April 2024
180 questions, booklet APR-30224.
Section I · General (compulsory) · Q1–20
Practise this section →- 1Sets — evaluating 1+(-1)^nAnalysisElementary Set Theory, Real Number System, Supremum, InfimumEasy
- 2Combinatorics — subsets built from even elementsAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 3Topology — boundedness of continuous functions characterises compactnessTopologySeparation Axioms, Connectedness and CompactnessHard
- 4Integration — Riemann vs. Lebesgue, what discontinuity does not doAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 5Real functions — what survives under fmapstofAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 6Set sequences — liminf/limsup of shrinking irrational setsAnalysisSequences and SeriesMedium
- 7Piecewise functions — computing a preimage setAnalysisElementary Set Theory, Real Number System, Supremum, InfimumMedium
- 8Cardinality — sequences over a four-letter alphabetAnalysisSequences and SeriesHard
- 9Uniform continuity — checking three named functionsAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 10Double limits — the standard axes counterexampleAnalysisSequences and SeriesHard
- 11Linear algebra — real dimension of complex upper-triangular matricesLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 12Matrix rank — AB versus BA for rectangular A,BLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 13Linear systems — consistency of an underdetermined systemLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsHard
- 14Matrix rank — Sylvester's inequalityLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 15Linear transformations — rank–nullity for a functionalLinear AlgebraAlgebra of Linear TransformationsEasy
- 16Linear transformations — properties of a projectionLinear AlgebraAlgebra of Linear TransformationsMedium
- 17Matrices — singular values via AA^mathrm TLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 18Similar matrices — everything polynomial is preservedLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsHard
- 19Nilpotent matrices — shifted invertibilityLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsHard
- 20Orthogonal matrices — forcing an eigenvalue of 1Linear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsHard
Section II · Mathematics · Q21–100
Practise this section →- 21Complex analysis — where 1/z has a primitiveComplex AnalysisContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaHard
- 22Complex logarithm — two familiar-looking identities that breakAnalysisSequences and SeriesHard
- 23Roots of unity — which power sum failsComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsHard
- 24Harmonic conjugates — mutual conjugacy forces constantsComplex AnalysisAnalytic Functions, Cauchy-Riemann EquationsHard
- 25Recognising an automorphism of the unit diskComplex AnalysisConformal Mappings, Mobius TransformationsHard
- 26Recovering f from its values on a convergent sequenceAnalysisSequences and SeriesMedium
- 27Identity theorem — a sequence of zeros needs to accumulateAnalysisSequences and SeriesHard
- 28Liouville-type consequences for f=u+iv entireAnalysisSequences and SeriesHard
- 29Injective entire functions — forced to be linearComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 30Injectivity on a small punctured disk — e^z versus e^1/zComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesHard
- 31Argument principle — integrating P'/P around all the rootsComplex AnalysisContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaMedium
- 32Residue at a third-order poleComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesMedium
- 33Residue from the Laurent series of (e^z-1)/z^2Complex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 34Inversion of a circle through the poleComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 35Eisenstein's criterion — and its limitsAlgebraPolynomial Rings and Irreducibility CriteriaMedium
- 36Ring theory — four claims, one survivesAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 37Finite commutative rings — Sylow components as idealsAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 38Irreducible but not prime — the classic mathbf Z[sqrt-5] witnessAlgebraRings, Ideals, UFD, PID, Euclidean DomainHard
- 39Prime vs. maximal ideals — where the general claim breaksAlgebraRings, Ideals, UFD, PID, Euclidean DomainHard
- 40Coprime power maps on a finite abelian groupAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 41The inversion map — a homomorphism exactly on abelian groupsAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionHard
- 42Conjugacy classes — counting them, and what centrality really givesAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 43Conjugacy in An — when Sn-classes splitAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionHard
- 44Embedding finite groups in alternating groups; Sn is not nilpotentLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremHard
- 45Counting arrangements of distinct lettersAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 46Pigeonhole — forcing a shared day of the weekAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionMedium
- 47Finite fields — the sum and product of all elementsAlgebraFields, Finite Fields, Field Extensions, Galois TheoryHard
- 48Finite fields — the canonical splitting-field descriptionAlgebraFields, Finite Fields, Field Extensions, Galois TheoryHard
