{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Cumulative Mixed Practice Exam",
    "author": "Staples Education",
    "description": "The twelve-question timed cumulative exam from the tutoring whiteboard archive, reproduced with a targeted explanation for every choice. Deliberately jumps between topics — that mix is the point.",
    "created": "2026-08-06T04:14:56.156Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "cumulative-review",
      "title": "Cumulative Mixed Practice Exam",
      "blurb": "Twelve questions jumping across the whole course — polygons, solids, right triangles, circles, coordinates and scaling. Timed like the real thing: aim for under 18 minutes, no notes.",
      "meta": "19 questions · mixed topics",
      "categories": [
        "polygons",
        "circles",
        "coordinate-lines",
        "pythagorean-theorem",
        "surface-area-volume",
        "dilations"
      ],
      "questions": [
        {
          "id": "cum-001",
          "topic": "Polygons",
          "difficulty": "medium",
          "stem": "A regular polygon has an interior angle of 144°. How many sides does it have?",
          "choices": [
            "8",
            "10",
            "12",
            "20"
          ],
          "correctIndex": 1,
          "explanations": [
            "8 sides gives an interior angle of 135°, not 144°.",
            "Correct. The exterior angle is 180° − 144° = 36°, and exterior angles total 360°, so n = 360/36 = 10.",
            "12 sides gives an interior angle of 150°.",
            "20 sides gives an interior angle of 162°."
          ],
          "hint": "Go via the exterior angle — it is the supplement, and they always add to 360°."
        },
        {
          "id": "cum-002",
          "topic": "Solids",
          "difficulty": "medium",
          "stem": "Find the volume of a cone with radius 5 and height 12, in terms of π.",
          "choices": [
            "100π",
            "300π",
            "60π",
            "200π"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. V = ⅓πr²h = ⅓π(25)(12) = ⅓π(300) = 100π.",
            "300π is the cylinder of the same dimensions. A cone is a third of it.",
            "60π uses ⅓π(5)(12) without squaring the radius.",
            "200π takes two thirds rather than one third."
          ],
          "hint": "Base area × height × ⅓."
        },
        {
          "id": "cum-003",
          "topic": "Pythagorean Theorem",
          "difficulty": "easy",
          "stem": "A right triangle has legs 8 and 15. Find the hypotenuse.",
          "choices": [
            "17",
            "23",
            "√23",
            "289"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. c² = 64 + 225 = 289, so c = 17. The 8-15-17 triple.",
            "23 adds the legs.",
            "√23 adds before squaring.",
            "289 is c²."
          ],
          "hint": "8-15-17 is one of the triples worth memorising."
        },
        {
          "id": "cum-004",
          "topic": "Circles",
          "difficulty": "medium",
          "stem": "Two chords intersect inside a circle, forming intercepted arcs of 70° and 50°. Find the angle between them.",
          "choices": [
            "60°",
            "10°",
            "120°",
            "35°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. For chords meeting inside a circle the angle is half the SUM of the intercepted arcs: ½(70 + 50) = ½(120) = 60°.",
            "10° halves the difference. The difference rule applies when the vertex is outside the circle.",
            "120° is the sum without halving.",
            "35° halves only the 70° arc. Both arcs contribute."
          ],
          "hint": "Inside the circle means add the arcs; outside means subtract. Then halve."
        },
        {
          "id": "cum-005",
          "topic": "Coordinate Geometry",
          "difficulty": "easy",
          "stem": "Find the distance between (−1, 4) and (2, 0).",
          "choices": [
            "5",
            "7",
            "√7",
            "25"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Δx = 3 and Δy = −4, so d = √(9 + 16) = √25 = 5.",
            "7 adds |Δx| and |Δy|.",
            "√7 adds the differences before squaring.",
            "25 is d²."
          ],
          "hint": "Another 3-4-5. Signs disappear once you square."
        },
        {
          "id": "cum-006",
          "topic": "Dilations",
          "difficulty": "medium",
          "stem": "A figure with area 12 is dilated by k = 4. Find the new area.",
          "choices": [
            "48",
            "192",
            "16",
            "36"
          ],
          "correctIndex": 1,
          "explanations": [
            "48 multiplies by k rather than k².",
            "Correct. Area scales by k²: 12 × 4² = 12 × 16 = 192.",
            "16 is k² alone, ignoring the original area.",
            "36 multiplies by 3. The factor is 4."
          ],
          "hint": "Area is two-dimensional, so the factor is applied twice."
        },
        {
          "id": "cum-007",
          "topic": "Triangles",
          "difficulty": "medium",
          "stem": "The angles of a triangle are x, x + 20 and 2x. Find the largest angle.",
          "choices": [
            "40°",
            "60°",
            "80°",
            "100°"
          ],
          "correctIndex": 2,
          "explanations": [
            "40° is the value of x, which is also the smallest angle. The question asks for the largest.",
            "60° is x + 20, the middle angle.",
            "Correct. The angles sum to 180°: x + (x + 20) + 2x = 4x + 20 = 180, so 4x = 160 and x = 40. The largest is 2x = 80°.",
            "100° would exceed the total once the other two are added: 40 + 60 + 100 = 200."
