April 2017
145 questions, booklet APR-30317.
Section I · General (compulsory) · Q1–5
Practise this section →- 1nabla f=(2xy+7,x^2)=(3,1) at (1,-2)AnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsEasy
- 2Continuous with zero partials, but the directional derivatives are not linearAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 3x^2 on Q, 0 off Q: squeezed only at the originAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 4Block triangular: eigenvalues of the 2times2 corner plus the last diagonal entryLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremEasy
- 5The Perron eigenvector of a positive column-stochastic matrix is positiveLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
Section II · Mathematics · Q6–75
Practise this section →- 6A critical point with fxx<0 and positive Hessian determinantAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 7d'Alembert-type solutions f(x+2y)+g(x-2y) satisfy uyy=4uxxPartial Differential EquationsClassification of Second Order PDEsEasy
- 8The Jacobian of f at the originAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsEasy
- 9Chain rule: D(fcirc g)(1,2)=Df(0,0),Dg(1,2)AnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 10A positive definite matrix need not be orthogonalLinear AlgebraQuadratic Forms, Reduction and ClassificationEasy
- 116=2cdot3=(1+sqrt-5)(1-sqrt-5) in Z[sqrt-5]AlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 12Real quadratic rings have infinitely many units: (1+sqrt2)^nAlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 13mathbb Fp^n exists for every n, so irreducibles of every degree existAlgebraPolynomial Rings and Irreducibility CriteriaMedium
- 14Express bi' in the basis b1,b2,b3 and use the coordinates as columnsLinear AlgebraAlgebra of Linear TransformationsMedium
- 15Gap series sum z^n^2: coefficients 0 or 1, so R=1Complex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 16Only an essential singularity lets f(zn) have no limit, even inftyComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesMedium
- 171/z is analytic and non-constant on Csetminus0, hence openComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremEasy
- 18cz^2+(d-a)z-b=0 has at most two rootsComplex AnalysisConformal Mappings, Mobius TransformationsEasy
- 19Maximum modulus principleComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremEasy
- 20Adding the limit point 0 makes 1/n compactAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 21xsin(1/x)to0 but sin(1/x) oscillatesAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 22Path connected Rightarrow connected; the converses failTopologySeparation Axioms, Connectedness and CompactnessEasy
- 23Borel subsetneq Lebesgue measurableAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 24f measurable does not make f measurableAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 25Lebesgue: functions of bounded variation are differentiable a.e.AnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 26In a non-abelian group of order p^3, G'=Z(G) has order pAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 27A4 has the normal Klein four-groupAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 28Abelian groups of order 3^4 correspond to the 5 partitions of 4AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 291/(1+alpha)inQ[alpha] exactly when alpha is algebraicAlgebraPolynomial Rings and Irreducibility CriteriaMedium
- 30sqrt[3]2 has degree 3, not a power of 2AlgebraFields, Finite Fields, Field Extensions, Galois TheoryEasy
- 31A finite-dimensional space has infinitely many subspaces (over R)AnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 32Bessel's inequality: only countably many Fourier coefficients are non-zeroLinear AlgebraInner Product Spaces, Orthonormal BasisMedium
- 33A complete orthonormal set has no non-zero vector orthogonal to itLinear AlgebraInner Product Spaces, Orthonormal BasisMedium
- 34X=Moplus M^perp holds for every closed subspace of a Hilbert spaceLinear AlgebraInner Product Spaces, Orthonormal BasisMedium
- 35All four statements are standard theorems — nothing is "not true"AnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 36"Open in Y" does not transfer to X unless Y is openTopologyBasis, Dense Sets, Subspace and Product TopologyEasy
- 37In a Hausdorff space every option holds for a limit pointTopologySeparation Axioms, Connectedness and CompactnessMedium
- 38An open surjection is a quotient map only if it is also continuousTopologyBasis, Dense Sets, Subspace and Product TopologyMedium
- 39The irrationals: every compact subset has empty interiorTopologySeparation Axioms, Connectedness and CompactnessMedium
- 40Stars and bars: binom10+6-16-1=binom155AlgebraPermutations, Combinations, Pigeon-Hole Principle, Inclusion-ExclusionEasy
- 41The divisor lattice is graded: every maximal chain has the same lengthAlgebraLattice Theory (Distributive/Modular/Boolean Lattices)Medium
- 42A Gray code is a Hamiltonian path through all 2^10 verticesGraph TheoryGraphs, Trees, ConnectivityMedium
- 43For odd m, 2^varphi(m)equiv1pmod mAlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremMedium
- 44Eliminating f from z=f(x^2+y^2+z^2)Partial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsEasy
- 45A product pq of derivatives makes equation (ii) non-linearPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsEasy
- 46Orthogonal surfaces: z(3z+1)(p+q)=x+y, through the given circlePartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsHard
- 47frac ddt(x^2-y^2)=2xy-2yx=0: hyperbolasOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 48Power series: 2C2=-C0 from y"=-e^xy at x=0Ordinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 49ax+bequiv0pmod m is solvable iff gcd(a,m)mid bAlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 50Primes equiv3pmod4 must occur to even powers: 180=2^2cdot3^2cdot5AlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 51sumdmid nmu(d)=0 for n1AlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremEasy
- 52Velocity in polar coordinates has components dot r and rdotthetaClassical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 53Pendulum: V=mgl(1-costheta) measured from the lowest pointClassical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 54Euler's theorem: a rigid displacement about a fixed point is a rotationClassical MechanicsMotion of Rigid Bodies, Euler's Dynamical Equations, Small OscillationsEasy
- 55The harmonic oscillator: ddot x+omega^2x=0Classical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 56Frenet–Serret with kappa=2, tau=s+1 (convention b'=tau n)Differential GeometryCurves and Surfaces, CurvatureMedium
- 57Twisted cubic at the origin: alpha'timesalpha"/alpha'^3=2Differential GeometryCurves and Surfaces, CurvatureEasy
- 58Helicoid: E=v^2+a^2, F=0, G=1Differential GeometryCurves and Surfaces, CurvatureEasy
- 59Euler–Poisson with fy=0 integrates once: fy'-frac ddxfy"= constCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 60y^(4)=y, and sin x meets all four end conditionsCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 61The Dirichlet integral leads to Laplace's equationCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaEasy
- 62Variable upper limit, x outside the integral, kernel depending on t-sLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 63With u=y", the BVP becomes u=2x-1+int0^1k1(x,t)u(t),dtLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelHard
- 64Substitute x=e^t^2w: then w=1+int0^tw, so w=e^tLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelMedium
- 65Delta+nabla=E-E^-1, not ENumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 66The linear Lagrange interpolantNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationEasy
- 67Two Euler steps: 0to0.1to0.19Numerical AnalysisNumerical Solutions of ODEs (Picard, Euler, Runge-Kutta)Easy
- 68Divided differences of 1/x alternate: f[a,b,c,d]=-1/(abcd)Numerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationMedium
- 69First shifting theoremIntegral TransformsLaplace TransformEasy
- 70LJ0=1/sqrts^2+1, so J0J0 has transform 1/(s^2+1)Integral TransformsLaplace TransformMedium
- 71The Fourier transform of e^-as is a Cauchy-type kernelIntegral TransformsFourier TransformMedium
- 72mu(A,Delta,B) is a pseudometric: distinct sets can be at distance 0AnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 73Fatou's lemma and the monotone convergence theoremAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 74The countable-cover infimum defines the induced outer measureAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralMedium
- 75Minkowski's inequalityAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
Section III · Statistics · Q76–145
Practise this section →- 76A node is fathomed when it is infeasible or integer — not when it is fractionalOperations ResearchDynamic and Integer ProgrammingEasy
- 77The shadow price is the rate of change of the optimal objective valueOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 78Only (A) and (D) are feasible, and (A) gives the larger Z=100Operations ResearchLinear Programming Problem, Simplex Method, DualityHard
- 79The revised simplex carries B^-1 and the multipliers cBB^-1 every iterationOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
- 80-X has characteristic function phi(-t)=overlinephi(t), not -phi(t)ProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 81A product of independent mean-one factors is a martingaleStochastic ProcessesMartingalesEasy
- 82Doob's martingale E(Zmid Fn)Stochastic ProcessesMartingalesEasy
- 83Integer values give period 2pi, not piProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 84Flipping the sign outside [-alpha,alpha] preserves the N(0,1) lawProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 85The average winnings settle near m<m/2ProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 86Atoms tfrac14 at -1 and tfrac12 at 2, plus density tfrac14 on (2,3)ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 87Atoms of tfrac13 at 0 and 1 with density tfrac13 betweenProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 88Mixture mean: 0.4times5+0.6times7ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 89phi(t)=1/(1-it) for the unit exponentialProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 901-X is Bernoulli(0.4), so F(0.65)=P(1-X=0)=0.6ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 91Density x+y on the unit square: Cov(X,Y)=-tfrac1144ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 92The options are statistics, not distributions; the conditional law is N(T/3,1/6)Multivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsMedium
- 93A multinomial count is binomial: success = category 1Distribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 94For the shifted exponential, X(1) is complete sufficient and Basu appliesStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 95Given T=t, each Xi is Bin(t,1/n)Statistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 96T=sumlog(1+Xi)sim Gamma(n,lambda), and E[(n-1)/T]=lambdaStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 97U(theta,2theta): the likelihood falls in theta on [X(n)/2,X(1)]Statistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 98Uniform scale family: a UMP two-sided test exists (misprinted hypotheses)Statistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 99The UMPU normal test is the symmetric two-sided z testStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 100LMP Cauchy test: sum Yi has null variance n/8, so the cut-off is sqrtn/8,zalphaStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 101Binomial minimax estimator (X+sqrt n/2)/(n+sqrt n)Statistical InferenceElementary Bayesian InferenceHard
