{
  "title": "Area of Segment - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Area of segment of circle radius 7cm with central angle 60°:", "options": {"A": "49/6(π/3 - √3/2) cm²", "B": "49/3(π/3 - √3/2) cm²", "C": "49(π/3 - √3/2) cm²", "D": "7(π/3 - √3/2) cm²"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "Formula for area of minor segment (θ in degrees):", "options": {"A": "(θπ/360 - sinθ/2)r²", "B": "(θπ/180 - sinθ)r²", "C": "r²θ/2", "D": "πr²θ/360"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "For radius 14cm, angle 90°, area of segment:", "options": {"A": "196(π/2 - 1) cm²", "B": "98(π/2 - 1) cm²", "C": "392(π/2 - 1) cm²", "D": "49(π/2 - 1) cm²"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "Area of segment = area of sector - area of:", "options": {"A": "Triangle", "B": "Square", "C": "Rectangle", "D": "Circle"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "For angle 180°, area of segment equals:", "options": {"A": "Area of semicircle", "B": "Area of triangle", "C": "Zero", "D": "Area of circle"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "Radius 10cm, angle 60°. Area of segment ≈", "options": {"A": "9.06 cm²", "B": "18.12 cm²", "C": "27.18 cm²", "D": "36.24 cm²"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "For very small angle, segment area ≈", "options": {"A": "(1/24)r²θ³", "B": "r²θ", "C": "πr²θ/360", "D": "0"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "Radius 7cm, angle 30°. Area of segment:", "options": {"A": "49/12(π/6 - sin30°) cm²", "B": "49(π/6 - 0.5) cm²", "C": "7(π/6 - 0.5) cm²", "D": "49/6(π/6 - 0.5) cm²"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "Area of major segment = area of circle - area of:", "options": {"A": "Minor segment", "B": "Sector", "C": "Triangle", "D": "Diameter"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "For radius r, angle θ, area of triangle in segment:", "options": {"A": "(1/2)r²sinθ", "B": "r²sinθ", "C": "(1/2)r²cosθ", "D": "r²θ"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "When θ=60°, sinθ=", "options": {"A": "√3/2", "B": "1/2", "C": "1", "D": "0"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "Radius 21cm, angle 120°. Area of segment:", "options": {"A": "441/2(2π/3 - √3/2) cm²", "B": "441(2π/3 - √3/2) cm²", "C": "220.5(2π/3 - √3/2) cm²", "D": "882(2π/3 - √3/2) cm²"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "For θ=90°, area of segment = r²(π/4 - )", "options": {"A": "1/2", "B": "√2/2", "C": "1", "D": "√3/2"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Chord length for radius r, angle θ:", "options": {"A": "2r sin(θ/2)", "B": "2r cos(θ/2)", "C": "rθ", "D": "r sinθ"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "Area of segment is maximum when θ =", "options": {"A": "π", "B": "π/2", "C": "π/3", "D": "π/4"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "Radius 14cm, chord length 14cm. Area of segment:", "options": {"A": "196(π/3 - √3/4) cm²", "B": "98(π/3 - √3/4) cm²", "C": "392(π/3 - √3/4) cm²", "D": "49(π/3 - √3/4) cm²"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "When θ=180°, area of segment =", "options": {"A": "πr²/2", "B": "πr²", "C": "r²", "D": "0"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "For same radius, which angle gives largest segment area?", "options": {"A": "180°", "B": "90°", "C": "60°", "D": "30°"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "Radius 10cm, area of segment 25(π/3 - √3/4) cm². Angle:", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "30°"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "Area of segment formula (θ in radians):", "options": {"A": "(1/2)r²(θ - sinθ)", "B": "r²(θ - sinθ)", "C": "(1/2)r²θ", "D": "r²sinθ"}, "correct_answer": "A"},

