{
  "title": "Perpendicular Bisector Theorem - Grade 9 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Perpendicular bisector of chord passes through:", "options": {"A": "One endpoint", "B": "Center of circle", "C": "Any point on chord", "D": "Midpoint of arc"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "If OM is perpendicular bisector of chord AB, then:", "options": {"A": "AM = MB", "B": "OA = OB", "C": "OM ⟂ AB", "D": "All of these"}, "correct_answer": "D"},
    {"id": 3, "difficulty": "easy", "question": "Chord of length 24cm is at distance 5cm from center. Radius?", "options": {"A": "10cm", "B": "12cm", "C": "13cm", "D": "15cm"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "In circle, equal chords are equidistant from:", "options": {"A": "Center", "B": "Any point", "C": "Circumference", "D": "Diameter"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "If chord is diameter, its perpendicular bisector is:", "options": {"A": "Another diameter", "B": "Radius", "C": "Tangent", "D": "Any line"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "Two chords equidistant from center are:", "options": {"A": "Parallel", "B": "Equal", "C": "Perpendicular", "D": "Tangents"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "For any chord AB and center O, triangle OAB is:", "options": {"A": "Right angled", "B": "Equilateral", "C": "Isosceles", "D": "Scalene"}, "correct_answer": "C"},
    {"id": 8, "difficulty": "easy", "question": "If perpendicular from center bisects chord, then it makes angle:", "options": {"A": "45° with chord", "B": "60° with chord", "C": "90° with chord", "D": "120° with chord"}, "correct_answer": "C"},
    {"id": 9, "difficulty": "easy", "question": "Chord nearer to center is:", "options": {"A": "Shorter", "B": "Longer", "C": "Equal", "D": "Parallel"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "In circle radius 10cm, chord distance 6cm from center has length:", "options": {"A": "8cm", "B": "12cm", "C": "16cm", "D": "20cm"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "If two chords have different distances from center, then:", "options": {"A": "Longer chord is closer", "B": "Shorter chord is closer", "C": "Equal chords", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "Perpendicular bisector of chord divides circle into:", "options": {"A": "Two equal arcs", "B": "Two unequal arcs", "C": "Three parts", "D": "Four parts"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "If chord length equals radius, distance from center is:", "options": {"A": "r/2", "B": "r√3/2", "C": "r√2/2", "D": "r/√2"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "Maximum distance of chord from center is:", "options": {"A": "0", "B": "r", "C": "2r", "D": "r/2"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "Minimum distance of chord from center is:", "options": {"A": "0", "B": "r", "C": "2r", "D": "r/2"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "Diameter's distance from center is:", "options": {"A": "0", "B": "r", "C": "2r", "D": "r/2"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Two chords at equal distances from center and on same side of center are:", "options": {"A": "Equal", "B": "Parallel", "C": "Congruent", "D": "A and C"}, "correct_answer": "D"},
    {"id": 18, "difficulty": "easy", "question": "If perpendicular from center to chord is 0, then chord is:", "options": {"A": "Diameter", "B": "Radius", "C": "Tangent", "D": "Secant"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "In circle, shortest chord through a point inside is:", "options": {"A": "Perpendicular to radius", "B": "Along diameter", "C": "Any chord", "D": "None of these"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "For fixed circle, as chord moves away from center, its length:", "options": {"A": "Increases", "B": "Decreases", "C": "Remains same", "D": "First increases then decreases"}, "correct_answer": "B"},
