Part A · Q1–20
Practise this section →- 1Data interpretation — largest volume from a density-mass radar chartGeneral AptitudeData InterpretationMedium
- 2Conditional probability — passing one subject given the otherGeneral AptitudeQuantitative AptitudeEasy
- 3Logical reasoning — minimum pourings between three vesselsGeneral AptitudeLogical ReasoningMedium
- 4Sequences — a three-term recurrence forces period threeGeneral AptitudeQuantitative AptitudeMedium
- 5Logical reasoning — what a below-replacement fertility rate impliesGeneral AptitudeLogical ReasoningMedium
- 6Counting — equal totals from three question valuesGeneral AptitudeQuantitative AptitudeHard
- 7Series — bounding a grouped harmonic sumGeneral AptitudeQuantitative AptitudeMedium
- 8Descriptive statistics — geometric mean under scalingGeneral AptitudeQuantitative AptitudeEasy
- 9Logical reasoning — analogy between a state and its industrial cityGeneral AptitudeLogical ReasoningEasy
- 10Data interpretation — cooling curves of two identical barsGeneral AptitudeData InterpretationMedium
- 11Counting — solutions of a linear equation in two unknownsGeneral AptitudeQuantitative AptitudeMedium
- 12Logical reasoning — what a set of syllogistic premises forcesGeneral AptitudeLogical ReasoningMedium
- 13Data interpretation — path of least change on a temperature mapGeneral AptitudeData InterpretationMedium
- 14Counting — five-digit numbers divisible by 5General AptitudeQuantitative AptitudeEasy
- 15Mixtures — combining two alloys in equal quantitiesGeneral AptitudeQuantitative AptitudeMedium
- 16Geometry — minimum height of a plane mirrorGeneral AptitudeQuantitative AptitudeMedium
- 17Probability — comparing two colour sequences of a dieGeneral AptitudeQuantitative AptitudeMedium
- 18Data interpretation — largest relative change in a bar chartGeneral AptitudeData InterpretationEasy
- 19Logical reasoning — ordering five students by seniorityGeneral AptitudeLogical ReasoningMedium
- 20Number theory — next coincidence of three periodic eventsGeneral AptitudeQuantitative AptitudeEasy
Section B · Part B · Q21–60
Practise this section →- 21Set theory — which countability statement always holdsAnalysisElementary Set Theory, Real Number System, Supremum, InfimumMedium
- 22Cardinality — a Cantor-type set of restricted decimalsAnalysisElementary Set Theory, Real Number System, Supremum, InfimumMedium
- 23Sequences — limit superior of a recursively defined sequenceAnalysisSequences and SeriesMedium
- 24Limits — a difference of square roots at infinityAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 25Uniform continuity — compositions with a piecewise powerAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 26Series — convergence of a Bertrand seriesAnalysisSequences and SeriesHard
- 27Linear algebra — subspaces containing a fixed null spaceLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionHard
- 28Eigenvalues — a 3x3 matrix with given trace and determinantLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 29Matrices — equal traces of powers up to the thirdLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 30Polynomials in matrices — annihilating three matrices at onceLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremHard
- 31Inner product spaces — which property of a weighted inner product failsLinear AlgebraInner Product Spaces, Orthonormal BasisMedium
- 32Quadratic forms — what an isotropic vector forces on the signatureLinear AlgebraQuadratic Forms, Reduction and ClassificationHard
- 33Complex analysis — a functional equation with one holomorphic solutionComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsHard
- 34Residues — a vanishing contour integral around a simple poleComplex AnalysisContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaMedium
- 35Complex analysis — convergence of a series of rescaled functionsComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesHard
- 36Contour integration — integrating the tangent over two circlesComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesMedium
- 37Group theory — counting solutions of a fifteenth-power equationAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 38Group theory — how many subgroups a finite group can haveAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 39Group actions — Burnside's count of orbitsAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 40Topology — the tube lemmaTopologySeparation Axioms, Connectedness and CompactnessMedium
- 41Differential equations — Abel's identity for the WronskianOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionMedium
- 42Differential equations — uniqueness and the range of a solutionOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsHard
- 43Partial differential equations — a transport problem on a half-linePartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsHard
- 44Partial differential equations — steady state of a heat equation with a sourcePartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Medium
- 45Numerical analysis — value of an interpolating polynomialNumerical AnalysisFinite DifferencesMedium
- 46Calculus of variations — an extremal of a logarithmic functionalCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 47Integral equations — a Volterra equation with a difference kernelLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 48Partial differential equations — a forced wave equationPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Hard
- 49Probability — a strong law limit for an indicatorProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 50Probability — a random series with Bernoulli digitsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 51Markov chains — how many stationary distributionsStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionMedium
- 52Distribution theory — building a standard Cauchy from normalsDistribution TheoryStandard Discrete and Continuous Univariate DistributionsHard
- 53Confidence intervals — a pivot from a triangular densityStatistical InferenceConfidence IntervalsHard
- 54Estimation — maximum likelihood over a finite parameter setStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 55Testing — size of a test based on a sum of uniformsStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 56Bayesian inference — posterior mode for a uniform scaleStatistical InferenceElementary Bayesian InferenceMedium
- 57Linear models — variances of uncorrelated least squares estimatorsLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsHard
- 58Multivariate analysis — a multiple correlation coefficientMultivariate AnalysisPartial and Multiple Correlation CoefficientsMedium
- 59Sampling theory — comparing sampling with and without replacementSampling TheorySimple Random, Stratified and Systematic SamplingMedium
- 60Linear programming — optimum over a bounded feasible regionOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
Section C · Part C · Q61–120
Practise this section →- 61Real numbers — digits of a sequence approaching a terminating decimalAnalysisElementary Set Theory, Real Number System, Supremum, InfimumHard
