July 2025
120 questions, booklet 5629542.
Section A · Part A · Q1–20
Practise this section →- 1Syllogism — chaining "all" statementsGeneral AptitudeLogical ReasoningEasy
- 2Venn diagram — exactly two sportsGeneral AptitudeData InterpretationEasy
- 3Value of a company from a part-sale of sharesGeneral AptitudeQuantitative AptitudeEasy
- 4Wheel rpm from vehicle speedGeneral AptitudeQuantitative AptitudeEasy
- 5Water displaced by a sphere in a cylinderGeneral AptitudeQuantitative AptitudeEasy
- 6Profit proportional to market share per unit productionGeneral AptitudeData InterpretationMedium
- 7Distances covered on a circular trackGeneral AptitudeQuantitative AptitudeEasy
- 8Displacement after a path with turnsGeneral AptitudeQuantitative AptitudeMedium
- 9Salaries before and after a fixed incrementGeneral AptitudeQuantitative AptitudeMedium
- 10New ratio after percentage increasesGeneral AptitudeQuantitative AptitudeEasy
- 11Unit digit of a sum of cubesGeneral AptitudeQuantitative AptitudeEasy
- 12Identify the person in the photoGeneral AptitudeLogical ReasoningEasy
- 13Diameter of identical semicircles from a figureGeneral AptitudeQuantitative AptitudeMedium
- 14Effect of dividing instead of multiplying by 2General AptitudeQuantitative AptitudeEasy
- 15Matching categories to a four-set Venn diagramGeneral AptitudeLogical ReasoningMedium
- 16Letter-to-digit code for LOTIONGeneral AptitudeLogical ReasoningEasy
- 17Two-digit number minus its digit sumGeneral AptitudeQuantitative AptitudeEasy
- 18Area of a figure made from four quarter-arcs of a unit circleGeneral AptitudeQuantitative AptitudeMedium
- 19Loss and partial recovery of an investmentGeneral AptitudeQuantitative AptitudeEasy
- 20Matching persons with shirt coloursGeneral AptitudeLogical ReasoningEasy
Section B · Part B · Q21–60
Practise this section →- 21Trichotomy for non-negative integersAnalysisElementary Set Theory, Real Number System, Supremum, InfimumEasy
- 22Dyadic fractions versus terminating decimalsAnalysisElementary Set Theory, Real Number System, Supremum, InfimumEasy
- 23Cardinality of sets of injections and surjectionsAnalysisElementary Set Theory, Real Number System, Supremum, InfimumHard
- 24Limit of a Mobius-type recursionAnalysisSequences and SeriesMedium
- 25Monotonicity of xlog(1+1/x)AnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremMedium
- 26Pointwise limit of tent functionsAnalysisSequences and SeriesEasy
- 27Characteristic polynomial from trace dataLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremEasy
- 28Counting rank-one matrices over mathbb F5Linear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsMedium
- 29Dimension of the quotient by the kernel of an evaluation mapLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 30Largest eigenvalue of a matrix with equal row sumsLinear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremMedium
- 31Orthonormality of sin(nx)/sin x under a weighted inner productLinear AlgebraInner Product Spaces, Orthonormal BasisHard
- 32Integral quadratic form on cubic polynomials without constant termLinear AlgebraQuadratic Forms, Reduction and ClassificationMedium
- 33Polynomial with a bounded contour-integral of modulusComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremHard
- 34Image of [0,1] under (z-i)/(z+i)Complex AnalysisConformal Mappings, Mobius TransformationsMedium
- 35Existence of entire functions with prescribed derivatives at 0Complex AnalysisAlgebra of Complex Numbers, Power Series, Transcendental FunctionsMedium
- 36Entire function with image in a lineComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 37Wilson, Fermat and Legendre-type divisibility statementsAlgebraFundamental Theorem of Arithmetic, Congruences, Chinese Remainder TheoremMedium
- 38Subrings of mathbb Q with the valuation propertyAlgebraRings, Ideals, UFD, PID, Euclidean DomainHard
- 39Which ideal is maximalAlgebraRings, Ideals, UFD, PID, Euclidean DomainMedium
- 40Homeomorphisms of a five-point space with topology emptyset,1,STopologyBasis, Dense Sets, Subspace and Product TopologyMedium
- 41General solution given one solutionOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionMedium
- 42Sum of reciprocals of eigenvalues of a Sturm-Liouville problemOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionMedium
- 43Quasilinear Cauchy problem by characteristicsPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsHard
- 44d'Alembert solution of the 1-D wave equationPartial Differential EquationsSeparation of Variables (Laplace, Heat and Wave Equations)Easy
- 45Determining constants in a cubic splineNumerical AnalysisFinite Differences, Lagrange/Hermite/Spline InterpolationMedium