    {"id": 21, "difficulty": "medium", "question": "In circle radius 13cm, chord length 24cm. Distance from center?", "options": {"A": "5cm", "B": "7cm", "C": "10cm", "D": "12cm"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Two chords AB and CD are at distances 3cm and 4cm from center. Which is longer if radius is 5cm?", "options": {"A": "AB", "B": "CD", "C": "Equal", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Chord of length 16cm is at distance 6cm from center. Another chord at distance 8cm from center has length:", "options": {"A": "12cm", "B": "14cm", "C": "16cm", "D": "18cm"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "In circle radius 10cm, two parallel chords lengths 12cm and 16cm are on same side of center. Distance between chords?", "options": {"A": "1cm", "B": "2cm", "C": "3cm", "D": "4cm"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "AB and CD are two equal chords intersecting at P. Then:", "options": {"A": "AP = CP", "B": "BP = DP", "C": "AP = DP", "D": "All of these"}, "correct_answer": "D"},
    {"id": 26, "difficulty": "medium", "question": "Two chords at distances 5cm and 12cm from center of radius 13cm. Ratio of lengths?", "options": {"A": "5:12", "B": "12:5", "C": "24:10", "D": "10:24"}, "correct_answer": "C"},
  {"id": 27, "difficulty": "medium", "question": "Chord length 30cm, distance 8cm from center. Another chord length 34cm, its distance from center?", "options": {"A": "0 cm", "B": "√15 cm", "C": "√17 cm", "D": "√21 cm"}, "correct_answer": "A"},    {"id": 28, "difficulty": "medium", "question": "Through point P inside circle, chord AB is drawn. For shortest chord through P, OP is _____ to AB:", "options": {"A": "Parallel", "B": "Perpendicular", "C": "At 45°", "D": "At 60°"}, "correct_answer": "B"},
    {"id": 29, "difficulty": "medium", "question": "In circle radius r, chord at distance r/2 from center has length:", "options": {"A": "r√2", "B": "r√3", "C": "2r√(3/4)", "D": "r√5"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "Two parallel chords lengths 48cm and 14cm on opposite sides of center, distance between them 17cm. Radius?", "options": {"A": "25cm", "B": "26cm", "C": "27cm", "D": "28cm"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Chord length 2l, distance d from center. Radius?", "options": {"A": "√(l²+d²)", "B": "√(l²-d²)", "C": "√(4l²+d²)", "D": "√(l²+4d²)"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "Two chords AB=8cm, CD=6cm equidistant from center of radius 5cm. Distance from center?", "options": {"A": "3cm", "B": "4cm", "C": "√21 cm", "D": "√29 cm"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "In semicircle, perpendicular from center to chord of full circle divides chord in ratio 3:1. If radius is 10cm, chord length?", "options": {"A": "12cm", "B": "16cm", "C": "18cm", "D": "20cm"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "AB is diameter, CD is chord perpendicular to AB meeting AB at P. If AP:PB=1:3 and radius=10cm, CD=?", "options": {"A": "8√5 cm", "B": "10√3 cm", "C": "12√2 cm", "D": "15 cm"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "Two concentric circles radii 5cm and 13cm. Chord of larger circle touches smaller circle. Length of chord?", "options": {"A": "12cm", "B": "18cm", "C": "24cm", "D": "30cm"}, "correct_answer": "C"},
    {"id": 36, "difficulty": "medium", "question": "In circle radius 25cm, chord distance 7cm from center. Perpendicular distance from chord to point on circumference nearest to chord?", "options": {"A": "18cm", "B": "20cm", "C": "22cm", "D": "24cm"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "Chord length 40cm, distance 9cm from center. Perpendicular bisector of chord divides chord into segments each:", "options": {"A": "20cm", "B": "√481 cm", "C": "√400 cm", "D": "√441 cm"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Two chords AB=30cm, CD=16cm in circle radius 17cm. Which is closer to center?", "options": {"A": "AB", "B": "CD", "C": "Equidistant", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Point P inside circle radius 13cm, shortest chord through P is 10cm. Distance OP?", "options": {"A": "5cm", "B": "8cm", "C": "10cm", "D": "12cm"}, "correct_answer": "D"},
