January 2018
145 questions, booklet JAN-30318.
Section I · General (compulsory) · Q1–5
Practise this section →- 1Along the y-axis f=pm1: no partials, no continuity, no option trueAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 2Directional derivative =nabla fcdothat v with hat v=(12,5)/13AnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsEasy
- 3f=x on Q, 0 off Q: squeezed to 0 only at the originAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 4The difference quotient at 0 lowers each degree by one: a shift matrixLinear AlgebraAlgebra of Linear TransformationsMedium
- 5Orthonormal columns make M orthogonal — not symmetricLinear AlgebraInner Product Spaces, Orthonormal BasisEasy
Section II · Mathematics · Q6–75
Practise this section →- 6The Laplacian in polar coordinatesAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsEasy
- 7A critical point with a positive definite HessianAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 8inta^b(x-a)^-ndx behaves like a p-integral: finite iff n<1AnalysisRiemann Sums and Riemann Integral, Improper IntegralsEasy
- 9x^2+y^2+z^2=(x+y+z)^2-2(xy+yz+zx)AnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 10A finite Boolean ring is a power of Z2AlgebraRings, Ideals, UFD, PID, Euclidean DomainMedium
- 11In a finite ring the additive order of 1 is finiteAlgebraRings, Ideals, UFD, PID, Euclidean DomainMedium
- 12mathbb Fp(x) is an infinite field containing ZpAlgebraFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 13Z[x] is a UFD in which (2,x) is not principalAlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 14Q[sqrt2] sits inside RsubsetCAlgebraRings, Ideals, UFD, PID, Euclidean DomainMedium
- 15Ratio test: the 2^n in the denominator sets the radius to 2Complex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 16Casorati–Weierstrass: only an essential singularity comes arbitrarily close to every valueComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesMedium
- 17z takes only real values, so its image of an open set is never open in CComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremEasy
- 18A M\"obius map omits a/c, its value at infinityComplex AnalysisConformal Mappings, Mobius TransformationsEasy
- 19sec z has a pole at every zero of cos zComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 20The topologist's sine curve: connected but not path connectedTopologySeparation Axioms, Connectedness and CompactnessMedium
- 21Heine–Borel in R^2: the parabola is unbounded, the sine curve not closedAnalysisSequences and SeriesEasy
- 22Compact iff complete and totally boundedAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 23Jordan decomposition, and monotone functions are Borel measurableAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 24Dirichlet's function is Lebesgue- but not Riemann-integrable; 1/x is neitherAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 25int0^1x^-1/3dx=tfrac32AnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 26Groups of order p^2 are abelianAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 27S5 contains the simple non-abelian group A5AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 28Infinitely many variables give a strictly increasing chain of idealsAlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 29Q(sqrt[3]2,omega) has degree 3cdot2=6AlgebraFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 30A regular heptagon is not constructible: 7 is not a Fermat primeAlgebraFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 31A linear span need not be closed in an infinite-dimensional spaceAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 32Riesz: a compact closed unit ball forces finite dimensionAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 33In a separable space an orthonormal set is countableLinear AlgebraInner Product Spaces, Orthonormal BasisMedium
- 34Every separable infinite-dimensional Hilbert space is isometric to ell^2Linear AlgebraInner Product Spaces, Orthonormal BasisEasy
- 35A separable dual forces a separable space — not converselyAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 36The intermediate value theorem gives a fixed point on [0,1]AnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 37An indiscrete space with two points cannot separate themTopologySeparation Axioms, Connectedness and CompactnessEasy
- 38Boundedness is not topological: arctan makes R boundedTopologyBasis, Dense Sets, Subspace and Product TopologyMedium
- 39Local connectedness means components of open sets are openTopologySeparation Axioms, Connectedness and CompactnessMedium
- 40Multinomial coefficient 7!/2!,2!,3!AlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 41The dual swaps wedge and vee after removing the implicationAlgebraLattice Theory (Distributive/Modular/Boolean Lattices)Easy
- 42Inclusion–exclusion with the lcms 30, 40, 24 and 120AlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 43an^2=4cdot5^n, so a12=2cdot5^6AnalysisSequences and SeriesEasy
- 44X(0)=0, X'(1)=0 picks sinsqrtlambda,t with cossqrtlambda=0Ordinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 45Insulated ends ux=0 become Neumann conditions on XPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Easy
- 46Eliminate a using x-a=2y/y'Ordinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsMedium
- 47Lagrange's method: xy and x^2+y^2+z^2 are first integralsPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
- 48Coefficients of zx,zy involving z make the PDE quasi-linearPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsEasy
- 49Legendre's three-square theorem excludes 4^a(8b+7)AlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremMedium
- 502 is a square mod p iff pequivpm1pmod8AlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremMedium
- 51Gauss's identity sumdmid nvarphi(d)=nAlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 52Three coordinates fix a point, three angles fix the orientationClassical MechanicsMotion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsEasy
- 53An attractive m/r^2 force has potential -m/r, so L=T+m/rClassical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 54Atwood's machine: (m1+m2)ddot x=(m1-m2)gClassical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 55Legendre transform of a pendulum-type LagrangianClassical MechanicsHamilton's Canonical Equations, Hamilton's PrincipleEasy
- 56The curve is not unit-speed, so alpha"=2 is not its curvatureDifferential GeometryCurves and Surfaces, CurvatureMedium
- 57A curve lying in the plane x+y=0 has zero torsionDifferential GeometryCurves and Surfaces, CurvatureMedium
- 58The unit cylinder is locally isometric to the plane: E=G=1, F=0Differential GeometryCurves and Surfaces, CurvatureEasy
- 59Euler–Lagrange for a double integral: one divergence term per variableCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 60F free of y: the first integral Fy'= const gives y'propto xCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 61An isoperimetric constraint adds a constant to y": the extremal is a parabolaCalculus of VariationsVariational Methods for Boundary Value ProblemsMedium
- 62Fixed limits and x only under the integral: Fredholm, first kindLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 63Differentiate the Volterra equation twice: x"-x=cos tLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 64Abel's equation: the kernel (t-s)^-1/2 inverts to x=2sqrt tLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelHard
- 65Delta^2E^-1x^2=Delta^2(x-1)^2=2Numerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 66A repeated node and a misprinted option: the intended cubic is 2x^3-7x^2+6x+1Numerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationMedium
- 67Simpson's one-third rule with eight stripsNumerical AnalysisNumerical Differentiation and IntegrationEasy
- 68Delta^5u0=sumk(-1)^5-kbinom5kukNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 69Differentiating a Laplace transform multiplies by -tIntegral TransformsLaplace TransformEasy
- 70A convolution equation: X(s)=(s^2+1)/(s^2(s+1))Integral TransformsLaplace TransformMedium
- 71Modulation: cos at splits the spectrum into shifts pm aIntegral TransformsFourier TransformEasy
- 72Carath\'eodory's criterion: E must split every test set additivelyAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 73Statement I is Halmos's definition of completeness; statement II omits mu(En)<inftyAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 74Sets that differ by a null set have the same integralsAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 75Singularity is closed under sums; nullmu and nuperpmu force nu=0AnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
Section III · Statistics · Q76–145
Practise this section →- 76Sensitivity analysis gives ranges, and a new constraint can only hurtOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 77The optimum is at the intersection of the two constraint linesOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 78Cutting planes add constraints; B&B lower bounds never decreaseOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 79Wolfe's method: stationarity of the Lagrangian in x1 and x2Operations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 80A normal scale mixture with a chi^21 precision: X is CauchyProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 81(1-Y/n^alpha)^n tends to 1-Y, e^-Y or 1 according as alpha<1, =1, 1ProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremMedium
- 82Rare large jumps break Lindeberg: the normalised sum is not GaussianProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremHard
- 83A martingale remains one for its own (smaller) filtrationStochastic ProcessesMartingalesEasy
- 840 on finite sets, infty on infinite ones: finitely but not countably additiveAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 85E(Umid F) and E(Vmid F) are both X — hardly independentProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 86Given X1+X2=0, exactly one of the two is +1ProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 87Shrinking towards 1 with weight tfrac12 dominates X when Var X=mu^2+1Statistical InferenceElementary Bayesian InferenceMedium
- 88Gamma(2,2) prior plus Poisson data gives a Gamma(6,5) posteriorStatistical InferenceElementary Bayesian InferenceEasy
- 89Charts matched on in-control ARL are compared on out-of-control ARLStatistical Quality ControlControl ChartsEasy
- 90Little's law in the queue: Wq=Lq/lambdaOperations ResearchElementary Queuing and Inventory ModelsEasy
- 91The Weibull hazard alphalambda^alpha t^alpha-1 increases iff alphage1Survival AnalysisHazard Function and Failure Rates, Censoring and Life TestingEasy
- 92A parallel system fails only when every unit failsSurvival AnalysisSeries and Parallel SystemsEasy
- 93Joint Poisson likelihood: 2lambda^2-3lambda-2=0 gives hatlambda=2Statistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 94With sigma^2 known, X1+4X2 is sufficient — the printed T1=X1-4X2 is not, and is correlated with T2Statistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 95For U(theta,theta^2) the likelihood falls in theta, so take the least feasible value sqrtX(3)Statistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 96Sufficiency is joint: bar X alone is not sufficient for mu when sigma^2 is unknownStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 97X is Bernoulli(theta) and complete; X has mean zero whatever thetaStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 98The range of a U(theta,theta+1) sample has mean (n-1)/(n+1)Distribution TheoryDistribution of Order Statistics and RangeEasy
- 99Regression through the origin: hatalpha=sum xizi/sum zi^2Linear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 100For simple hypotheses the NP test is a likelihood-ratio testStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 101Poisson MLR in T=sum Xi: reject for large totalsStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 102Small totals favour a smaller exponential mean; 2T/theta0simchi^22nStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 103Unbiased: rejection probability at most alpha under H0, at least alpha under H1Statistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 104For the Cauchy, the median is asymptotically normal and the mean is not even consistentDistribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsEasy
- 105nFn(x)simBin(n,F(x)), so Fn(x) is asymptotically normalDistribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsEasy
- 106Delta method with g(lambda)=lambda e^-lambdaDistribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsEasy
- 107Wilks's theorem: -2loglambdaoversetdlongrightarrowchi^21 under H0Statistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 108Many functions of the data estimate 1-e^-lambda consistentlyStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 109Squared canonical correlations are the eigenvalues of Sigma11^-1Sigma12Sigma22^-1Sigma21Multivariate AnalysisData Reduction: PCA, Discriminant, Cluster AnalysisMedium
- 110The scaled T^2 is F-distributed — central only under H0Multivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsMedium
- 111(n-1)Ssim Wp(n-1,Sigma), so Ssim Wp(n-1,Sigma/(n-1))Multivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 112The Mahalanobis form of bar X is chi^2pMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 113Conditional covariance: the Schur complement Sigma11-Sigma12Sigma22^-1Sigma21Multivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 114Ymid X=5 has mean 10 and s.d. 5sqrt1-rho^2; the sign of rho is not identifiedMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsMedium
- 115Adjusted R^2=1-(1-R^2)n-1/n-k-1approx0.960 — not among the optionsLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 116Fitted values are centred at Xbeta, not at betaLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsEasy
- 117Under sigma^2Omega, the fitted values do not have covariance sigma^2OmegaLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsMedium
- 118hatbeta is unique exactly when X has full column rankLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsMedium
- 119The Box–Cox family is continuous in lambda, with log Y as its lambda=0 limitLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionMedium
- 120A tiny R^2 rules out only a linear relationLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 121The exact bias of the regression estimator is -Cov(b,bar x)Sampling TheoryRatio and Regression MethodsMedium
- 122The ratio estimator's bias is O(1/n)Sampling TheoryRatio and Regression MethodsEasy
- 123Cluster sampling loses to SRSWOR exactly when (M-1)rho0Sampling TheorySimple Random, Stratified and Systematic SamplingMedium
- 124For a fixed-size design, sumjne ipiij=(n-1)piiSampling TheoryProbability Proportional to Size SamplingMedium
- 125Equal selection probabilities turn PPSWR into SRSWRSampling TheoryProbability Proportional to Size SamplingEasy
- 126OC-curve parameter for factor A: Phi^2=nbsumtaui^2/(asigma^2)Linear Models & RegressionAnalysis of Variance and Covariance, Fixed/Random/Mixed EffectsHard
- 127Statement 2 has k and a swapped; statements 1 and 3 are true — no option says soDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDMedium
- 128Three textbook facts on factorials, orthogonal contrasts and confoundingDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionEasy
- 129Resolution counts the shortest word: in resolution V, two- and three-factor interactions are aliasedDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionMedium
- 130Replicate 1 confounds AB, replicate 2 confounds BCDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionMedium
- 131AR roots outside the unit circle but the MA root inside: causal, not invertibleIndex Numbers & Time SeriesTime Series ModelsMedium
- 132An ACF that cuts off after lag 5 with an unremarkable PACF points to MA(5)Index Numbers & Time SeriesTime Series ModelsMedium
- 133Interpolating an AR(1): both neighbours get weight phi/(1+phi^2)Index Numbers & Time SeriesTime Series ModelsMedium
- 134The "1" in ARIMA(2,1,2) is a unit root of the AR operatorIndex Numbers & Time SeriesTime Series ModelsEasy
- 135States 0, 1, 2 all communicate: the chain is irreducibleStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionEasy
- 136The two-state chain's limit is (beta,alpha)/(alpha+beta)Stochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionEasy
- 137An incomplete birth-process event with no correct optionStochastic ProcessesPoisson and Birth-and-Death ProcessesEasy
- 138PGF tfrac12(s+s^2): X is 1 or 2 with equal chanceProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 139TFR =5times the sum of age-specific fertility ratesDemography / Vital StatisticsLife Tables, Mortality RateEasy
- 140Sensitivity analysis: which changes leave the current basis optimalOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 141By symmetry E(tfracX1+X22mid T)=bar X, with variance theta(1-theta)/nStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 142F(1.5)=P(Xle1)=(1+lambda)e^-lambdaDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 143Density tfrac12 on (0,1) plus atoms of tfrac14 at 1 and 2ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 144Uniform on the triangle 0<x<y<1: given Y=y, Xsim U(0,y)ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 145A point mass at c has characteristic function e^itcProbabilityCharacteristic Functions, Probability InequalitiesEasy