- 46Extremal of a quadratic functional with fixed end valuesCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 47Fredholm equation with a separable kernelLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelEasy
- 48Conjugate momenta and equations of motionClassical MechanicsGeneralized Coordinates, Lagrange's EquationsEasy
- 49Mean, median and mode of a two-valued data setDescriptive StatisticsDescriptive Statistics, Exploratory Data AnalysisMedium
- 50Biased die with prime faces twice as likelyProbabilitySample Space, Discrete Probability, Independent Events, Bayes TheoremEasy
- 51Random walk on mathbb Z2025 with uniform nonzero stepsStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 52Survival probability of a mixture of exponential and gamma lifetimesProbabilityRandom Variables and Distribution Functions, Expectation and MomentsMedium
- 53MLE for a one-parameter trinomial-type pmfStatistical InferenceMethods of Estimation, Properties of EstimatorsMedium
- 54Confidence coefficient of an interval based on X/thetaStatistical InferenceConfidence IntervalsMedium
- 55UMP test for the Pareto shape parameterStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 56Posterior under a flat prior for a normal meanStatistical InferenceElementary Bayesian InferenceEasy
- 57R^2 and adjusted R^2 from the overall F-statisticLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionMedium
- 58Partial correlation for a linear transform of independent normalsMultivariate AnalysisPartial and Multiple Correlation CoefficientsMedium
- 59Unbiased estimator of the population mean under stratified samplingSampling TheorySimple Random, Stratified and Systematic SamplingEasy
- 60Maximise x+y over a polygonOperations ResearchLinear Programming Problem, Simplex Method, DualityMedium
Section C · Part C · Q61–120
Practise this section →- 61limsup and liminf of two classical sequencesAnalysisSequences and SeriesMedium
- 62Limit of a geometric mean of n+1,dots,2n over nAnalysisSequences and SeriesMedium
- 63Uniform limits: continuity, boundedness and differentiabilityAnalysisSequences and SeriesHard
- 64Convex bounded function on (0,infty)AnalysisContinuity, Uniform Continuity, Differentiability, Mean Value TheoremHard
- 65Uniform convergence of e^-n^2x^2/n and its derivativesAnalysisSequences and SeriesMedium
- 66Riemann integrability of compositions, quotients and square rootsAnalysisRiemann Sums and Riemann Integral, Improper IntegralsHard
- 67Lebesgue integrability via a partition of the rangeAnalysisMonotonic Functions, Bounded Variation, Lebesgue Measure & IntegralHard
- 68Euler relations for a homogeneous function of degree 3AnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsMedium
- 69Surjectivity of the derivative of Xmapsto XX^TAnalysisFunctions of Several Variables, Inverse & Implicit Function TheoremsHard
- 70Volterra integration operator on C[0,1]AnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesHard
- 71Linear independence of shifts of a quadratic polynomialLinear AlgebraVector Spaces, Subspaces, Linear Dependence, Basis, DimensionMedium
- 72Counting subspaces and linear maps over mathbb F3Linear AlgebraAlgebra of Linear TransformationsMedium
- 73Minimal polynomial and eigenvalues of products of 2times2 matricesLinear AlgebraAlgebra of Matrices, Rank, Determinant, Linear EquationsHard
- 74Commutators on mathbb C^2Linear AlgebraEigenvalues and Eigenvectors, Cayley-Hamilton TheoremHard
- 75Jordan structure of an operator with a given annihilating polynomialLinear AlgebraMatrix Representation, Change of Basis, Canonical and Jordan FormsHard
- 76Matrices similar to J2(-1)oplus J2(1)Linear AlgebraMatrix Representation, Change of Basis, Canonical and Jordan FormsHard
- 77Adjoint, normal and self-adjoint operators on a complex inner product spaceLinear AlgebraInner Product Spaces, Orthonormal BasisHard
- 78Isotropic vector of a nondegenerate form on mathbb R^2Linear AlgebraQuadratic Forms, Reduction and ClassificationMedium
- 79Points where no derivative of an entire non-polynomial vanishesComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesHard
- 80Periodicity and behaviour at infinity of entire functionsComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremMedium
- 81Contour integral of e^ze^1/z around the unit circleComplex AnalysisTaylor Series, Laurent Series, Calculus of ResiduesMedium
- 82Automorphisms of the punctured unit discComplex AnalysisLiouville's Theorem, Maximum Modulus Principle, Open Mapping TheoremHard
- 83Kernel of the coset action (core of a subgroup)AlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 84Automorphism groupsAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremHard
- 85Isomorphism theorems and normalityAlgebraGroups, Subgroups, Sylow Theorems, Cayley's TheoremMedium