    {"id": 40, "difficulty": "medium", "question": "In circle radius 15cm, chord subtends 120° at center. Distance from center?", "options": {"A": "7.5cm", "B": "10cm", "C": "12.5cm", "D": "15cm"}, "correct_answer": "A"},
   {"id": 41, "difficulty": "medium", "question": "Chord length equal to distance from center. Angle subtended at center?", "options": {"A": "2 arcsin(1/√5)", "B": "60°", "C": "90°", "D": "120°"}, "correct_answer": "A"},    {"id": 42, "difficulty": "medium", "question": "Two parallel chords on same side of center lengths 10cm and 24cm, radius 13cm. Distance between chords?", "options": {"A": "5cm", "B": "7cm", "C": "9cm", "D": "11cm"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "medium", "question": "In circle, chord at distance 12cm from center is 5cm longer than chord at distance 5cm from center. Radius?", "options": {"A": "13cm", "B": "15cm", "C": "17cm", "D": "19cm"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "medium", "question": "Perpendicular bisectors of two chords intersect at center. Chords are:", "options": {"A": "Parallel", "B": "Equal", "C": "Perpendicular", "D": "Not necessarily parallel"}, "correct_answer": "D"},
    {"id": 45, "difficulty": "medium", "question": "Chord length 2a, distance b from center. Distance from midpoint of chord to center?", "options": {"A": "b", "B": "√(a²+b²)", "C": "√(a²-b²)", "D": "√(b²-a²)"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "medium", "question": "Two chords intersect at right angles. Perpendiculars from center to these chords are 3cm and 4cm. Radius if chords bisect each other?", "options": {"A": "5cm", "B": "6cm", "C": "7cm", "D": "8cm"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "In circle radius 10cm, chord distance x from center has length 2√(100-x²). Maximum length occurs when x=?", "options": {"A": "0", "B": "5", "C": "10", "D": "10√2"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "medium", "question": "Two chords AB and AC make equal angles with radius through A. Then:", "options": {"A": "AB=AC", "B": "B and C equidistant from center", "C": "Both A and B", "D": "Neither"}, "correct_answer": "C"},
    {"id": 49, "difficulty": "medium", "question": "Chord length 48cm, distance 7cm from center. Area between chord and minor arc?", "options": {"A": "(625π/2 - 168) cm²", "B": "(625π/2 - 336) cm²", "C": "(625π - 168) cm²", "D": "(625π - 336) cm²"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "medium", "question": "Point P divides chord AB in ratio 2:1. Line through P perpendicular to AB passes through center if:", "options": {"A": "P is midpoint", "B": "Always", "C": "Never", "D": "AP=2PB"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "Two circles radii 5cm and 12cm intersect. Common chord length 8cm. Distance between centers?", "options": {"A": "3+√105 cm", "B": "3+√135 cm", "C": "√105+√135 cm", "D": "√105-√135 cm"}, "correct_answer": "B"},
    {"id": 52, "difficulty": "medium", "question": "From point 17cm from center of circle radius 15cm, chord of length 16cm drawn. How many such chords?", "options": {"A": "0", "B": "1", "C": "2", "D": "Infinite"}, "correct_answer": "C"},
    {"id": 53, "difficulty": "medium", "question": "In circle radius 26cm, chord distance 10cm from center. Chord divides circle into two arcs. Ratio of arc lengths?", "options": {"A": "2:3", "B": "3:2", "C": "5:8", "D": "8:5"}, "correct_answer": "C"},