    {"id": 21, "difficulty": "medium", "question": "Radius 7cm, chord length 7cm. Area of segment:", "options": {"A": "49/12(π/3 - √3/4) cm²", "B": "49/6(π/3 - √3/4) cm²", "C": "49/3(π/3 - √3/4) cm²", "D": "49(π/3 - √3/4) cm²"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Area of major segment radius 14cm, angle 60°:", "options": {"A": "196π - 49(π/3 - √3/4) cm²", "B": "196π - 98(π/3 - √3/4) cm²", "C": "196π - 147(π/3 - √3/4) cm²", "D": "196π - 24.5(π/3 - √3/4) cm²"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Chord subtends 90° at center. Area of minor segment:", "options": {"A": "r²(π/4 - 1/2)", "B": "r²(π/2 - 1)", "C": "r²(π - 1)", "D": "r²(π/8 - 1/4)"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "Radius 10cm, area of segment 25(π/2 - 1) cm². Angle:", "options": {"A": "90°", "B": "60°", "C": "120°", "D": "45°"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Area of segment = (1/2)r²(θ - sinθ). For θ=π/3, area:", "options": {"A": "(1/2)r²(π/3 - √3/2)", "B": "r²(π/3 - √3/2)", "C": "(1/2)r²(π/6 - 1/2)", "D": "r²(π/6 - 1/2)"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Chord length 14√2 cm in circle radius 14cm. Area of segment:", "options": {"A": "196(π/2 - 1) cm²", "B": "98(π/2 - 1) cm²", "C": "392(π/2 - 1) cm²", "D": "49(π/2 - 1) cm²"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Two segments from chords of length 14cm each in circle radius 14cm. Their areas:", "options": {"A": "Equal", "B": "Different", "C": "One zero", "D": "Can't say"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "Area of segment radius r, chord length c. Height of segment h =", "options": {"A": "r - √(r² - c²/4)", "B": "r + √(r² - c²/4)", "C": "c/2", "D": "√(4r² - c²)/2"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "Radius 10cm, chord length 10cm. Angle:", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "30°"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "Area of segment = area of sector when:", "options": {"A": "θ=0", "B": "θ=π", "C": "θ=π/2", "D": "Never"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "For radius 7cm, area of segment 14.5 cm² (approx). Angle about:", "options": {"A": "90°", "B": "60°", "C": "45°", "D": "30°"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "Chord distance from center 3cm, radius 5cm. Area of segment:", "options": {"A": "25(cos⁻¹(3/5) - 4/5) cm²", "B": "25(2cos⁻¹(3/5) - 4/5) cm²", "C": "25/2(2cos⁻¹(3/5) - 4/5) cm²", "D": "25/2(cos⁻¹(3/5) - 4/5) cm²"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "Ratio area segment : area sector for θ=60°:", "options": {"A": "(π/3 - √3/2):(π/3)", "B": "1:1", "C": "√3:π", "D": "π-√3:π"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Area of major segment radius 10cm, chord length 10√2 cm:", "options": {"A": "100π - 50(π/2 - 1) cm²", "B": "100π - 25(π/2 - 1) cm²", "C": "100π - 100(π/2 - 1) cm²", "D": "100π - 75(π/2 - 1) cm²"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "Radius 14cm, area of segment 49(π - √3/2) cm². Segment is:", "options": {"A": "Major", "B": "Minor", "C": "Semicircle", "D": "Can't say"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "For θ=120°, area segment = r²( )", "options": {"A": "(π/3 - √3/4)", "B": "(2π/3 - √3/2)", "C": "(π/3 - √3/2)", "D": "(2π/3 - √3/4)"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "Chord length equal to radius. Area segment/r² =", "options": {"A": "(π/3 - √3/4)", "B": "(π/6 - √3/8)", "C": "(π/3 - 1/2)", "D": "(π/6 - 1/4)"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Radius 10cm, chord length 16cm. Area segment:", "options": {"A": "100(2sin⁻¹(0.8) - 1.2) cm²", "B": "50(2sin⁻¹(0.8) - 1.2) cm²", "C": "100(sin⁻¹(0.8) - 0.6) cm²", "D": "50(sin⁻¹(0.8) - 0.6) cm²"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Two chords equal length, different distances from center. Their segment areas:", "options": {"A": "Different", "B": "Same", "C": "Equal if chords equal", "D": "Can't compare"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "Area of segment = (1/6) area circle. θ ≈", "options": {"A": "120°", "B": "90°", "C": "60°", "D": "30°"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "medium", "question": "Radius 21cm, chord length 21cm. Area segment ≈", "options": {"A": "220.5(π/3 - √3/4) cm²", "B": "441(π/3 - √3/4) cm²", "C": "110.25(π/3 - √3/4) cm²", "D": "882(π/3 - √3/4) cm²"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "medium", "question": "For same chord length, area segment maximum when:", "options": {"A": "Radius minimum", "B": "Radius maximum", "C": "Radius = chord/2", "D": "Always same"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "medium", "question": "Area segment = (1/4) area circle. θ ≈", "options": {"A": "180°", "B": "120°", "C": "90°", "D": "60°"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "medium", "question": "Chord length 24cm, distance from center 5cm. Area segment:", "options": {"A": "169(2sin⁻¹(12/13) - 120/169) cm²", "B": "169/2(2sin⁻¹(12/13) - 120/169) cm²", "C": "169(sin⁻¹(12/13) - 60/169) cm²", "D": "169/2(sin⁻¹(12/13) - 60/169) cm²"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "Radius r, area segment = area of equilateral triangle side r. θ =", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "150°"}, "correct_answer": "C"},
