Worked examples
Every question, solved in full.
Free worked GRE and GMAT questions: the correct answer, the reasoning that gets there, and why each other option is wrong. No account needed to read them.
Worked questions by exam
- Heat flow through a thicker wallUse proportional reasoning for steady heat conduction while keeping material, area and temperature difference fixed.
- Linear thermal expansionDistinguish a small length change from the final length and convert metres to millimetres correctly.
- A collision that conserves momentum but loses kinetic energyChoose the system before a sticking collision, then check both momentum and kinetic energy.
- Net Force and AccelerationA public mechanics question that shows how to combine opposing horizontal forces before applying Newton's second law.
- Energy Conservation on a Frictionless TrackA public energy-conservation question showing why mass cancels when gravitational potential energy converts into speed on a frictionless track.
- Perfectly Inelastic CollisionA public momentum question showing how to find the shared final speed after two carts couple in a perfectly inelastic collision.
- Mass-Spring Oscillation PeriodA public oscillation question showing how mass and spring constant set the period of simple harmonic motion independent of amplitude.
- Gauss's Law and Sphere RadiusA public Gauss's law question contrasting how the enclosed-charge flux and the surface electric field respond when a Gaussian sphere is enlarged.
- Series-Parallel Branch CurrentA public DC-circuit question showing how to reduce a series-parallel network before tracing the current through a single branch.
- Infinite Square Well TransitionsA public quantum question showing how infinite-square-well energies scale with the square of the quantum number and set photon transition energies.
- Photoelectric EffectA public photoelectric-effect question showing how a photon's energy splits between the work function and the electron's maximum kinetic energy.
- Carnot Engine Work OutputA public heat-engine question showing how absolute reservoir temperatures set a Carnot engine's efficiency and per-cycle work output.
- Time DilationA public special-relativity question showing how to identify the proper time and apply the Lorentz factor to a moving clock.
- Double-Slit Bright FringesA public interference question showing how wavelength, slit separation, and screen distance set the position of a double-slit bright fringe.
- De Broglie Wavelength of an Accelerated ElectronA public atomic-physics question on turning an accelerating voltage into a de Broglie wavelength, using the momentum of the electron rather than the photon energy relation.
- Counting Time and Relative UncertaintyA public laboratory-methods question on Poisson counting statistics: how much longer a detector must run to halve the relative uncertainty in its total count.
- Two draws without replacementUpdate both the favourable count and the total after the first draw instead of assuming independence.
- Part-to-Whole RatiosA public ratio question that shows how to convert a part-to-part ratio into a part-to-whole share before scaling to a real quantity.
- Exponent RulesA public exponents question that combines the power-of-a-product rule with the quotient rule to simplify an algebraic fraction.
- Addition Rule of ProbabilityA public probability question that shows why the probability of an overlap must be subtracted when combining two non-mutually-exclusive events.
- CombinationsA public counting question that shows when to use a combination rather than a permutation for an unordered selection.
- Median of a Data SetA public descriptive-statistics question that shows how to find the median of an even-sized data set and how it differs from the mean.
- Z-Scores on a Normal DistributionA public statistics question that shows how to standardize a raw score into a z-score using the mean and standard deviation.
- A limit at a missing pointFactor before cancelling, retain the domain restriction, and distinguish a limit from a function value.
- Eigenvalues of a triangular matrixUse a characteristic polynomial and a trace check rather than assuming all matrix entries are eigenvalues.
- Limits of Rational FunctionsA public limits question that resolves a $0/0$ indeterminate form by factoring and canceling a common factor before substituting.
- Geometric Series ConvergenceA public series question that checks the common-ratio convergence condition and applies the closed-form sum of an infinite geometric series.
- Determinants of Scaled, Transposed, and Inverted MatricesA public linear algebra question that chains three determinant properties — scalar multiples in dimension n, transposes, and inverses — into a single evaluation.
- Counting Elements of a Given OrderA public group theory question that counts the elements of order 6 in a cyclic group by using the order formula n/gcd(k, n) rather than by listing elements.
- Is the Set Open, Compact or Connected? A Planar AnnulusA public topology question that runs the open, closed, bounded, and connected checks on a half-open annulus in the plane and reads off compactness from Heine-Borel.
- A Second-Order Initial-Value Problem with Complex RootsA public differential equations question that solves a constant-coefficient initial-value problem whose characteristic roots are complex, including the product rule needed for the second condition.
- A Contour Integral at a Third-Order PoleA public complex analysis question that evaluates a contour integral with a pole of order three, where the factorial in Cauchy's formula for derivatives decides the answer.
- Negative reinforcement or punishment?Classify a consequence from what changes in the environment and what happens to the behaviour afterward.
- Retrieval Cues and Encoding SpecificityA public GRE Psychology question showing why a recall failure is often a retrieval-cue problem rather than evidence that the memory was never stored.
- The All-or-None Law and Stimulus IntensityA public GRE Psychology question on how a neuron encodes a stronger stimulus, and why the answer cannot be a bigger action potential.
- Conservation and the Preoperational ChildA public GRE Psychology question on a classic conservation failure, and how to name the Piagetian stage and the specific construct without confusing centration with egocentrism.
- Two-Factor Theory and Misattributed ArousalA public GRE Psychology question that separates the two-factor theory of emotion from the James-Lange, Cannon-Bard, appraisal, and facial-feedback accounts.
- Positive and Negative Symptoms of SchizophreniaA public GRE Psychology question on sorting a case description into positive symptoms, negative symptoms, and medication side effects.
- IQ Scores and the Normal DistributionA public GRE Psychology question converting a Wechsler IQ score to a percentile with a z-score, and naming the empirical-rule errors that produce the wrong percentiles.
- Identifying Schools of PsychologyA public GRE Psychology question that reads a stated methodological commitment back to the school of psychology it defines.
- Successive percentage changesTrack the changing base in a price increase followed by a discount, with an explanation for every answer.
- Find an alternative causeIdentify evidence that weakens a causal conclusion without disputing the reported observation.
- Is a ratio enough to determine a count?Separate a relative quantity from an absolute count in an original data-sufficiency exercise.
- Combined Work RatesA public work-rate question that shows why individual completion times must be converted to rates before combining two workers.
- Dilution MixturesA public mixture question that shows how to track the unchanged amount of solute with a proportion when a solution is diluted.