- 49Riesz's theorem — compactness of the unit ball characterises finite dimensionAnalysisSequences and SeriesMedium
- 50The circle is not simply connected; the square disk isAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 51A homeomorphic image of a disconnected unionTopologySeparation Axioms, Connectedness and CompactnessHard
- 52Homeomorphism type of plane curves — compact loops versus unbounded arcsComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsMedium
- 53The precise definition of a regular spaceTopologyBasis, Dense Sets, Subspace and Product TopologyHard
- 54What every metric space guarantees — and what it doesn'tAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 55Connectedness — what a shared point buys you (and what it doesn't)TopologySeparation Axioms, Connectedness and CompactnessHard
- 56Constant functions inside C[0,1] — closed, not dense, not compactAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 57Subspace topology — what transfers cleanly, and what doesn'tTopologyBasis, Dense Sets, Subspace and Product TopologyHard
- 58The union of the two coordinate axes is connectedTopologySeparation Axioms, Connectedness and CompactnessMedium
- 59Continuous image of a disk — can't be two disjoint piecesAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 60Half-open squares — when a reflection gives a homeomorphism, and when nothing doesTopologySeparation Axioms, Connectedness and CompactnessHard
- 61Euler equation — no value of K actually gives c1+c2ln xOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionHard
- 62A textbook Peano non-uniqueness exampleAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 63Series solution — the printed key does not match the recurrenceOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsHard
- 64Linear independence on subintervals — ODE solutions are specialLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionHard
- 65Lipschitz condition — bounded rectangle versus unbounded stripAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 66Wronskian of a triple root — Abel's formula, computed directlyLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionHard
- 67An IVP with a knife-edge domain restrictionAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 68Lagrange's PDE — spotting the invariant that isn't actually invariantPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Hard
- 69The PDE of every surface of revolution about the x-axisPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 70Compatibility of a pair of first-order PDEsPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Hard
- 71Envelope of a family of unit spheres — the characteristic is a great circlePartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsHard
- 72Naming the arbitrary-function solution of a first-order PDEPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 73Classifying a variable-coefficient second-order PDEPartial Differential EquationsClassification of Second Order PDEsEasy
- 74Classifying uxx+yuyy=0 on y0Partial Differential EquationsClassification of Second Order PDEsEasy
- 75The compatibility condition for two first-order PDEsPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Hard
- 76Finite differences — the averaging operator in terms of the shiftNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 77Picard's method — the first successive approximationOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 78The first divided difference of 1/x^2Numerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 79Runge–Kutta order 4 — identifying the K3 stageOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsMedium
- 80Diagonal dominance — the precise (absolute-value) definitionNumerical AnalysisSolution of Linear Systems: Gauss Elimination and Gauss-SeidelHard
- 81Finite differences — the nth difference of a degree-n polynomialNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationMedium
- 82Minimal surface of revolution — the catenary equationCalculus of VariationsVariational Methods for Boundary Value ProblemsMedium
- 83Isoperimetric problem — Euler–Lagrange with a Lagrange multiplierCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 84First integral of a second-order Euler–Lagrange (Euler–Poisson) equationCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaHard
- 85Solving a boundary value problem via Euler–LagrangeCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 86Solutions of a y"-dependent functional form a four-parameter familyOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionMedium
- 87Coercive functionals on a Banach spaceAnalysisSequences and SeriesMedium
- 88Classifying an integral equation — fixed limits, unknown outside the integralLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 89A separable-kernel eigenvalue problemLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 90Volterra equation — converting to an initial value problemAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 91Fredholm alternative — the two faces of the Fredholm determinantLinear AlgebraAlgebra of Linear TransformationsHard
- 92Solving a Volterra equation by differentiating twiceAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 93Eigenvalues from a separable trigonometric kernelLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 94Projectile motion with linear air resistanceClassical MechanicsMotion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsHard
- 95Rotating-frame Lagrangian — the Hamiltonian is conserved, but isn't the energyClassical MechanicsGeneralized Coordinates, Lagrange's EquationsHard
- 96Lagrange's equation of motion — rate of change of momentumClassical MechanicsGeneralized Coordinates, Lagrange's EquationsHard
- 97Gauge freedom in the Lagrangian — adding a total time derivativeClassical MechanicsGeneralized Coordinates, Lagrange's EquationsHard
- 98Circular motion on a string — energy conservation from the lowest pointClassical MechanicsMotion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsMedium
- 99The simple pendulum — when H and E genuinely coincideClassical MechanicsGeneralized Coordinates, Lagrange's EquationsMedium
- 100Orthogonal matrix — the inverse is the transposeLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsHard
Section III · Statistics · Q101–180
Practise this section →- 101Exponential distribution — what order statistics do (and don't) inheritDistribution TheoryStandard Discrete and Continuous Univariate DistributionsHard
- 102Correlation — geometric vs. arithmetic mean of the regression coefficientsLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionHard
- 103Grouped data — ogive intersection is the medianDescriptive StatisticsDescriptive Statistics, Exploratory Data AnalysisMedium
- 104Inclusion–exclusion — minimum possible joint attendanceAlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionHard
- 105Set theory — the field generated by one setAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 106Measurability — what forward images do not preserveProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 107Measurability — functions relative to a coarse sigma-fieldProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 108Classical probability — pairwise independence and a unionProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 109Joint distributions — a table that happens to factorProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 110Independent testers — a false "all-of-the-above"ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 111Events with P(E)<P(F) and overlap — no implication survivesProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremHard
- 112Markov's inequality — applied to the fourth momentProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 113Characteristic functions — recognising the standard Laplace lawProbabilityCharacteristic Functions, Probability InequalitiesMedium
- 114Almost-sure vs. pointwise convergence — a single exceptional pointProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 115A vanishing-probability spike — Borel–Cantelli settles a.s. convergenceProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 116Scaling a fixed random variable — convergence is to X, not to 0ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 117Continuous mapping theorem — probability + distribution still falls short of a.s.ProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 118Strong law — applied to the squared observationsProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 119Markov chains — the two equivalent faces of null persistenceStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 120Doubly stochastic matrices — solving for a missing entryStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 121Markov chain simulation — inverting the cumulative rowStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 122Markov chains — a self-contained pair of statesStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 123Poisson process — probability of a silent intervalStochastic ProcessesPoisson and Birth-and-Death ProcessesEasy
- 124Pure birth process with constant rate — recognising a Poisson processStochastic ProcessesPoisson and Birth-and-Death ProcessesHard
- 125Matching means — Binomial is always less dispersed than PoissonProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 126Order statistics — the maximum of two uniforms is Beta(2,1)Distribution TheoryDistribution of Order Statistics and RangeEasy
- 127Binomial splitting — the hypergeometric conditional lawDistribution TheoryStandard Discrete and Continuous Univariate DistributionsHard
- 128Order statistics of Uniform(0,theta) — a theta-free conditional lawDistribution TheoryStandard Discrete and Continuous Univariate DistributionsHard
- 129A shift identity — the mean of a truncated variableProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 130Geometric distribution — solving for p from the varianceProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 131Unbiased estimation — lambda^2 from a Poisson sampleDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionHard
- 132Cramer–Rao bound — Fisher information of a shifted CauchyStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 133Size of a critical region — summing a binomial tailStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 134Sufficiency, ancillarity and the Cramer–Rao bound — reading Basu's theorem carefullyStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 135Hypothesis testing — the one statement that survives scrutinyStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 136MVUE for sigma^2 — known mean removes the Bessel correctionStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 137Setting up a one-sided test — the claim goes in H1ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 138Exponential families — spotting the curved caseProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 139Consistency — rescaling makes the extremes behaveStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 140Asymptotic normality — independence of bar Xn and Sn^2 under normalityMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 141Testing a Bernoulli proportion — the Wald statisticStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 142MLE — the invariance propertyStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 143Contingency tables — expected frequency under independenceStatistical InferenceAnalysis of Discrete Data and Chi-Square Test of Goodness of FitMedium
- 144Choosing a test for the median — nonparametric, small sampleStatistical InferenceNonparametric Tests, Rank Correlation, Test for IndependenceHard
- 145Bayesian binomial — Beta posterior from a uniform priorStatistical InferenceElementary Bayesian InferenceEasy
- 146Bayesian Poisson — a Gamma prior and its posterior meanStatistical InferenceElementary Bayesian InferenceHard
- 147Estimability in the general linear model — the row-space characterisationLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsHard
- 148Estimable functions — a rank-deficient three-parameter modelProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 149Reading a partially filled ANOVA tableLinear Models & RegressionAnalysis of Variance and Covariance, Fixed/Random/Mixed EffectsMedium
- 150Overparametrised one-way ANOVA — what is and isn't estimableLinear Models & RegressionAnalysis of Variance and Covariance, Fixed/Random/Mixed EffectsHard
- 151Regression residuals — correlated and heteroscedastic even when the errors aren'tLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionHard
- 152Covariance of y and hat y under least squaresLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 153Orthogonality of LS residuals to the regressorDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDMedium
- 154Logistic regression — odds ratio for a pm1-coded covariateLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 155Multivariate normal — correlation between two linear combinationsMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsMedium
- 156Bivariate normal — shape of the conditional mean and varianceProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 157Quadratic forms in normal vectors — when A collapses to the identityStatistical InferenceAnalysis of Discrete Data and Chi-Square Test of Goodness of FitHard
- 158Quadratic forms — checking positive definiteness via leading minorsProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 159A quadratic form that collapses to a perfect squareProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 160Partial correlation — the residual-correlation identityMultivariate AnalysisPartial and Multiple Correlation CoefficientsMedium
- 161Euclidean distance to a cluster centreMultivariate AnalysisData Reduction: PCA, Discriminant, Cluster AnalysisEasy
- 162What PCA actually reducesMultivariate AnalysisData Reduction: PCA, Discriminant, Cluster AnalysisMedium
- 163Sampling theory — spotting a formula with a missing factorSampling TheorySimple Random, Stratified and Systematic SamplingHard
- 164Inclusion probabilities — they always sum to the expected sample sizeProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 165Ratio estimation — from a size-2 SRSWOR sampleSampling TheoryRatio and Regression MethodsMedium
- 166Optimum (Neyman–cost) allocation in stratified samplingOperations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 167Systematic sampling — decomposing total variabilitySampling TheorySimple Random, Stratified and Systematic SamplingMedium
- 168BIBD parameters — underdetermined without lambdaDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDHard
- 169CRD — error degrees of freedom with unequal group sizesDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsEasy
- 170Two-level ANOVA — equivalent to a two-sided t-testLinear Models & RegressionAnalysis of Variance and Covariance, Fixed/Random/Mixed EffectsHard
- 171The C-matrix of a block design — rank deficient by constructionDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDHard
- 172CRD — variance of a group mean minus the grand meanDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsMedium
- 173Exponential reliability — a scale-free coincidenceSurvival AnalysisHazard Function and Failure Rates, Censoring and Life TestingMedium
- 174Reliability of a series-parallel systemSurvival AnalysisSeries and Parallel SystemsMedium
- 175Type-II censored exponential data — the MLE of the meanSurvival AnalysisHazard Function and Failure Rates, Censoring and Life TestingHard
- 176Simplex sensitivity — re-checking optimality after a cost changeOperations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 177Graphical LPP — identifying a redundant constraintOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 178Assignment problem — solving a 4times4 cost matrixOperations ResearchTransportation and Assignment ProblemsHard
- 179Inventory costs — which component moves with order quantityOperations ResearchElementary Queuing and Inventory ModelsMedium
- 180M/M/1 queue — computing the standard performance measuresOperations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Hard