    {"id": 46, "difficulty": "medium", "question": "Two segments from perpendicular chords radius r. Sum areas:", "options": {"A": "r²(π/2 - 1)", "B": "r²(π - 2)", "C": "r²(π/4 - 1/2)", "D": "r²(π - 1)"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "Area segment radius 10cm equals 25(π - 2) cm². Chord length:", "options": {"A": "10√2 cm", "B": "10 cm", "C": "10√3 cm", "D": "20 cm"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "medium", "question": "For θ=30°, area segment ≈", "options": {"A": "0.024r²", "B": "0.12r²", "C": "0.024πr²", "D": "0.12πr²"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "medium", "question": "Radius 14cm, area segment 98(π/2 - 1) cm². Chord length:", "options": {"A": "14√2 cm", "B": "14 cm", "C": "28 cm", "D": "7√2 cm"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "medium", "question": "Area segment = (1/2)r²(θ - sinθ). Maximum when θ =", "options": {"A": "π", "B": "π/2", "C": "2π", "D": "π/3"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "Chord divides circle area 1:3. Area segment/area circle =", "options": {"A": "1/4", "B": "3/4", "C": "1/2", "D": "1/3"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "Radius 7cm, area triangle in segment 49√3/4 cm². θ =", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "30°"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "Area major segment = area circle when:", "options": {"A": "θ=0", "B": "θ=π", "C": "θ=2π", "D": "Never"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "Two circles radii 7cm, 14cm have segments with same chord length 14cm. Areas ratio:", "options": {"A": "Different, need angle", "B": "1:4", "C": "1:2", "D": "1:√2"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "Radius 10cm, height of segment 2cm. Area segment:", "options": {"A": "100(2cos⁻¹(4/5) - 24/25) cm²", "B": "50(2cos⁻¹(4/5) - 24/25) cm²", "C": "100(cos⁻¹(4/5) - 12/25) cm²", "D": "50(cos⁻¹(4/5) - 12/25) cm²"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "Area segment radius r, chord length c. Formula using c only:", "options": {"A": "(r²/2)(2sin⁻¹(c/2r) - (c/2r)√(1 - c²/4r²))", "B": "Complex", "C": "No simple form", "D": "c³/12r"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "For θ=90°, area segment/area sector =", "options": {"A": "(π/2 - 1)/(π/2)", "B": "1/2", "C": "√2/π", "D": "π/4"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "Radius 5cm, area segment 6.25(π - 2) cm². This is:", "options": {"A": "Major segment", "B": "Minor segment", "C": "Semicircle", "D": "Can't say"}, "correct_answer": "B"},
    {"id": 59, "difficulty": "medium", "question": "Chord length equal to radius. Area segment/area circle =", "options": {"A": "(π/3 - √3/4)/π", "B": "1/6", "C": "1/12", "D": "√3/4π"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "Two segments from complementary angles. Sum areas:", "options": {"A": "(π/2 - 1)r²", "B": "(π - 2)r²", "C": "r²/2", "D": "πr²/4"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "medium", "question": "Area segment = area of square with side r/2. θ ≈", "options": {"A": "120°", "B": "90°", "C": "60°", "D": "30°"}, "correct_answer": "B"},