    {"id": 54, "difficulty": "medium", "question": "Chord length 84cm, distance 13cm from center. Perpendicular distance from chord to parallel tangent?", "options": {"A": "26cm", "B": "30cm", "C": "34cm", "D": "38cm"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "Two chords AB=2x, CD=2y equidistant from center. Then:", "options": {"A": "x=y", "B": "x²=y²", "C": "x²+y²=r²", "D": "x²-y²=r²"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "Through point at distance d from center, shortest chord has length 2√(r²-d²). Longest chord through point has length?", "options": {"A": "2r", "B": "2√(r²+d²)", "C": "2√(r²-d²)", "D": "2d"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "In circle, chord makes 60° with radius at endpoint. Distance from center if radius=r?", "options": {"A": "r/2", "B": "r√3/2", "C": "r/√2", "D": "r√2/2"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "Two concentric circles, chord of larger touches smaller. If radii R and r, chord length?", "options": {"A": "2√(R²-r²)", "B": "√(R²-r²)", "C": "2√(R²+r²)", "D": "√(R²+r²)"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "Chord length 2√3 r, distance from center?", "options": {"A": "r", "B": "r/2", "C": "r√3/2", "D": "r/√2"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "medium", "question": "Three parallel chords lengths 16cm, 18cm, 20cm in circle radius 15cm. Distances between them if on same side?", "options": {"A": "√11-√5, √5-√2", "B": "√15-√11, √11-√5", "C": "√15-√11, √5-√2", "D": "√15-√5, √5-√2"}, "correct_answer": "B"},
    {"id": 61, "difficulty": "hard", "question": "Two chords intersect at right angles. Segments of one chord are 6cm and 8cm, of other chord are 5cm and x cm. Radius if chords are equidistant from center?", "options": {"A": "√61 cm", "B": "√65 cm", "C": "√73 cm", "D": "√85 cm"}, "correct_answer": "B"},
    {"id": 62, "difficulty": "hard", "question": "Chord length 2a divides circle radius R into two segments. Height of smaller segment?", "options": {"A": "R-√(R²-a²)", "B": "R+√(R²-a²)", "C": "√(R²-a²)-R", "D": "√(R²+a²)-R"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "hard", "question": "From point P inside circle, chords PA, PB, PC drawn with PA perpendicular to PB. If PA=8, PB=6, PC=10, distance OP where O is center?", "options": {"A": "√13", "B": "√15", "C": "√17", "D": "√19"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "hard", "question": "Two circles radii 13cm and 15cm intersect, common chord length 24cm. Distance between centers if circles intersect orthogonally?", "options": {"A": "√194", "B": "√200", "C": "√205", "D": "√218"}, "correct_answer": "B"},
    {"id": 65, "difficulty": "hard", "question": "Chord AB of circle x²+y²=25 has midpoint M(3,1). Length AB?", "options": {"A": "4", "B": "6", "C": "8", "D": "10"}, "correct_answer": "C"},
    {"id": 66, "difficulty": "hard", "question": "Maximum number of chords of length 20cm in circle radius 15cm?", "options": {"A": "2", "B": "4", "C": "6", "D": "Infinite"}, "correct_answer": "D"},
    {"id": 67, "difficulty": "hard", "question": "Three chords equal length in circle radius R subtend angles 2α,2β,2γ at center with α+β+γ=π. Sum of distances from center to chords?", "options": {"A": "0", "B": "R", "C": "2R", "D": "3R"}, "correct_answer": "A"},
   {"id": 68, "difficulty": "hard", "question": "Chord length 2r sin θ at distance r cos θ from center. If chord subtends 90° at point on circumference, then θ=?", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "D"},    {"id": 69, "difficulty": "hard", "question": "Two chords AB and AC contain angle 60° at A. If AB=AC and distance from center to BC is 5cm, radius?", "options": {"A": "10cm", "B": "10√3 cm", "C": "15cm", "D": "20cm"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "hard", "question": "Chord length l divides circle area in ratio 1:2. Distance from center?", "options": {"A": "r cos(π/6)", "B": "r sin(π/6)", "C": "r cos(π/3)", "D": "r sin(π/3)"}, "correct_answer": "A"},
