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
GRE Physics Subject Test
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Heat flow through a thicker wall
Use proportional reasoning for steady heat conduction while keeping material, area and temperature difference fixed.
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Linear thermal expansion
Distinguish a small length change from the final length and convert metres to millimetres correctly.
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A collision that conserves momentum but loses kinetic energy
Choose the system before a sticking collision, then check both momentum and kinetic energy.
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Net Force and Acceleration
A public mechanics question that shows how to combine opposing horizontal forces before applying Newton's second law.
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Energy Conservation on a Frictionless Track
A public energy-conservation question showing why mass cancels when gravitational potential energy converts into speed on a frictionless track.
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Perfectly Inelastic Collision
A public momentum question showing how to find the shared final speed after two carts couple in a perfectly inelastic collision.
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Mass-Spring Oscillation Period
A public oscillation question showing how mass and spring constant set the period of simple harmonic motion independent of amplitude.
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Gauss's Law and Sphere Radius
A public Gauss's law question contrasting how the enclosed-charge flux and the surface electric field respond when a Gaussian sphere is enlarged.
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Series-Parallel Branch Current
A public DC-circuit question showing how to reduce a series-parallel network before tracing the current through a single branch.
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Infinite Square Well Transitions
A public quantum question showing how infinite-square-well energies scale with the square of the quantum number and set photon transition energies.
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Photoelectric Effect
A public photoelectric-effect question showing how a photon's energy splits between the work function and the electron's maximum kinetic energy.
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Carnot Engine Work Output
A public heat-engine question showing how absolute reservoir temperatures set a Carnot engine's efficiency and per-cycle work output.
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Time Dilation
A public special-relativity question showing how to identify the proper time and apply the Lorentz factor to a moving clock.
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Double-Slit Bright Fringes
A public interference question showing how wavelength, slit separation, and screen distance set the position of a double-slit bright fringe.
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De Broglie Wavelength of an Accelerated Electron
A 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.
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Counting Time and Relative Uncertainty
A public laboratory-methods question on Poisson counting statistics: how much longer a detector must run to halve the relative uncertainty in its total count.
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GRE General Test
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Two draws without replacement
Update both the favourable count and the total after the first draw instead of assuming independence.
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Part-to-Whole Ratios
A 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.
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Exponent Rules
A public exponents question that combines the power-of-a-product rule with the quotient rule to simplify an algebraic fraction.
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Addition Rule of Probability
A public probability question that shows why the probability of an overlap must be subtracted when combining two non-mutually-exclusive events.
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Combinations
A public counting question that shows when to use a combination rather than a permutation for an unordered selection.
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Median of a Data Set
A public descriptive-statistics question that shows how to find the median of an even-sized data set and how it differs from the mean.
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Z-Scores on a Normal Distribution
A public statistics question that shows how to standardize a raw score into a z-score using the mean and standard deviation.
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GRE Mathematics Subject Test
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A limit at a missing point
Factor before cancelling, retain the domain restriction, and distinguish a limit from a function value.
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Eigenvalues of a triangular matrix
Use a characteristic polynomial and a trace check rather than assuming all matrix entries are eigenvalues.
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Limits of Rational Functions
A public limits question that resolves a $0/0$ indeterminate form by factoring and canceling a common factor before substituting.
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Geometric Series Convergence
A public series question that checks the common-ratio convergence condition and applies the closed-form sum of an infinite geometric series.
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Determinants of Scaled, Transposed, and Inverted Matrices
A public linear algebra question that chains three determinant properties — scalar multiples in dimension n, transposes, and inverses — into a single evaluation.
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Counting Elements of a Given Order
A 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.
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Testing a Planar Set for Openness, Compactness, and Connectedness
A 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.
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A Second-Order Initial-Value Problem with Complex Roots
A 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.
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A Contour Integral at a Third-Order Pole
A 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.
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GRE Psychology Subject Test
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Negative reinforcement or punishment?
Classify a consequence from what changes in the environment and what happens to the behaviour afterward.
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Retrieval Cues and Encoding Specificity
A public GRE Psychology question showing why a recall failure is often a retrieval-cue problem rather than evidence that the memory was never stored.
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The All-or-None Law and Stimulus Intensity
A public GRE Psychology question on how a neuron encodes a stronger stimulus, and why the answer cannot be a bigger action potential.
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Conservation and the Preoperational Child
A 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.
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Two-Factor Theory and Misattributed Arousal
A public GRE Psychology question that separates the two-factor theory of emotion from the James-Lange, Cannon-Bard, appraisal, and facial-feedback accounts.
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Positive and Negative Symptoms of Schizophrenia
A public GRE Psychology question on sorting a case description into positive symptoms, negative symptoms, and medication side effects.
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IQ Scores and the Normal Distribution
A 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.
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Identifying Schools of Psychology
A public GRE Psychology question that reads a stated methodological commitment back to the school of psychology it defines.
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GMAT Exam
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Successive percentage changes
Track the changing base in a price increase followed by a discount, with an explanation for every answer.
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Find an alternative cause
Identify evidence that weakens a causal conclusion without disputing the reported observation.
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Is a ratio enough to determine a count?
Separate a relative quantity from an absolute count in an original data-sufficiency exercise.
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Combined Work Rates
A public work-rate question that shows why individual completion times must be converted to rates before combining two workers.
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Dilution Mixtures
A public mixture question that shows how to track the unchanged amount of solute with a proportion when a solution is diluted.
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