    {"id": 62, "difficulty": "medium", "question": "Radius 14cm, chord length 14√3 cm. Area segment:", "options": {"A": "196(π/3 - √3/4) cm²", "B": "98(π/3 - √3/4) cm²", "C": "392(π/3 - √3/4) cm²", "D": "49(π/3 - √3/4) cm²"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "medium", "question": "For small θ, area segment ≈ (1/24)r²θ³ (θ in radians). For θ=10° ≈ 0.1745 rad, area ≈", "options": {"A": "0.00022r²", "B": "0.0022r²", "C": "0.022r²", "D": "0.22r²"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "medium", "question": "Area major segment radius 7cm, chord length 7cm:", "options": {"A": "49π - 49/12(π/3 - √3/4) cm²", "B": "49π - 49/6(π/3 - √3/4) cm²", "C": "49π - 49/3(π/3 - √3/4) cm²", "D": "49π - 49(π/3 - √3/4) cm²"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "Chord divides circle into two segments areas ratio 1:2. θ for smaller segment ≈", "options": {"A": "120°", "B": "90°", "C": "60°", "D": "30°"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "medium", "question": "Radius 10cm, area segment 25(2π/3 - √3/2) cm². θ =", "options": {"A": "120°", "B": "90°", "C": "60°", "D": "30°"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "medium", "question": "Area segment maximum possible for given perimeter occurs when:", "options": {"A": "θ = π", "B": "θ = 2 rad", "C": "Chord = diameter", "D": "Can't maximize"}, "correct_answer": "C"},
    {"id": 68, "difficulty": "medium", "question": "Two chords parallel, equal length. Areas between them:", "options": {"A": "Equal", "B": "Different", "C": "Sum zero", "D": "Can't say"}, "correct_answer": "B"},
    {"id": 69, "difficulty": "medium", "question": "Radius r, area segment = (1/4) area circle. Then θ satisfies:", "options": {"A": "θ - sinθ = π/2", "B": "θ - sinθ = π/4", "C": "θ - sinθ = π", "D": "θ - sinθ = 1"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "medium", "question": "Chord length c, distance from center d. Area segment:", "options": {"A": "r²cos⁻¹(d/r) - d√(r² - d²)", "B": "2[r²cos⁻¹(d/r) - d√(r² - d²)]", "C": "r²sin⁻¹(d/r) - d√(r² - d²)", "D": "2[r²sin⁻¹(d/r) - d√(r² - d²)]"}, "correct_answer": "A"},
    {"id": 71, "difficulty": "medium", "question": "Area segment = area of inscribed square side r/√2. θ ≈", "options": {"A": "90°", "B": "120°", "C": "60°", "D": "45°"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "Radius 7cm, area major segment = 150 cm² (approx). Area minor segment ≈", "options": {"A": "4 cm²", "B": "40 cm²", "C": "100 cm²", "D": "154 cm²"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "For θ=180°, area segment/area semicircle =", "options": {"A": "1", "B": "2", "C": "π/2", "D": "π"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "medium", "question": "Two segments from chords making right angle at center. Sum areas:", "options": {"A": "r²(π/2 - 1)", "B": "r²(π - 2)", "C": "r²(π/4 - 1/2)", "D": "2r²(π/4 - 1/2)"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "Area segment radius 14cm = 98 cm². Possible θ:", "options": {"A": "About 90°", "B": "About 60°", "C": "About 120°", "D": "About 30°"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "medium", "question": "Chord length equal to radius√2. Area segment:", "options": {"A": "r²(π/2 - 1)", "B": "r²(π/4 - 1/2)", "C": "r²(π - √2)", "D": "r²(π/2 - √2/2)"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "Area segment minimal (non-zero) occurs when:", "options": {"A": "θ→0", "B": "θ=π/2", "C": "θ=π", "D": "θ=2π"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "Radius 10cm, chord divides circle into segments areas 50π cm² and 50π cm². Chord is:", "options": {"A": "Diameter", "B": "Radius", "C": "Side of hexagon", "D": "Not possible"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "medium", "question": "Area segment = (1/2) area sector. Then θ satisfies:", "options": {"A": "θ - sinθ = θ/2", "B": "θ - sinθ = 0", "C": "θ - sinθ = θ", "D": "sinθ = θ/2"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "medium", "question": "Two circles radii R,r (R>r) with same chord length c. Ratio areas segments:", "options": {"A": "R²/r² × (θ_R - sinθ_R)/(θ_r - sinθ_r)", "B": "R/r", "C": "1:1", "D": "c² dependent"}, "correct_answer": "A"},

    {"id": 81, "difficulty": "hard", "question": "Area of segment radius r, chord length c. Exact formula:", "options": {"A": "(r²/2)(2sin⁻¹(c/2r) - (c/r)√(1 - c²/4r²))", "B": "r²sin⁻¹(c/2r) - (c/2)√(r² - c²/4)", "C": "Both equivalent", "D": "Neither"}, "correct_answer": "C"},
    {"id": 82, "difficulty": "hard", "question": "Maximum area of segment for given perimeter P occurs when:", "options": {"A": "Arc length = 2×chord length", "B": "θ = 2 rad", "C": "r = P/(2+π)", "D": "All wrong"}, "correct_answer": "B"},