- 86Endomorphisms of mathbb Q[X]AlgebraRings, Ideals, UFD, PID, Euclidean DomainHard
- 87Irreducibility of X^5pm X+1 over mathbb QAlgebraPolynomial Rings and Irreducibility CriteriaHard
- 88Finite fields, splitting fields and sums of squaresAlgebraFields, Finite Fields, Field Extensions, Galois TheoryHard
- 89Preimages and images under a nonconstant real polynomialAnalysisMetric Spaces, Compactness, Connectedness, Normed Linear SpacesMedium
- 90Topology of mathbb Q+mathbb Qsqrt2 inside mathbb RTopologyBasis, Dense Sets, Subspace and Product TopologyHard
- 91Non-Lipschitz right-hand side 1+sqrty+ and Peano existenceOrdinary Differential EquationsExistence & Uniqueness, Singular Solutions, Systems of First Order ODEsHard
- 92Solutions of a linear ODE with 2pi-periodic coefficientsOrdinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionHard
- 93Wronskian test for y"+q(x)y=0Ordinary Differential EquationsGeneral Theory of Linear ODEs, Variation of Parameters, Sturm-Liouville, Green's FunctionHard
- 94Cauchy problem whose data curve is characteristicPartial Differential EquationsLagrange and Charpit Methods, Cauchy Problem for First Order PDEsHard
- 95Maximum principle for Delta u+ux=0 on the unit discPartial Differential EquationsClassification of Second Order PDEsHard
- 96Forward Euler for y'=-yNumerical AnalysisNumerical Solutions of ODEs (Picard, Euler, Runge-Kutta)Medium
- 97Two-point Gauss rule on [0,3]Numerical AnalysisNumerical Differentiation and IntegrationMedium
- 98Extremals of int1^2 (y')^2x^-alphadxCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaMedium
- 99Extremals versus minimizers for a quadratic functional with a free endCalculus of VariationsVariation of a Functional, Euler-Lagrange Equation, Conditions for ExtremaHard
- 100Volterra equation with kernel (t-x)^2Linear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelHard
- 101Converting an initial value problem into a Volterra equationLinear Integral EquationsFredholm and Volterra Type, Separable Kernels, Resolvent KernelHard
- 102Properties of Poisson bracketsClassical MechanicsHamilton's Canonical Equations, Hamilton's PrincipleMedium
- 103Poisson variables conditioned on their sumDistribution TheoryStandard Discrete and Continuous Univariate DistributionsMedium
- 104Moments and Chebyshev bound for a mixed distributionProbabilityRandom Variables and Distribution Functions, Expectation and MomentsHard
- 105Classification of states of a five-state chainStochastic ProcessesMarkov Chains: Classification of States, Stationary DistributionHard
- 106CLT and Slutsky for power sums of i.i.d. variablesProbabilityModes of Convergence, Laws of Large Numbers, Central Limit TheoremsHard
- 107Distribution of X^2+Y^2 and e^-(X^2+Y^2)/2Distribution TheorySampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 108Sampling distributions built from a bivariate normal sampleMultivariate AnalysisMultivariate Normal Distribution, Wishart Distribution, Distribution of Quadratic FormsHard
- 109Confidence intervals for a location parameter of a shifted exponentialStatistical InferenceConfidence IntervalsHard
- 110UMVUE for the centre and half-width of a uniform distributionStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 111Estimators for the scale of a Laplace distribution with location thetaStatistical InferenceMethods of Estimation, Properties of EstimatorsHard
- 112Likelihood ratio test for the rate of a shifted exponentialStatistical InferenceTests of Hypotheses: MP and UMP Tests, Likelihood Ratio TestsHard
- 113Wilcoxon rank-sum statistic for sample sizes 7 and 9Nonparametric InferenceSimple Nonparametric Tests for One and Two Sample ProblemsHard
- 114Bayes estimators for a normal meanStatistical InferenceElementary Bayesian InferenceHard
- 115Regression of x^2 on x with and without an interceptLinear Models & RegressionSimple and Multiple Linear Regression, Logistic RegressionMedium
- 116General linear hypothesis for a restricted regression modelLinear Models & RegressionGauss-Markov Models, Estimability, BLUE, Tests for Linear HypothesesHard
- 117Principal components of an equicorrelation matrixMultivariate AnalysisData Reduction: Principal Component Analysis, Discriminant, Cluster, Canonical CorrelationHard
- 118Properties of a block design from its incidence matrixDesign of ExperimentsCompletely Randomized, Randomized Block and Latin-Square Designs; Connectedness, Orthogonality, BIBDHard
- 119Reliability, mean life and hazard rate of a two-component parallel systemReliabilityHazard Function and Failure Rates, Censoring and Life Testing, Series and Parallel SystemsHard
- 120Long-run measures of an M/M/1 queueQueueing TheorySteady-State Solutions of Markovian Queuing Models: M/M/1, M/M/C, M/G/1Medium