{
  "title": "Chord Properties - Grade 9 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Perpendicular from center to chord:", "options": {"A": "Bisects the chord", "B": "Bisects the arc", "C": "Both A and B", "D": "Neither"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "Equal chords are equidistant from:", "options": {"A": "Center", "B": "Any point", "C": "Circumference", "D": "Diameter"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "If two chords intersect inside circle, products of segments are:", "options": {"A": "Equal", "B": "Unequal", "C": "Complementary", "D": "Supplementary"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "Chord nearer to center is:", "options": {"A": "Shorter", "B": "Longer", "C": "Equal", "D": "Cannot determine"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "Longest chord in circle is:", "options": {"A": "Radius", "B": "Diameter", "C": "Any chord", "D": "Tangent"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "If OM ⟂ chord AB, then:", "options": {"A": "AM = MB", "B": "OA = OB", "C": "∠OAM = ∠OBM", "D": "All of these"}, "correct_answer": "D"},
    {"id": 7, "difficulty": "easy", "question": "Two chords at equal distances from center are:", "options": {"A": "Parallel", "B": "Equal", "C": "Perpendicular", "D": "Tangents"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "In circle, chord AB=10cm, distance from center 6cm, radius?", "options": {"A": "8cm", "B": "√61 cm", "C": "√89 cm", "D": "10cm"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "If chord is diameter, its distance from center is:", "options": {"A": "0", "B": "r", "C": "2r", "D": "r/2"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "As chord moves away from center, its length:", "options": {"A": "Increases", "B": "Decreases", "C": "First increases then decreases", "D": "Remains constant"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "Chord subtending right angle at center has length:", "options": {"A": "r", "B": "r√2", "C": "2r", "D": "r√3"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "If two chords are parallel, they are:", "options": {"A": "Always equal", "B": "Always equidistant from center", "C": "Sometimes equal", "D": "Never equal"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "Angle subtended by chord at center is _____ angle subtended at circumference.", "options": {"A": "Half", "B": "Double", "C": "Equal", "D": "Complementary"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "Perpendicular bisector of chord passes through:", "options": {"A": "One endpoint", "B": "Center", "C": "Midpoint of arc", "D": "Any point"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "If chords AB and CD intersect at E, then:", "options": {"A": "AE×EB = CE×ED", "B": "AE×EC = BE×ED", "C": "AE×ED = BE×EC", "D": "AB×CD = AD×BC"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "In circle radius 13cm, chord length 24cm, distance from center?", "options": {"A": "5cm", "B": "7cm", "C": "10cm", "D": "12cm"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Chord length equal to radius subtends angle at center:", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 18, "difficulty": "easy", "question": "Two chords intersect at right angles. If one chord is diameter, other is:", "options": {"A": "Also diameter", "B": "Bisected by diameter", "C": "Perpendicular bisector", "D": "Tangent"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "Equal chords cut off equal:", "options": {"A": "Arcs", "B": "Segments", "C": "Sectors", "D": "All of these"}, "correct_answer": "D"},
    {"id": 20, "difficulty": "easy", "question": "Through point inside circle, shortest chord is:", "options": {"A": "Perpendicular to radius through point", "B": "Parallel to given direction", "C": "Along diameter", "D": "Any chord"}, "correct_answer": "A"},
  {"id": 21, "difficulty": "medium", "question": "In circle radius 13cm, two parallel chords lengths 24cm and 10cm are on same side of center. If distance between chords is 7cm, find distance of longer chord from center.", "options": {"A": "5cm", "B": "8cm", "C": "12cm", "D": "15cm"}, "correct_answer": "A"}, {"id": 21, "difficulty": "medium", "question": "Two chords AB and CD intersect at E. If AE=4, EB=6, CE=3, find ED.", "options": {"A": "4", "B": "6", "C": "8", "D": "10"}, "correct_answer": "C"},