    {"id": 71, "difficulty": "hard", "question": "From point P on chord AB, perpendicular to AB meets circle at Q,R. If AP=3, PB=4, and PQ=2, radius?", "options": {"A": "√13", "B": "√15", "C": "√17", "D": "√19"}, "correct_answer": "C"},
    {"id": 72, "difficulty": "hard", "question": "Two chords intersect inside circle. Products of segments are equal. If one chord passes through center, the other is:", "options": {"A": "Diameter", "B": "Bisected by first", "C": "Perpendicular to first", "D": "Parallel to tangent"}, "correct_answer": "B"},
    {"id": 73, "difficulty": "hard", "question": "Regular hexagon inscribed in circle. Distance from center to side (apothem)?", "options": {"A": "r√3/2", "B": "r/2", "C": "r√2/2", "D": "r/√2"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "hard", "question": "Chord length 2√(2Rr-r²) where r is distance from center. For fixed R, maximum l occurs when r=?", "options": {"A": "0", "B": "R/2", "C": "R", "D": "2R"}, "correct_answer": "B"},
    {"id": 75, "difficulty": "hard", "question": "Two parallel chords lengths 30cm and 16cm on opposite sides of center distance 23cm apart. Radius?", "options": {"A": "17cm", "B": "18cm", "C": "19cm", "D": "20cm"}, "correct_answer": "A"},
   {"id": 76, "difficulty": "hard", "question": "Chord length 2R sin(θ/2) subtends angle θ at center. For chord equal to distance from center, θ satisfies?", "options": {"A": "sin(θ/2)=cos θ", "B": "sin(θ/2)=2 cos θ", "C": "2 sin(θ/2)=cos(θ/2)", "D": "sin(θ/2)=cos(θ/2)"}, "correct_answer": "C"},    {"id": 77, "difficulty": "hard", "question": "From external point P, tangent PT and secant PAB drawn. If PA=4, AB=5, distance from center to chord AB?", "options": {"A": "√13", "B": "√15", "C": "√17", "D": "√19"}, "correct_answer": "B"},
    {"id": 78, "difficulty": "hard", "question": "Two circles intersect orthogonally. Common chord is of length 2c. Distance between centers 2d. Relation?", "options": {"A": "c²+d²=r1²+r2²", "B": "c²+d²=r1r2", "C": "c²+d²=(r1²+r2²)/2", "D": "c²-d²=r1²-r2²"}, "correct_answer": "D"},
    {"id": 79, "difficulty": "hard", "question": "Chord divides circumference in ratio 1:9. Ratio of segments of diameter perpendicular to chord?", "options": {"A": "1:3", "B": "1:4", "C": "2:3", "D": "3:4"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "hard", "question": "Maximum area of triangle formed by chord length l and center?", "options": {"A": "l√(r²-l²/4)/2", "B": "l√(4r²-l²)/4", "C": "l²√(4r²-l²)/4", "D": "l²/4"}, "correct_answer": "B"},
    {"id": 81, "difficulty": "hard", "question": "Two chords perpendicular to each other have lengths 14cm and 48cm. Distance from intersection to center 5cm. Radius?", "options": {"A": "25cm", "B": "26cm", "C": "27cm", "D": "28cm"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "Chord length 2a moves parallel to itself at distance b from center. Locus of midpoint?", "options": {"A": "Circle radius √(a²+b²)", "B": "Circle radius √(a²-b²)", "C": "Line parallel to chord", "D": "Circle concentric with given"}, "correct_answer": "D"},
    {"id": 83, "difficulty": "hard", "question": "Three equal chords of circle radius R intersect at common point inside circle dividing each other in ratio 1:2. Distance from center to intersection?", "options": {"A": "R/2", "B": "R/√3", "C": "R√3/2", "D": "R/√2"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "From point on concentric circle radius r1, chord to outer circle radius r2 touches inner circle. Length?", "options": {"A": "2√(r2²-r1²)", "B": "√(r2²-r1²)", "C": "2√(r2²+r1²)", "D": "√(r2²+r1²)"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "Chord length l, distance d from center. Distance from midpoint of chord to nearest point on circumference?", "options": {"A": "R-√(R²-l²/4)", "B": "√(R²-l²/4)-d", "C": "R-d", "D": "√(R²-d²)-l/2"}, "correct_answer": "B"},