          ],
          "hint": "Sum to 180°, solve for x, then compare all three before answering."
        },
        {
          "id": "cum-008",
          "topic": "Special Right Triangles",
          "difficulty": "medium",
          "stem": "A 30-60-90 triangle has hypotenuse 14. Find the shorter leg.",
          "choices": [
            "7",
            "7√3",
            "14/√3",
            "7√2"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. In the ratio x : x√3 : 2x the hypotenuse is 2x, so 2x = 14 gives x = 7 as the short leg.",
            "7√3 ≈ 12.12 is the LONGER leg, x√3.",
            "14/√3 would follow from treating 14 as the long leg rather than the hypotenuse.",
            "7√2 uses the 45-45-90 constant."
          ],
          "hint": "The hypotenuse is exactly twice the short leg in a 30-60-90 triangle."
        },
        {
          "id": "cum-009",
          "topic": "Coordinate Geometry",
          "difficulty": "medium",
          "stem": "Find the equation of the line through (0, 3) perpendicular to y = −¼x.",
          "choices": [
            "y = 4x + 3",
            "y = −4x + 3",
            "y = ¼x + 3",
            "y = −¼x + 3"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The negative reciprocal of −¼ is 4. Since the point is on the y-axis, the intercept is 3, giving y = 4x + 3.",
            "y = −4x + 3 inverts the slope but keeps it negative. The product (−¼)(−4) = 1, not −1.",
            "y = ¼x + 3 flips the sign without inverting. The product is −1/16.",
            "y = −¼x + 3 is parallel to the given line, not perpendicular."
          ],
          "hint": "Negative reciprocal: flip it and change the sign. The original slope is already negative."
        },
        {
          "id": "cum-010",
          "topic": "Solids",
          "difficulty": "medium",
          "stem": "A sphere has radius 6. Find its volume in terms of π.",
          "choices": [
            "144π",
            "288π",
            "216π",
            "864π"
          ],
          "correctIndex": 1,
          "explanations": [
            "144π is the surface area, 4π(36).",
            "Correct. V = (4/3)πr³ = (4/3)π(216) = 288π.",
            "216π is r³ times π, missing the 4/3.",
            "864π is 4πr³, forgetting to divide by 3."
          ],
          "hint": "Cube the radius first, then apply the 4/3."
        },
        {
          "id": "cum-011",
          "topic": "Circles",
          "difficulty": "easy",
          "stem": "An inscribed angle intercepts a 130° arc. Find the angle.",
          "choices": [
            "65°",
            "260°",
            "130°",
            "50°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. An inscribed angle is half its intercepted arc: ½(130°) = 65°.",
            "260° doubles the arc instead of halving it.",
            "130° repeats the arc — that would be the central angle on the same arc.",
            "50° is the supplement of 130° divided by nothing obvious. The rule here is simply to halve."
          ],
          "hint": "Inscribed angle equals half its arc."
        },
        {
          "id": "cum-012",
          "topic": "Area",
          "difficulty": "easy",
          "stem": "Find the area of a trapezoid with parallel sides 8 and 14 and height 5.",
          "choices": [
            "110",
            "55",
            "27",
            "70"
          ],
          "correctIndex": 1,
          "explanations": [
            "110 is (b₁ + b₂)h without halving.",
            "Correct. A = ½(b₁ + b₂)h = ½(8 + 14)(5) = ½(22)(5) = 55.",
            "27 adds all three numbers.",
            "70 multiplies only 14 by 5, using one base instead of the average of both."
          ],
          "hint": "Average the two parallel sides, then multiply by the height."
        },
        {
          "id": "cum-013",
          "topic": "Regular Polygons",
          "difficulty": "medium",
          "stem": "A regular polygon has an interior angle of 144°. How many sides does it have?",
          "choices": [
            "8",
            "10",
            "12",
            "20"
          ],
          "correctIndex": 1,
          "explanations": [
            "8 sides gives an exterior angle of 45° and an interior angle of 135°, not 144°.",
            "Correct. Interior 144° means exterior 180 − 144 = 36°. Since each exterior angle of a regular n-gon is 360/n, we get n = 360/36 = 10.",
            "12 sides gives an exterior angle of 30° and an interior angle of 150°. Close, but the arithmetic does not match 144°.",
            "20 sides comes from dividing 360 by 18 rather than 36 — likely from halving the exterior angle somewhere along the way."
          ],
          "hint": "Route through the exterior angle. It converts a fraction equation into a single division."
        },
        {
          "id": "cum-014",
          "topic": "Cone Volume",
          "difficulty": "easy",
          "stem": "Find the volume of a cone with radius 5 and height 12, in terms of π.",
          "choices": [
            "60π",
            "100π",
            "300π",
            "20π"
          ],
          "correctIndex": 1,
          "explanations": [
            "60π multiplies the radius and height without squaring the radius or applying the ⅓. The base is a circle, so r must be squared.",
            "Correct. V = ⅓πr²h = ⅓π(25)(12) = ⅓π(300) = 100π.",
            "300π is the CYLINDER volume πr²h with the same dimensions. A cone holds exactly one third of the cylinder that contains it — forgetting the ⅓ is a listed high-risk error.",
            "20π comes from using the diameter or otherwise mis-substituting. Substitute r = 5 and square it before multiplying."