   {"id": 22, "difficulty": "medium", "question": "Chord length 30cm at 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": 23, "difficulty": "medium", "question": "Two parallel chords lengths 16cm and 30cm on same side of center, distance between chords 7cm. Radius?", "options": {"A": "17cm", "B": "18cm", "C": "19cm", "D": "20cm"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "Chord AB subtends 60° at center. Distance from center to AB if radius R?", "options": {"A": "R/2", "B": "R√3/2", "C": "R/√2", "D": "R√2/2"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "Two chords intersect at right angles. Segments of one chord 6 and 8, of other chord 5 and x. Find x.", "options": {"A": "9.6", "B": "10", "C": "12", "D": "15"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Through point P inside circle, chord AB drawn. Shortest chord through P has length 12cm, longest chord 20cm. Distance OP?", "options": {"A": "4cm", "B": "5cm", "C": "6cm", "D": "8cm"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Two concentric circles radii 5cm and 13cm. Chord of larger touching smaller has length?", "options": {"A": "12cm", "B": "18cm", "C": "24cm", "D": "26cm"}, "correct_answer": "C"},
    {"id": 28, "difficulty": "medium", "question": "Chord length 24cm, distance 5cm from center. Area between chord and minor arc?", "options": {"A": "25π - 60", "B": "169π/2 - 60", "C": "169π/2 - 120", "D": "169π - 120"}, "correct_answer": "B"},
    {"id": 29, "difficulty": "medium", "question": "Chord length equal to radius subtends angle θ at center. Another chord at distance R/2 from center subtends angle φ. Find θ+φ.", "options": {"A": "90°", "B": "120°", "C": "150°", "D": "180°"}, "correct_answer": "D"}, {"id": 30, "difficulty": "medium", "question": "Two concentric circles radii 5cm and 13cm. Chord of larger circle touches smaller circle. Find length of this chord.", "options": {"A": "12cm", "B": "18cm", "C": "24cm", "D": "26cm"}, "correct_answer": "C"},  {"id": 31, "difficulty": "medium", "question": "Two chords intersect at E. AE=9, EB=4, CE=6, ED=6. Are chords perpendicular?", "options": {"A": "Yes", "B": "No", "C": "Can't say", "D": "If AE=EC"}, "correct_answer": "B"},
   {"id": 32, "difficulty": "medium", "question": "Two equal chords on opposite sides of center are each at distance 12cm from center. If radius is 13cm, find chord length.", "options": {"A": "5cm", "B": "10cm", "C": "12cm", "D": "24cm"}, "correct_answer": "B"}, {"id": 33, "difficulty": "medium", "question": "Three parallel chords lengths 10,12,14 in circle radius √65. Distances between them if on same side?", "options": {"A": "1,1", "B": "1,2", "C": "2,2", "D": "2,3"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "From point distance 5 from center, chords drawn making angles 30°,60° with line through center. Lengths ratio?", "options": {"A": "√3:1", "B": "2:√3", "C": "√2:1", "D": "√3:√2"}, "correct_answer": "A"},
   {"id": 35, "difficulty": "easy", "question": "A circle has infinite lines of symmetry. True or False?", "options": {"A": "True", "B": "False", "C": "Sometimes true", "D": "Never true"}, "correct_answer": "A"},{"id": 36, "difficulty": "medium", "question": "In circle radius 10cm, two parallel equal chords each length 12cm are on opposite sides of center. Find distance between chords.", "options": {"A": "8cm", "B": "12cm", "C": "16cm", "D": "20cm"}, "correct_answer": "C"},   {"id": 37, "difficulty": "medium", "question": "Regular hexagon inscribed. Distance from center to side?", "options": {"A": "R√3/2", "B": "R/2", "C": "R√2/2", "D": "R/√2"}, "correct_answer": "A"},
   {"id": 38, "difficulty": "medium", "question": "Chord length 2R sin θ at distance R cos θ from center. For chord subtending 90° at circumference, θ=?", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "D"}, {"id": 39, "difficulty": "medium", "question": "Two chords divide circle into four arcs with measures 70°,110°,70°,110°. Chords are?", "options": {"A": "Equal", "B": "Perpendicular", "C": "Parallel", "D": "Diameter and chord"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "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": 41, "difficulty": "medium", "question": "In circle x²+y²=25, chord has midpoint (3,1). Length?", "options": {"A": "4", "B": "6", "C": "8", "D": "10"}, "correct_answer": "C"},
    {"id": 42, "difficulty": "medium", "question": "Two chords intersect inside circle. Products of segments equal. If one chord passes through center, other is:", "options": {"A": "Diameter", "B": "Bisected by first", "C": "Perpendicular to first", "D": "Parallel to tangent"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "medium", "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": 44, "difficulty": "medium", "question": "Two chords AB and CD intersect at E outside circle. Then:", "options": {"A": "AE×BE = CE×DE", "B": "AE×CE = BE×DE", "C": "AE×DE = BE×CE", "D": "AB×CD = AD×BC"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "Chord length 2√(R²-d²). For fixed R, as d increases, chord length:", "options": {"A": "Increases", "B": "Decreases", "C": "First increases then decreases", "D": "Constant"}, "correct_answer": "B"},
    {"id": 46, "difficulty": "medium", "question": "Three equal chords of circle radius R intersect at common point 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": 47, "difficulty": "medium", "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": 48, "difficulty": "medium", "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": 49, "difficulty": "medium", "question": "Two parallel chords lengths 10 and 24 on same side distance 5 apart. Radius if chords subtend complementary angles?", "options": {"A": "13", "B": "15", "C": "17", "D": "19"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "medium", "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": 51, "difficulty": "medium", "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": 52, "difficulty": "medium", "question": "Two chords divide circle into four arcs in arithmetic progression. If smallest chord subtends 60°, largest subtends?", "options": {"A": "90°", "B": "120°", "C": "150°", "D": "180°"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "medium", "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": 54, "difficulty": "medium", "question": "From point on circle, chords to endpoints of diameter. Sum of squares of chords?", "options": {"A": "2R²", "B": "4R²", "C": "6R²", "D": "8R²"}, "correct_answer": "B"},
    {"id": 55, "difficulty": "medium", "question": "Chord length 2R sin 18°. Distance from center?", "options": {"A": "R cos 18°", "B": "R sin 18°", "C": "R cos 36°", "D": "R sin 36°"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "medium", "question": "Two intersecting chords divided into segments proportional to adjacent sides of cyclic quadrilateral. If chords perpendicular, ratio?", "options": {"A": "1:1", "B": "depends on angle", "C": "equal to ratio of chords", "D": "not determined"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "Chord equal to side of inscribed square. Angle subtended at center?", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "varies"}, "correct_answer": "B"},
    {"id": 58, "difficulty": "medium", "question": "Through fixed point inside circle, chords drawn. Locus of midpoints?", "options": {"A": "Circle", "B": "Line", "C": "Parabola", "D": "Ellipse"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "Two chords AB and CD intersect at E inside. If AE=EB and CE:ED=1:3, distance from center to AB half that to CD, ratio AB:CD?", "options": {"A": "1:2", "B": "2:1", "C": "1:√2", "D": "√2:1"}, "correct_answer": "D"},
    {"id": 60, "difficulty": "medium", "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"},
    {"id": 61, "difficulty": "hard", "question": "Two equal chords intersect at right angles. Segments a,b of one chord. Radius in terms a,b?", "options": {"A": "√((a²+b²)(a+b)²/(4ab))", "B": "√((a²+b²)(a+b)²/(2ab))", "C": "√((a²+b²)(a+b)²/ab)", "D": "√(ab(a²+b²)/(a+b)²)"}, "correct_answer": "A"},
    {"id": 62, "difficulty": "hard", "question": "Through fixed point inside, chords drawn. Product of segment lengths constant. For equal chords angle θ, product?", "options": {"A": "R²-d²", "B": "(R²-d²) sin²(θ/2)", "C": "(R²-d²) cos²(θ/2)", "D": "(R²-d²) tan²(θ/2)"}, "correct_answer": "C"},
    {"id": 63, "difficulty": "hard", "question": "Two equal chords contain angle 2α at common endpoint. Distance from center to chord joining other endpoints?", "options": {"A": "R cos α", "B": "R sin α", "C": "R tan α", "D": "R cot α"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "hard", "question": "From point on circle, two equal chords divide circle into three equal arcs. Angle between chords?", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "150°"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "hard", "question": "Three equal chords divide circumference into arcs ratio 1:2:3. Angles subtended at center?", "options": {"A": "60°,120°,180°", "B": "60°,90°,210°", "C": "60°,100°,200°", "D": "Not possible"}, "correct_answer": "D"},