    {"id": 86, "difficulty": "hard", "question": "Regular polygon n sides inscribed in circle. Distance from center to side is R cos(π/n). For n=12, distance?", "options": {"A": "R√3/2", "B": "R√2/2", "C": "R/2", "D": "R√6/2"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "Two chords AB and CD intersect at E. If AE=4, EB=6, CE=3, ED=8, and distance from center to AB is 2, radius?", "options": {"A": "√13", "B": "√17", "C": "√21", "D": "√29"}, "correct_answer": "B"},
   {"id": 88, "difficulty": "hard", "question": "Chord subtends right angle at point on circumference. Distance from center if radius R?", "options": {"A": "0", "B": "R/2", "C": "R/√2", "D": "R"}, "correct_answer": "A"},    {"id": 89, "difficulty": "hard", "question": "Two parallel chords lengths 10cm and 24cm on same side of center distance 5cm apart. Radius if chords subtend complementary angles at center?", "options": {"A": "13cm", "B": "15cm", "C": "17cm", "D": "19cm"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "From point distance d from center, shortest chord makes angle θ with line joining point to center. Then sin θ=?", "options": {"A": "d/R", "B": "R/d", "C": "√(R²-d²)/R", "D": "√(R²-d²)/d"}, "correct_answer": "C"},
    {"id": 91, "difficulty": "hard", "question": "Chord of circle x²+y²-4x-6y-12=0 is bisected at (1,1). Length?", "options": {"A": "4", "B": "6", "C": "8", "D": "10"}, "correct_answer": "C"},
    {"id": 92, "difficulty": "hard", "question": "Two chords divide circle into four arcs with measures in arithmetic progression. If smallest chord subtends 60° at center, largest chord subtends?", "options": {"A": "90°", "B": "120°", "C": "150°", "D": "180°"}, "correct_answer": "B"},
    {"id": 93, "difficulty": "hard", "question": "Equal chords AB and AC contain angle 120° at A. Distance from A to midpoint of BC if radius R?", "options": {"A": "R√3/2", "B": "R", "C": "R√2", "D": "2R"}, "correct_answer": "B"},
    {"id": 94, "difficulty": "hard", "question": "From point on circle, chords are drawn to endpoints of diameter. Sum of squares of these chords?", "options": {"A": "2R²", "B": "4R²", "C": "6R²", "D": "8R²"}, "correct_answer": "B"},
    {"id": 95, "difficulty": "hard", "question": "Chord length 2R sin 18° in circle radius R. Distance from center?", "options": {"A": "R cos 18°", "B": "R sin 18°", "C": "R cos 36°", "D": "R sin 36°"}, "correct_answer": "C"},
    {"id": 96, "difficulty": "hard", "question": "Two intersecting chords are divided into segments proportional to adjacent sides of cyclic quadrilateral. If chords are perpendicular, ratio of segments?", "options": {"A": "1:1", "B": "depends on angle", "C": "equal to ratio of chords", "D": "not determined"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "Chord length equal to side of inscribed square. Angle subtended at center?", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "varies with radius"}, "correct_answer": "B"},
    {"id": 98, "difficulty": "hard", "question": "Through fixed point inside circle, chords drawn. Locus of midpoints?", "options": {"A": "Circle", "B": "Line", "C": "Parabola", "D": "Ellipse"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "Two chords AB and CD intersect at E inside circle. If AE=EB and CE:ED=1:3, and distance from center to AB is half that to CD, ratio of lengths AB:CD?", "options": {"A": "1:2", "B": "2:1", "C": "1:√2", "D": "√2:1"}, "correct_answer": "D"},
    {"id": 100, "difficulty": "hard", "question": "Chord length 2√(R²-d²) moves keeping distance d from center. Locus of midpoint of chord joining endpoints of fixed diameter and moving chord?", "options": {"A": "Circle", "B": "Line", "C": "Parabola", "D": "Hyperbola"}, "correct_answer": "A"}
  ]
}