          ],
          "hint": "A cone is a fraction of the cylinder with the same base and height. Do not lose that fraction."
        },
        {
          "id": "cum-015",
          "topic": "Special Right Triangles",
          "difficulty": "medium",
          "stem": "A 30-60-90 triangle has a hypotenuse of 14. Find the shorter leg.",
          "choices": [
            "7",
            "7√3",
            "14√3",
            "7√2"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. In a 30-60-90 triangle the sides are x : x√3 : 2x, so the hypotenuse is twice the short leg. From 2x = 14 we get x = 7.",
            "7√3 is the LONG leg, x√3 with x = 7. It sits opposite the 60° angle, not the 30° one.",
            "14√3 multiplies the hypotenuse by √3, which is not part of this ratio. The √3 attaches to the short leg to produce the long leg.",
            "7√2 uses the 45-45-90 relationship, where the hypotenuse is a leg times √2. That is the other special right triangle."
          ],
          "hint": "In this ratio the hypotenuse has the simplest relationship to one particular leg. Identify which."
        },
        {
          "id": "cum-016",
          "topic": "Mixed Review",
          "difficulty": "medium",
          "stem": "The angles of a triangle measure x, x + 20, and 2x. Find the largest angle.",
          "choices": [
            "40°",
            "60°",
            "80°",
            "100°"
          ],
          "correctIndex": 2,
          "explanations": [
            "40° is the value of x and the smallest of the three angles. Solving x + (x + 20) + 2x = 180 gives 4x = 160 and x = 40, but the question asks for the largest.",
            "60° is the middle angle, x + 20 = 40 + 20. The three angles are 40°, 60° and 80°, so this is not the largest.",
            "Correct. The angle sum gives x + (x + 20) + 2x = 180, so 4x + 20 = 180, then 4x = 160 and x = 40. The angles are 40°, 60° and 80°, making 80° the largest.",
            "100° would break the angle sum: 40 + 60 + 100 = 200, which exceeds 180°. No triangle has these angles."
          ],
          "hint": "Solve for the variable, evaluate all three expressions, and only then compare."
        },
        {
          "id": "cum-017",
          "topic": "Trapezoid Area",
          "difficulty": "easy",
          "stem": "Find the area of a trapezoid with bases 8 and 14 and height 5.",
          "choices": [
            "55",
            "110",
            "70",
            "27"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A = ½(b₁ + b₂)h = ½(8 + 14)(5) = ½(22)(5) = 55. The formula averages the two bases and multiplies by the height.",
            "110 is (8 + 14)(5) with the ½ omitted. Without halving you get the area of a parallelogram whose base is the sum of both — twice too much.",
            "70 is 14 × 5, using only the longer base. A trapezoid's bases differ, so both must enter the calculation.",
            "27 adds the three given numbers, 8 + 14 + 5. Area is a product-based quantity, not a sum of the given measurements."
          ],
          "hint": "The trapezoid formula takes the average of the two parallel sides, then treats that as a single base."
        },
        {
          "id": "cum-018",
          "topic": "Reasoning: Strategy",
          "difficulty": "hard",
          "stem": "You must find a side of a triangle. You know two sides and the angle BETWEEN them. Which tool applies?",
          "choices": [
            "The Pythagorean theorem, since two sides are known.",
            "The Law of Cosines, which handles two sides and their included angle.",
            "The Law of Sines, since an angle is known.",
            "Not enough information."
          ],
          "correctIndex": 1,
          "explanations": [
            "The Pythagorean theorem needs a RIGHT angle. Nothing says the included angle is 90°.",
            "Correct. Two sides and the included angle (SAS) is exactly the Law of Cosines case: c² = a² + b² − 2ab·cos C.",
            "The Law of Sines needs an angle paired with its OPPOSITE side. Here the known angle sits between the known sides.",
            "SAS determines a triangle completely, so the third side is fully determined."
          ],
          "hint": "Ask which parts are paired. The Law of Sines needs an angle and the side across from it.",
          "kind": "reasoning"
        },
        {
          "id": "cum-019",
          "topic": "Reasoning: Error Analysis",
          "difficulty": "medium",
          "stem": "A student proves two triangles congruent by AAA and concludes the triangles are the same size. What is wrong?",
          "choices": [
            "AAA is not a congruence criterion — it establishes SIMILARITY, which fixes shape but not size.",
            "AAA is valid; the student made an arithmetic error.",
            "AAA only works for right triangles.",
            "The student should have written AAS, which means the same thing."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Three pairs of congruent angles guarantee the same SHAPE but say nothing about size — a small triangle and a large one can have identical angles. Congruence needs at least one side: SSS, SAS, ASA, AAS or HL.",
            "AAA is not a congruence criterion at all, so there is no arithmetic that would rescue it.",
            "AAA fails to prove congruence for right triangles too. HL is the right-triangle shortcut.",
            "AAS includes a SIDE, which is exactly what AAA lacks. They are not the same."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}