    {"id": 83, "difficulty": "hard", "question": "Area segment = k × area circle. For k=1/4, θ approximately:", "options": {"A": "2.31 rad (132.5°)", "B": "1.57 rad (90°)", "C": "1.05 rad (60°)", "D": "0.52 rad (30°)"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Two segments from intersecting chords dividing circle into 4 parts. If chords perpendicular, sum opposite segment areas:", "options": {"A": "πr²/2", "B": "r²(π - 2)", "C": "r²", "D": "πr²/4"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "Segment area as function of height h: A = r²cos⁻¹((r-h)/r) - (r-h)√(2rh - h²). For small h, A ≈", "options": {"A": "(2/3)√(2r) h^(3/2)", "B": "(4/3)√(r) h^(3/2)", "C": "√(2r) h^(3/2)", "D": "2√(r) h^(3/2)"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "Maximum ratio area segment/area sector occurs when:", "options": {"A": "θ→0", "B": "θ=π", "C": "θ where derivative zero", "D": "θ=π/2"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "Area segment = area of triangle with same chord and height. Then:", "options": {"A": "θ = 2 rad", "B": "h = 2r/3", "C": "c = r√3", "D": "No solution"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "Two segments from parallel chords distance d apart. Area between them:", "options": {"A": "2[r²cos⁻¹(d/r) - d√(r² - d²)]", "B": "r²[cos⁻¹(d/r) - (d/r)√(1 - d²/r²)]", "C": "Complex", "D": "No formula"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "Segment area = (1/6)πr². Solve θ - sinθ = π/3. Approx θ:", "options": {"A": "2.0 rad (114.6°)", "B": "2.3 rad (131.8°)", "C": "1.8 rad (103.1°)", "D": "2.5 rad (143.2°)"}, "correct_answer": "B"},
    {"id": 90, "difficulty": "hard", "question": "For given chord length c, area segment maximum when radius:", "options": {"A": "→∞", "B": "= c/2", "C": "= c/√2", "D": "= c"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "Segment formed by chord and arc has perimeter P. Maximum area when:", "options": {"A": "Arc length = 2×chord length", "B": "r = P/(2+π)", "C": "θ = 2 rad", "D": "All equivalent"}, "correct_answer": "D"},
    {"id": 92, "difficulty": "hard", "question": "Area segment = area of rectangle with sides chord and h. Then:", "options": {"A": "h = (r/2)(θ - sinθ)/sin(θ/2)", "B": "h = r(1 - cos(θ/2))", "C": "Both", "D": "Neither"}, "correct_answer": "C"},
    {"id": 93, "difficulty": "hard", "question": "Two circles radii R,r with segments of equal area. If chords equal, R/r =", "options": {"A": "√[(θ_r - sinθ_r)/(θ_R - sinθ_R)]", "B": "r/R", "C": "1", "D": "Depends on θ"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "Segment area A, chord length c, height h. Relation: c² =", "options": {"A": "8rh - 4h²", "B": "4h(2r - h)", "C": "Both same", "D": "4r² - (2r-h)²"}, "correct_answer": "C"},
    {"id": 95, "difficulty": "hard", "question": "Maximum area segment inscribed in semicircle occurs when:", "options": {"A": "Chord parallel to diameter", "B": "Chord makes 60° with diameter", "C": "Chord is radius", "D": "Isosceles right triangle"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "Area segment = k × r². For small k, θ ≈", "options": {"A": "(6k)^(1/3)", "B": "(24k)^(1/3)", "C": "(12k)^(1/3)", "D": "(3k)^(1/3)"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "Segment folded to form cone. If segment radius R, angle θ, cone dimensions:", "options": {"A": "Slant height = R, radius base = Rθ/2π", "B": "Height = R√(1 - θ²/4π²)", "C": "Both", "D": "Neither"}, "correct_answer": "C"},
    {"id": 98, "difficulty": "hard", "question": "Two segments from chords dividing circle into n equal areas. Central angles:", "options": {"A": "Solve θ_i - sinθ_i = 2πi/n", "B": "Equal angles", "C": "Arithmetic progression", "D": "No simple relation"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "Area segment as function of θ: A(θ) = (1/2)r²(θ - sinθ). A'(θ) =", "options": {"A": "(1/2)r²(1 - cosθ)", "B": "r²(1 - cosθ)", "C": "(1/2)r²sinθ", "D": "r²θ"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "For segment with given area A, minimum perimeter occurs when:", "options": {"A": "θ = 2 rad", "B": "r = √(2A/θ)", "C": "Chord = 2r sin(θ/2)", "D": "All related"}, "correct_answer": "A"}
  ]
}