April 2017
84 questions, booklet APR-30217.
Section I · General (compulsory) · Q1–16
Practise this section →- 1A linear term plus a bounded periodic one is never periodicAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 2sin(npi/3) cycles through pmfracsqrt32 and 0AnalysisSequences and SeriesEasy
- 31/(3+3i)=(1-i)/6Complex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 4Only the 1/z term contributes: residue 1Complex AnalysisContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaEasy
- 5det(cA)=c^ndet A with c=det ALinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 6(AA^t)^t=AA^tLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 7Row 3 equals row 1 + row 2 on the left but not on the rightLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 8A'A=I gives (det A)^2=1Linear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 9R\'enyi's representation: the maximum of n unit exponentials equals sum Xi/i in lawDistribution TheoryDistribution of Order Statistics and RangeHard
- 10P(X2<X1)=theta1/(2theta2) when theta1<theta2ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 11P(Acup B)+P(Acup B^c)=1+P(A)ProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 12The remaining probability 0.3 splits 2:1ProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 13A missing inequality: the stem gives only one of the two comparisons it needsProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 14Setting n-m variables to zero gives a basic solutionOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 15A closed disc is convexOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 16x1+x2le1 and x1+x2ge2 cannot both holdOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
Section II · Mathematics · Q17–50
Practise this section →- 17xx has derivative 2x, which has a corner at 0AnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 18The Maclaurin series of cos zComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 19A-I has rank 1, so the eigenvalue 1 needs no Jordan block of size 2Linear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 20A^3=0ne A^2 forces one nilpotent Jordan block of size 3Linear AlgebraMatrix Representation, Change of Basis, Canonical and Jordan FormsEasy
- 21tan x blows up where cos x=0AnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 22x^2-y^2=x+yx-y: bounded slope locally, unbounded globallyAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 23fn keeps a bump of height tfrac12 at x=1/n; gn's bump shrinks like 1/sqrt nAnalysisSequences and SeriesMedium
- 24The interior critical point gives only a local maximum; the endpoint x=2 gives 2AnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 25f integrable does not make f integrable: pm1 on rationals and irrationalsAnalysisRiemann Sums and Riemann Integral, Improper IntegralsEasy
- 26The exponential series converges everywhereComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 27A real-valued function satisfies Cauchy–Riemann only where its gradient vanishesComplex AnalysisAnalytic Functions, Cauchy-Riemann EquationsEasy
- 28-e^xcos y+1=Re(1-e^z), so its conjugate is -e^xsin yComplex AnalysisAnalytic Functions, Cauchy-Riemann EquationsEasy
- 29Four points are concyclic or collinear iff their cross ratio is realComplex AnalysisConformal Mappings, Mobius TransformationsMedium
- 30Analytic functions are infinitely differentiableComplex AnalysisContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaEasy
- 31e^1/(z-2) has an essential singularity at z=2; z=1 is only a poleComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 32Zeros can accumulate at the boundary without fequiv0Complex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 33Matrices with all entries equal form an abelian groupAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 34exp is an isomorphism (R,+)to(R^+,cdot)AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 35If A kills no non-zero matrix from the left, A has trivial kernelAlgebraRings, Ideals, UFD, PID, Euclidean DomainMedium
- 36Aut(Z7)congZ7^, cyclic of order 6AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 37Cayley's theorem: G embeds in the symmetric group on its own elementsAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 38The multiplicative group of a finite field is cyclicAlgebraFields, Finite Fields, Field Extensions, Galois TheoryEasy
- 39A square-free minimal polynomial means diagonalisable; trace 8, determinant 12Linear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremEasy
- 40Exponent 2 on (x-1), exponent 1 on (x-2) in the minimal polynomialLinear AlgebraMatrix Representation, Change of Basis, Canonical and Jordan FormsEasy
- 41A linear equation defines a subspace; x1x2+x3=0 does notLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 42The rows of the matrix are the coefficients of each outputLinear AlgebraMatrix Representation, Change of Basis, Canonical and Jordan FormsEasy
- 43Diagonalisable means the eigenspace for 5 has full dimension 3Linear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremEasy
- 44Parallelogram law: u+v^2+u-v^2=2u^2+2v^2Linear AlgebraInner Product Spaces, Orthonormal BasisEasy
- 45B^2-AC=1-4<0: ellipticPartial Differential EquationsClassification of Second Order PDEsEasy
- 46Independence can be lost on a subinterval — but not for solutions of a linear ODEOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionMedium
- 47The equation is exact: d(x+ye^x/y)=0Ordinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsMedium
- 48y1^2-y2^2le2by1-y2 needs yle bOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 49z=ae^bxsin by is harmonicPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsEasy
- 50Check the complete integral: p=2a/(z+b), q=2/(z+b)Partial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsMedium
Section III · Statistics · Q51–84
Practise this section →- 51Regression coefficients: c/a=sigmaX^2/sigmaY^2, ac=r^2, and the lines meet at the meansLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionMedium
- 52r^2=bYXbXY=2ale1 with a of the same sign as 2Linear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 53Marginal convergence in distribution says nothing about sumsProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 54frac1n(n+1)sum jXj tends to E(X1)/2 — the option lacks a factor 2ProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremMedium
- 55Jensen with the strictly concave logarithmProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 56Y=X^2 collects both roots, with Jacobian 1/(2sqrt y)ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 57Independent summands of a constant must both be constantProbabilityCharacteristic Functions, Probability InequalitiesMedium
- 58The square of a standard normal is chi^21Distribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 592X1+X2 has mean 3lambda but variance 5lambda: not PoissonDistribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 60log(1+X) is exponential with rate lambda; two of them add to Gamma(2,lambda)Distribution TheoryStandard Discrete and Continuous Univariate DistributionsMedium
- 61P(max=x)=F(x)^m-F(x-1)^m for integer-valued variablesDistribution TheoryDistribution of Order Statistics and RangeMedium
- 62-log Xi is exponential with rate theta, so the sum is Gamma(n,theta)Distribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 63Coverage 2Phi(K)-1 grows with KStatistical InferenceConfidence IntervalsEasy
- 64MLR makes the MP test for one theta1theta0 UMP for all of themStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 65Ebar X^2=theta^2+sigma^2/n, so subtract sigma^2/nStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 66The geometric MLE 1/(1+bar x) is at most 1/(1+r) when every xige rStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 67Beta(3,1) against uniform: the likelihood ratio 3x^2 increases in xStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 68For the Laplace location family nothing smaller than the ordered sample is sufficientStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 69Small sum Xi^2 is evidence of a small sigmaStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 70alpha=P2(Xge1)=e^-2 and beta=P1(X<1)=1-e^-1Statistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 71A function of data and parameter with a parameter-free law is a pivotStatistical InferenceConfidence IntervalsEasy
- 72Divide the residual sum of squares by its n-K degrees of freedomLinear Models & RegressionGauss-Markov Models, Estimability, Best Linear Unbiased EstimatorsEasy
- 73Smirnov: 4nDn^+2oversetdlongrightarrowchi^22Statistical InferenceNonparametric Tests, Rank Correlation, Test for IndependenceMedium
- 74Fisher's z: the difference is 1.83 standard errors — significant at 10\%, not at 5\%Distribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 75M/M/1: Ls=rho/(1-rho)Operations ResearchSteady-State Solutions of Markovian Queuing Models (M/M/1, M/M/C, M/G/1)Easy
- 76Johnson's rule: short A-times first, short B-times lastOperations ResearchSequencing and Project SchedulingMedium
- 77Annual demand 3200, holding frac14times6: Q^=sqrt2cdot3200cdot150/1.5=800Operations ResearchElementary Queuing and Inventory ModelsEasy
- 78Weak duality: an unbounded dual leaves no feasible primalOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 79A transportation BFS has at most m+n-1 basic variablesOperations ResearchTransportation and Assignment ProblemsEasy
- 80Matching pennies: mix tfrac12,tfrac12, value 0Operations ResearchGame TheoryEasy
- 81Equal-width strata of a uniform cut the variance by the factor 1/L^2Sampling TheorySimple Random, Stratified and Systematic SamplingMedium
- 82Cluster sampling is usually the least efficient per elementSampling TheorySimple Random, Stratified and Systematic SamplingEasy
- 83Equal replication and lambda=3 for every pair, but unequal blocksDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDEasy
- 84Each missing plot costs one error degree of freedomDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square DesignsEasy