- 102For a symmetric population the mean and median are both unbiased and consistentStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 103sqrt n(2bar X-theta)oversetdlongrightarrow N(0,theta^2/3), so the square is theta^2chi^21/3Distribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 104Dropping one or two observations does not spoil consistencyStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 105nbar X^2 is chi^21 when theta=0 (non-central otherwise)Distribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 106X(1)to-theta and X(n)totheta, so the range estimates 2thetaDistribution TheoryDistribution of Order Statistics and RangeMedium
- 107Principal components redistribute, but preserve, the total varianceMultivariate AnalysisData Reduction: PCA, Discriminant, Cluster AnalysisEasy
- 108u'Sigma^-1u/u'A^-1usimchi^2n-p for Wishart A with n-1 d.f.Multivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsHard
- 109l'Alsim(l'Sigma l)chi^2n-1Multivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 110The Mahalanobis form is chi^2p, with MGF (1-2t)^-p/2Multivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 111Read off sigma1^2=4, sigma2^2=1, rho=0.8 from the bivariate normal densityMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsMedium
- 112The regression of X^(1) on X^(2) is the conditional meanMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Quadratic FormsEasy
- 113Reading "se(hatbeta,hatgamma)" as the s.e. of hatbetahatgamma (delta method)Linear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionHard
- 114Varhatbeta=sigma^2/sum(Xi-bar X)^2, not sigma^2/sum Xi^2Linear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 115Positive errors with mean theta: the slope is still unbiased, the intercept absorbs thetaLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionMedium
- 116A true linear restriction keeps the restricted estimator unbiased and more preciseLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsMedium
- 117Unbiasedness, sigma^2(X'X)^-1 and BLUE are exact; consistency needs conditions on XLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsMedium
- 118GLS weights by Omega^-1, not OmegaLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsEasy
- 119The ratio estimator wins when rhofrac12,CV(x)/CV(y)Sampling TheoryRatio and Regression MethodsEasy
- 120Regression gains rho^2; with rho=0 it gains nothingSampling TheoryRatio and Regression MethodsEasy
- 121Fixed sample size: sumpii=nSampling TheoryProbability Proportional to Size SamplingEasy
- 122Clusters of size one are just unitsSampling TheorySimple Random, Stratified and Systematic SamplingMedium
- 123With replacement, every draw uses the same probabilities piSampling TheoryProbability Proportional to Size SamplingEasy
- 124Latin square: rows, columns and treatments, all additiveDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsEasy
- 125Intrablock estimate: hattaui=kQi/(lambda v)=Qi/(rE)Design of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDMedium
- 126Confounding ABC means its contrast equals the block contrastDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionEasy
- 127Response surface designs are not always rotatableDesign of ExperimentsResponse Surface DesignsMedium
- 128Strip-plot EMS for A: error + whole-plot error + the A effectsDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsHard
- 129Started at 0, the AR(1) has mean mu1-phi^t/1-phi and variance frac1-phi^2t1-phi^2ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 130The AR(2) stationarity triangleIndex Numbers & Time SeriesTime Series ModelsMedium
- 131MA(1) with theta and 1/theta: identical autocorrelationsIndex Numbers & Time SeriesTime Series ModelsEasy
- 132psi(B)=1-0.3B/1-0.5B and pi(B)=1-0.5B/1-0.3BIndex Numbers & Time SeriesTime Series ModelsMedium
- 133An inessential state is always transient — statement 3 is true tooStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionMedium
- 134One-step transition probability p12=tfrac14Stochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionEasy
- 135A doubly stochastic two-state chain has the uniform stationary lawStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionEasy
- 136mathring ex=ex+tfrac12 and ex=px(1+ex+1)Demography / Vital StatisticsLife Tables, Mortality RateMedium
- 137l31=l30,p30 with p30=e30/(1+e31)Demography / Vital StatisticsLife Tables, Mortality RateMedium
- 138Poisson with mean 6: P(N4)=1-115e^-6Stochastic ProcessesPoisson and Birth-and-Death ProcessesEasy
- 139The residual Y-E(Ymid X) is uncorrelated with every function of XProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 140In-control ARL = mean run length before a false alarmStatistical Quality ControlControl ChartsEasy
- 141Holding cost is proportional to unit cost, so Q^propto C^-1/2Operations ResearchElementary Queuing and Inventory ModelsEasy
- 142rho=0.6: P(2 in the system)=0.4times0.36=0.144Operations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Easy
- 143Rectifying inspection: n always, plus N-n when the lot is rejectedStatistical Quality ControlAcceptance SamplingEasy
- 144Components 1 and 2 in parallel, then 3 in seriesSurvival AnalysisSeries and Parallel SystemsEasy
- 145EX-Y for uniform marginals never exceeds tfrac12ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium