January 2018
84 questions, booklet JAN-30218.
Section I · General (compulsory) · Q1–16
Practise this section →- 1The vanishing term 6/n^2 leaves the two limit points pm1AnalysisSequences and SeriesEasy
- 2Integrating the geometric series sum(-1)^kx^2k gives tan^-1xAnalysisSequences and SeriesEasy
- 31/(-1+i)=(-1-i)/2 lies in the third quadrantComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 4Residues pmtfrac12 cancel on a contour around both poles — but no contour is printedComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 5The 3times3 Fourier matrix squares to the index-reversal permutationLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 6A skew-symmetric matrix is fixed by its entries above the diagonalLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 7rank(AA^t)=rank A for real matricesLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 8Eigenvalue 1 has a two-dimensional eigenspace, so A is diagonalisableLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 9The distribution function has atoms of tfrac14 at 2 and 3 — no option matchesProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 10A discrete uniform on 2n+1 points: Y=4 means X=pm2Distribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 11The odd term integrates out, leaving X1sim U(-1,1)ProbabilityCharacteristic Functions, Probability InequalitiesMedium
- 12Chebyshev: P[X-EX1]leVar X/1^2ProbabilityCharacteristic Functions, Probability InequalitiesEasy
- 13Normal probabilities by standardising: P(B)<P(A)<P(C)Distribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 14Increasing x2 keeps every constraint satisfied and Z grows without boundOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 15A solid ellipse is a convex setOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 16The fundamental theorem of LP: an optimum is attained at an extreme pointOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
Section II · Mathematics · Q17–50
Practise this section →- 17e^-1/x glued to 0: smooth at the origin but not analyticAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 18A geometric series in -z^2Complex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 19The y-axis is fixed and the xz-plane turns through pi/4Linear AlgebraMatrix Representation, Change of Basis, Canonical and Jordan FormsEasy
- 20The trace is one linear condition on n^2 entriesLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionEasy
- 21Heine–Borel is special to R^n: closed and bounded need not be compactAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesEasy
- 22x^3 has derivatives 3xx and 6x, and stops thereAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 23sin x-sin ylex-y: Lipschitz, hence uniformly continuousAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 24An increasing function's total variation is f(b)-f(a)AnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 25log x/x peaks at x=e with value 1/eAnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremEasy
- 26Dirichlet's function: rationals and irrationals are both denseAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralEasy
- 27sin(iy)=isinh y and cos(iy)=cosh yComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 28Collinear points have proportional difference vectorsComplex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 29Liouville's theoremComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremEasy
- 30Only the rational function has a pole; the others are essential or entireComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesEasy
- 31Morera's theorem and its converse (for simply connected domains)Complex AnalysisContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaMedium
- 32sin z/z has a removable singularity at 0Complex AnalysisContour Integral, Cauchy's Theorem, Cauchy's Integral FormulaEasy
- 33Polynomials with zero constant term form a ring without 1AlgebraRings, Ideals, UFD, PID, Euclidean DomainEasy
- 34The product of the mth roots of unity is (-1)^m+1Complex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsEasy
- 35a^n=e makes the order of a a divisor of nAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremEasy
- 36Non-identity self-inverse elements pair up with the rest of G to force parityAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 37Groups of order p, p^2 and 15 are abelian; D5 has order 10AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 38Subfields of mathbb F2^7 correspond to divisors of the prime 7AlgebraFields, Finite Fields, Field Extensions, Galois TheoryMedium
- 39A cyclic vector does not make T nilpotent; it only rules out T=0Linear AlgebraAlgebra of Linear TransformationsMedium
- 40Minimal polynomial = characteristic polynomial: one Jordan block per eigenvalueLinear AlgebraMatrix Representation, Change of Basis, Canonical and Jordan FormsEasy
- 41A non-trivial kernel is an eigenspace for 0Linear AlgebraAlgebra of Linear TransformationsEasy
- 42Rotation by 90^circ swaps the two diagonal basis vectors up to signLinear AlgebraMatrix Representation, Change of Basis, Canonical and Jordan FormsMedium
- 43Multiplying by an invertible matrix preserves rankLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsEasy
- 44lambda^2-(tr A)lambda+det A=lambda^2+1Linear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremEasy
- 45The discriminant B^2-AC vanishes identically: parabolicPartial Differential EquationsClassification of Second Order PDEsEasy
- 46Integrability of a Pfaffian equation: XcdotcurlX=0Partial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsEasy
- 47Damped oscillations come from characteristic roots -alphapm iomegaOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 48Abel's formula: W=Ce^-a1tOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
- 49(My-Nx)/N=2x depends on x alone: integrating factor e^x^2Ordinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsEasy
- 50A linear homogeneous fourth-order equation has a four-dimensional solution spaceOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionEasy
Section III · Statistics · Q51–84
Practise this section →- 51A sample mean 1.96 standard errors above mu: odds of 39:1 againstDistribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsEasy
- 52Adding a regressor can never increase the residual sum of squaresLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionEasy
- 53A countable intersection of sure events is sureProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 54X+Y is 2X on Xlealpha and exactly 0 otherwise: an atom at the originProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 55Kolmogorov's 0–1 law for the tail sigma-fieldProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremEasy
- 56Change of variables for Y=1/X: the Jacobian is 1/y^2ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 57Exchangeability gives P(X1le X2)=tfrac12 for a continuous FProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 58Y=X^2 has density y^-1/2/(2theta) on (0,theta^2) — option (C) misprints the rangeProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 59If Xsim F(m,n) then 1/Xsim F(n,m)Distribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsEasy
- 60Counting failures instead of successes: n-Xsim B(n,1-p)Distribution TheoryStandard Discrete and Continuous Univariate DistributionsEasy
- 61Gamma variables with a common scale: X/(X+Y)sim Beta(alpha,beta)Distribution TheoryStandard Discrete and Continuous Univariate DistributionsMedium
- 62Given the total, X1/Tsim Beta(1,n-1)Distribution TheoryStandard Discrete and Continuous Univariate DistributionsMedium
- 63Cov(bar X,X1-X2)=(Var X1-Var X2)/n=0ProbabilityRandom Variables and Distribution Functions, Expectation and MomentsEasy
- 64With mu known the question should ask about sigma^2, and sum(Xi-mu)^2 is not offeredStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 65The Poisson log-likelihood sum xiloglambda-nlambda is concaveStatistical InferenceMethods of Estimation, Properties of EstimatorsEasy
- 66One Bernoulli observation: the size-0.05 MP test must randomise at X=1Statistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 67The UMVUE is uncorrelated with every unbiased estimator of zeroStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 68For a shifted exponential the minimum carries all the information about the locationStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsMedium
- 69Three standard facts about the partial correlation r12.3Multivariate AnalysisPartial and Multiple Correlation CoefficientsMedium
- 70Checking a claimed distribution is a goodness-of-fit problemStatistical InferenceAnalysis of Discrete Data and Chi-Square Test of Goodness of FitEasy
- 71K=mu0+z0.05,sigma/sqrt n with sigma=2Statistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsEasy
- 72The 2times2 shortcut chi^2=N(ad-bc)^2/(product of margins)Statistical InferenceAnalysis of Discrete Data and Chi-Square Test of Goodness of FitEasy
- 73Expected number of runs: 1+2n1n2/(n1+n2) with 20 men and 13 womenStatistical InferenceNonparametric Tests, Rank Correlation, Test for IndependenceMedium
- 74Error variance =SSE/(N-k)=399.6/27Linear Models & RegressionAnalysis of Variance and Covariance, Fixed/Random/Mixed EffectsEasy
- 75At the EOQ the ordering and holding costs are equalOperations ResearchElementary Queuing and Inventory ModelsEasy
- 76Gantt charts schedule jobs on machines over timeOperations ResearchSequencing and Project SchedulingEasy
- 77Little's law gives Ls=Lq+lambda/mu, not Lq+1/lambdaOperations ResearchElementary Queuing and Inventory ModelsEasy
- 78Strong duality: the primal has an optimum iff the dual doesOperations ResearchLinear Programming Problem, Simplex Method, DualityEasy
- 79A pure-strategy equilibrium is a saddle point of the payoff matrixOperations ResearchGame TheoryEasy
- 80A transportation BFS has m+n-1 basic variablesOperations ResearchTransportation and Assignment ProblemsEasy
- 81With rhowsy=0, systematic sampling has the SRSWR varianceSampling TheorySimple Random, Stratified and Systematic SamplingMedium
- 82Inclusion probability under SRSWOR: binomN-1n-1/binom Nn=n/NSampling TheorySimple Random, Stratified and Systematic SamplingEasy
- 832^5 in blocks of 8 means 4 blocks, so 4-1=3 confounded effectsDesign of Experiments2^K Factorial Experiments: Confounding and ConstructionEasy
- 84Five blocks of four from five treatments: each block must omit a different letterDesign of ExperimentsConnectedness and Orthogonality of Block Designs, BIBDMedium