- 62Supremum and infimum — bounds of a set defined by parityAnalysisElementary Set Theory, Real Number System, Supremum, InfimumMedium
- 63Integrals — properties of a function defined by an integralAnalysisRiemann Sums and Riemann Integral, Improper IntegralsMedium
- 64Sequences of functions — a spike that destroys pointwise convergenceAnalysisSequences and SeriesMedium
- 65Uniform convergence — where a steepening ramp converges uniformlyAnalysisSequences and SeriesMedium
- 66Monotone functions — limits, jumps and integrabilityAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 67Improper integrals — when convergence forces the integrand to vanishAnalysisRiemann Sums and Riemann Integral, Improper IntegralsHard
- 68Several variables — a homogeneous quotient at the originAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 69Implicit function theorem — which variable can be solved forAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsHard
- 70Topology — an uncountable product of intervalsTopologySeparation Axioms, Connectedness and CompactnessHard
- 71Vector spaces — spanning sets, independence and basesLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 72Linear operators — the algebra generated by an idempotentLinear AlgebraAlgebra of Linear TransformationsHard
- 73Diagonalizability — what the square of an operator revealsLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremHard
- 74Eigenvalues — rational polynomials vanishing on the spectrumLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremHard
- 75Matrices — commuting and anticommuting with an involutionLinear AlgebraAlgebra of Linear TransformationsHard
- 76Linear operators — eigenvectors and invariant subspaces over CLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 77Hilbert spaces — the Hilbert cube inside ell^2Linear AlgebraInner Product Spaces, Orthonormal BasisHard
- 78Matrix spaces — which ranks span under the trace inner productLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsHard
- 79Complex functions — mixing a holomorphic and a modulus termComplex AnalysisAnalytic Functions, Cauchy-Riemann EquationsMedium
- 80Schwarz-Pick — sharp bounds for a self-map of the discComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremHard
- 81Entire functions — subtracting a simple poleComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesHard
- 82Contour integration — a double pole with zero residueComplex AnalysisContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaMedium
- 83Finite groups — consequences of being non-abelianAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 84Pigeonhole — repeated borrowings in a calendar monthAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionHard
- 85Groups of order 8 — what elements of order 4 decideAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 86Rings — when divisibility is a total relationAlgebraRings, Ideals, UFD, PID, Euclidean DomainHard
- 87Polynomials — testing irreducibility in two ringsAlgebraPolynomial Rings and Irreducibility CriteriaMedium
- 88Galois theory — an irreducible degree-p polynomial with two complex rootsAlgebraPolynomial Rings and Irreducibility CriteriaHard
- 89Metric spaces — distance to a setAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 90Topology — comparing topologies on the real lineTopologySeparation Axioms, Connectedness and CompactnessMedium
- 91Frobenius method — comparing solutions at a regular singular pointOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionHard
- 92Oscillation — zeros of solutions of a second order equationOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionHard
- 93Series solutions — a smooth but non-analytic forcing termOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsHard
- 94Harmonic functions — the Dirichlet problem on the discPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Medium
- 95Damped waves — an energy that decreasesPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Hard
- 96Numerical analysis — convergence of Gauss-Seidel for a 2x2 systemNumerical AnalysisNumerical Solutions of Algebraic Equations, Newton-Raphson, Rate of ConvergenceMedium
- 97Newton-Raphson — order of convergence at a double rootNumerical AnalysisNumerical Solutions of Algebraic Equations, Newton-Raphson, Rate of ConvergenceHard
- 98Calculus of variations — an isoperimetric constraintCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaHard
- 99Calculus of variations — an extremal that is a straight lineCalculus of VariationsVariational Methods for Boundary Value ProblemsMedium
- 100Integral equations — a Fredholm equation with a constant kernelLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 101Integral equations — eigenfunction of a degenerate kernelLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelHard
- 102Classical mechanics — canonical transformations and generating functionsClassical MechanicsHamilton's Canonical Equations, Hamilton's PrincipleHard
- 103Descriptive statistics — symmetrising a datasetDescriptive StatisticsDescriptive StatisticsMedium
- 104Distribution functions — a mixture of a discrete and a continuous lawProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 105Markov chains — classifying states of a four-state chainStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionMedium
- 106Probability — sum and difference of two diceProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremMedium
- 107Extreme values — the Gumbel limit for exponential maximaDistribution TheoryDistribution of Order Statistics and RangeHard
- 108Moment generating functions — identifying two distributionsProbabilityCharacteristic Functions, Probability InequalitiesMedium
- 109Estimation — moments and likelihood for a shifted uniformStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 110Consistency — estimating a ratio in a shifted exponentialStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 111Likelihood ratio test — a one-sided test for an exponential scaleStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 112Sufficiency and ancillarity — a regression with one parameterStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 113Non-parametric inference — the sign test and its confidence intervalNon-parametric InferenceRank and Sign Tests, Run TestMedium
- 114Bayesian inference — posterior mean against prior meanStatistical InferenceElementary Bayesian InferenceHard
- 115Linear models — minimum mean squared error against least squaresLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsHard
- 116Linear models — distributions of the regression sums of squaresLinear Models & RegressionAnalysis of Variance and Covariance, Fixed/Random/Mixed EffectsHard
- 117Multivariate analysis — Wishart facts for a normal sampleMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 118Design of experiments — effects confounded in a 2^5 designDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionHard
- 119Survival analysis — a series system of four componentsSurvival AnalysisSeries and Parallel SystemsMedium
- 120Queueing theory — an M/M/2 system with finite capacityOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Medium