    {"id": 66, "difficulty": "hard", "question": "Two equal chords intersect at P. Line through P and center meets circle at Q,R. P midpoint QR if?", "options": {"A": "Chords perpendicular", "B": "Chords equal", "C": "OP=0", "D": "Always"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "hard", "question": "Regular polygon n sides inscribed. Distance center to side R cos(π/n). n=24, distance?", "options": {"A": "0.991R", "B": "0.966R", "C": "0.924R", "D": "0.866R"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "hard", "question": "Two equal chords in x²+y²=25 have midpoints on 3x+4y=0. Length each chord?", "options": {"A": "6", "B": "8", "C": "10", "D": "12"}, "correct_answer": "B"},
    {"id": 69, "difficulty": "hard", "question": "From point P, two equal chords PA,PB. PA=PB=l, ∠APB=θ. Distance OP?", "options": {"A": "√(R²-l²/4 sin²(θ/2))", "B": "√(R²-l²/4 cos²(θ/2))", "C": "√(R²-l² sin²(θ/2))", "D": "√(R²-l² cos²(θ/2))"}, "correct_answer": "B"},
    {"id": 70, "difficulty": "hard", "question": "Chord equal to side of inscribed regular n-gon. n→∞, distance from center→?", "options": {"A": "0", "B": "R", "C": "R/2", "D": "2R"}, "correct_answer": "B"},
    {"id": 71, "difficulty": "hard", "question": "Two equal chords AB,CD intersect at E. Circles on AE,CE as diameters intersect at E,F. F lies on:", "options": {"A": "Chord BD", "B": "Line through E ⟂ AC", "C": "Original circle", "D": "None"}, "correct_answer": "C"},
    {"id": 72, "difficulty": "hard", "question": "Three equal chords concurrent. Minimum area central region?", "options": {"A": "R²(π/3-√3/2)", "B": "R²(π/2-1)", "C": "R²(π/4-1/2)", "D": "0"}, "correct_answer": "D"},
    {"id": 73, "difficulty": "hard", "question": "From point P, two equal chords making angles α,β with OP. If chords equal, α,β relation?", "options": {"A": "α=β", "B": "α+β=90°", "C": "α+β=180°", "D": "No relation"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "hard", "question": "Two equal chords subtend angles 2α,2β at center. Distance between chords if on same side?", "options": {"A": "R(cos α - cos β)", "B": "R(sin α - sin β)", "C": "R(tan α - tan β)", "D": "R(cot α - cot β)"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "hard", "question": "Chord length 2R sin(π/2n) in regular n-gon. n=5, distance from center?", "options": {"A": "R cos(π/5)", "B": "R sin(π/5)", "C": "R cos(π/10)", "D": "R sin(π/10)"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "hard", "question": "Two equal chords intersect at right angles. Segments a,b of one chord. Radius in terms a,b?", "options": {"A": "√((a+b)(a²+b²)/(4ab))", "B": "√((a+b)(a²+b²)/(2ab))", "C": "√((a+b)²(a²+b²)/(4ab))", "D": "√((a+b)²(a²+b²)/(2ab))"}, "correct_answer": "C"},
    {"id": 77, "difficulty": "hard", "question": "Through point distance d from center, chord making angle θ with line through center. Length?", "options": {"A": "2√(R²-d² sin²θ)", "B": "2√(R²-d² cos²θ)", "C": "2√(R²-d² tan²θ)", "D": "2√(R²-d² cot²θ)"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "hard", "question": "Two equal chords intersect. Line joining intersection to center bisects angle between chords if?", "options": {"A": "Chords equal", "B": "Chords perpendicular", "C": "Always", "D": "Never"}, "correct_answer": "C"},
    {"id": 79, "difficulty": "hard", "question": "From point on circle, two equal chords divide circle into arcs ratio 1:2:1:2. Angle between chords?", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "150°"}, "correct_answer": "B"},
    {"id": 80, "difficulty": "hard", "question": "Regular polygon inscribed. Ratio of distances from center to side for n and 2n?", "options": {"A": "cos(π/2n):cos(π/n)", "B": "cos(π/n):cos(π/2n)", "C": "sin(π/2n):sin(π/n)", "D": "tan(π/2n):tan(π/n)"}, "correct_answer": "B"},
    {"id": 81, "difficulty": "hard", "question": "Two equal chords AB,AC with ∠BAC=60°. Area ΔABC maximum when?", "options": {"A": "AB diameter", "B": "AB=AC=R", "C": "AB=AC=R√3", "D": "Always same"}, "correct_answer": "B"},
