Rotation and reflection matrices
112 GSAT Mathematics A, Multiple select 11: compare images, inverses, and compositions of rotation and reflection matrices.
Exam visualizations
Start from one past exam problem. First see which step of reasoning trips up most students, then take that step apart with an interactive figure.
17 problems
Combining sine and cosine
112 GSAT Mathematics A, Multiple select 12: combine sine and cosine into a single wave, then read off the phase, the axes of symmetry, and the translation.
Skew lines, the common perpendicular, and distance in space
112 GSAT Mathematics A, Fill-in 17: use direction vectors and the common perpendicular to combine three mutually perpendicular lengths into a distance in space.
Same area, same volume for the solid of revolution?
114 AST Mathematics I, Non-multiple-choice 17: compare a family of functions with the same area, and find the maximum volume of the solid of revolution by the disk method.
A lottery wait and the geometric distribution
113 AST Mathematics I, Multiple select 4: the geometric distribution, counted as trials until the first win, gives the mean wait, the chance from finitely many tokens, and the nine-tenths cutoff.
Center of symmetry of a cubic
114 GSAT Mathematics A, Fill-in 13 uses polynomial division and the quotient's axis of symmetry to locate the cubic's center of symmetry.
A parabola under a restricted translation
115 GSAT Mathematics A, Fill-in 16: keep the vertex on one line and the graph through one fixed point, then discard the translation that leaves the parabola where it was.
The same row operations, as a linear combination
113 AST Mathematics I, Fill-in 9: the same row operations preserve a linear combination of the constant columns, which gives the solution of the new system.
A 120° isosceles construction and the cosine law
112 AST Mathematics I, Fill-in 9: read the vertex angle and the base angles, then use an angle-sum formula and the cosine law for the squared distance between the new vertices.
Two trigonometric ratios on a focal chord
111 AST Mathematics I, Multiple select 7: equal lengths from the definition of a parabola, and the right triangles from the projections, identify one and the same segment ratio.
Equal distances and a square on the unit circle
111 AST Mathematics I, Fill-in 11: read the complex absolute value as a distance, and find the complex number in quadrant I from the perpendicular bisector of a chord of the unit circle.
Center on the x-axis, a distance ratio, and tangent slopes
115 AST Mathematics I, Fill-in 10: a circle with center on the x-axis is fixed by a ratio of distances to two perpendicular lines, then the slopes of the tangents through the origin are found.
Distance between parallel traces of a plane on the coordinate planes
114 AST Mathematics I, Fill-in 10: a plane meets two coordinate planes in parallel lines; find the shortest distance between them.
Parallelogram area with the edge directions fixed
114 AST Mathematics I, Fill-in 11: the sides stay on two fixed directions, and the area is read from the vector between the center and one vertex.
Shortest path on a lampshade
2023 AMC 12B, Problem #21 unfolds the shade into a half-annulus, where the straight line leaves the paper and the shortest path is a tangent plus an inner arc.
Equilateral triangle on three concentric circles
2025 AMC 12B, Problem #25 turns the radius-2 circle 60° about a vertex so that it is internally tangent to the outer circle, and the squared side is 7.
Area probability for two random points
2024 AMC 12A, Problem #20 turns the area ratio into the product xy and pins the probability between 3/4 and 7/8 with the unit square.
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