    {"id": 82, "difficulty": "hard", "question": "Chord length equal to distance from center. Midpoint traces:", "options": {"A": "Circle", "B": "Line", "C": "Ellipse", "D": "Parabola"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "Two equal chords intersect. Lines joining endpoints form quadrilateral. Cyclic?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Three equal chords divide circle into 6 arcs alternating α,β with α+β=120°. Chords concurrent?", "options": {"A": "Yes", "B": "No", "C": "Only if α=β", "D": "Only if α=60°"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "From point P inside, two equal chords PA,PB. ∠APB=120°, OP=d. Chord length?", "options": {"A": "√(3(R²-d²))", "B": "√(2(R²-d²))", "C": "√(4(R²-d²)/3)", "D": "√(3(R²-d²)/2)"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "Two equal chords subtend 90° and 60° at point on circumference. Ratio distances from center?", "options": {"A": "√2:√3", "B": "√3:√2", "C": "1:√3", "D": "√3:1"}, "correct_answer": "B"},
    {"id": 87, "difficulty": "hard", "question": "Chord length 2R sin θ. Two equal chords angles θ,φ at center. Sum distances from center?", "options": {"A": "R(cos θ+cos φ)", "B": "R(sin θ+sin φ)", "C": "R(tan θ+tan φ)", "D": "R(cot θ+cot φ)"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "Regular n-gon inscribed. n increases, distance center to side:", "options": {"A": "Increases to R", "B": "Decreases to 0", "C": "Increases to R/2", "D": "Decreases to R/2"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "Two equal chords intersect. Lines joining endpoints form two triangles. These are:", "options": {"A": "Similar", "B": "Congruent", "C": "Equal area", "D": "All of these"}, "correct_answer": "D"},
    {"id": 90, "difficulty": "hard", "question": "From point distance d from center, chord drawn. Midpoint traces circle radius √(R²-d²)/2. For equal chords angle 2α, distance between midpoints?", "options": {"A": "√(R²-d²) sin α", "B": "√(R²-d²) cos α", "C": "√(R²-d²) tan α", "D": "√(R²-d²) cot α"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "Three equal chords divide circle into 4 equal area regions if?", "options": {"A": "They form equilateral triangle", "B": "They are diameters", "C": "They concur at center", "D": "Impossible"}, "correct_answer": "D"},
    {"id": 92, "difficulty": "hard", "question": "Chord length 2√(R²-d²). Two equal chords distances d1,d2. If chords perpendicular, d1²+d2²=?", "options": {"A": "R²", "B": "2R²", "C": "R²/2", "D": "√2 R²"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "Two equal chords AB,AC. Circle with diameter BC meets AB at D. AD=?", "options": {"A": "AB/2", "B": "AB/3", "C": "AB/√2", "D": "AB/√3"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "Regular polygon inscribed. Distance center to side R cos(π/n). n=7, approximate?", "options": {"A": "0.901R", "B": "0.875R", "C": "0.822R", "D": "0.782R"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "Two equal chords intersect at P. Line through P perpendicular to one chord bisects other chord if?", "options": {"A": "Chords equal", "B": "P is midpoint", "C": "Always", "D": "Never"}, "correct_answer": "C"},
    {"id": 96, "difficulty": "hard", "question": "Chord equal to side of inscribed square. Another chord equal to side of inscribed equilateral triangle. Ratio distances from center?", "options": {"A": "√2:√3", "B": "√3:√2", "C": "1:√2", "D": "1:√3"}, "correct_answer": "B"},
    {"id": 97, "difficulty": "hard", "question": "From point on circle, two equal chords divide circumference into arcs ratio 2:3:2:3. Angle between chords?", "options": {"A": "72°", "B": "90°", "C": "108°", "D": "120°"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "Two equal chords AB,CD intersect at E. Circles on AB,CD as diameters intersect at E,F. EF is:", "options": {"A": "Perpendicular to line joining centers", "B": "Parallel to AC", "C": "Diameter of original circle", "D": "None"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "Regular n-gon inscribed. Distance center to side n=10?", "options": {"A": "R cos 18°", "B": "R sin 18°", "C": "R cos 36°", "D": "R sin 36°"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "Two equal chords intersect. Lines joining endpoints form kite. Diagonals intersect at:", "options": {"A": "Center of circle", "B": "Point of intersection of chords", "C": "Midpoint of chords", "D": "None"}, "correct_answer": "B"}
  ]
}