{
  "title": "Angle Subtended by Chord - Grade 9 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "The angle subtended by a chord at the center is ________ the angle subtended at any point on the remaining part of the circle.", "options": {"A": "equal to", "B": "half of", "C": "twice", "D": "four times"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "Angles in the same segment of a circle are:", "options": {"A": "equal", "B": "complementary", "C": "supplementary", "D": "none of these"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "If AB is a chord of a circle with center O, and ∠AOB = 80°, then angle subtended by AB at any point on the major arc is:", "options": {"A": "40°", "B": "80°", "C": "160°", "D": "20°"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "The angle subtended by a diameter at any point on the circle is:", "options": {"A": "60°", "B": "90°", "C": "180°", "D": "360°"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "If ∠ACB and ∠ADB are angles subtended by chord AB on the same side of AB, then:", "options": {"A": "∠ACB = ∠ADB", "B": "∠ACB + ∠ADB = 90°", "C": "∠ACB + ∠ADB = 180°", "D": "∠ACB = 2∠ADB"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "A chord AB subtends 40° at the center. The angle subtended by AB at a point on the minor arc is:", "options": {"A": "20°", "B": "40°", "C": "80°", "D": "160°"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "In a circle, if a chord subtends 60° at the center, it subtends ______ at any point on the major arc.", "options": {"A": "30°", "B": "60°", "C": "120°", "D": "240°"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "If two chords subtend equal angles at the center, then the chords are:", "options": {"A": "parallel", "B": "equal", "C": "perpendicular", "D": "none of these"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "The angle in a semicircle is:", "options": {"A": "acute", "B": "right", "C": "obtuse", "D": "straight"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "If a chord subtends 90° at the center, the angle subtended at any point on the minor arc is:", "options": {"A": "45°", "B": "90°", "C": "135°", "D": "180°"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "Equal chords subtend equal angles at the center of a circle. True or False?", "options": {"A": "True", "B": "False", "C": "Sometimes true", "D": "Never true"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "If chord AB = chord CD, then ∠AOB = ∠COD where O is center. True or False?", "options": {"A": "True", "B": "False", "C": "Sometimes true", "D": "Never true"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "The angle subtended by a chord at the center is 120°. The angle subtended at a point on the major arc is:", "options": {"A": "60°", "B": "120°", "C": "240°", "D": "30°"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Angles in the same segment are equal because they are subtended by the same chord. True or False?", "options": {"A": "True", "B": "False", "C": "Sometimes true", "D": "Never true"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "A diameter subtends an angle of 90° at any point on the circle. This is called:", "options": {"A": "Thales theorem", "B": "Pythagoras theorem", "C": "Midpoint theorem", "D": "Apollonius theorem"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "If a chord subtends 30° at a point on the circle, it subtends ______ at the center.", "options": {"A": "15°", "B": "30°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 17, "difficulty": "easy", "question": "Two chords AB and CD are equal. If ∠AOB = 70°, then ∠COD =", "options": {"A": "35°", "B": "70°", "C": "140°", "D": "110°"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "The angle subtended by a chord at the center is always ______ the angle subtended at any point on the alternate segment.", "options": {"A": "equal to", "B": "half of", "C": "twice", "D": "four times"}, "correct_answer": "C"},
    {"id": 19, "difficulty": "easy", "question": "If two angles subtended by the same chord at different points on the same segment are 45° and 45°, this illustrates:", "options": {"A": "angles in same segment are equal", "B": "angles at center are twice", "C": "angles in semicircle are 90°", "D": "none of these"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "A chord AB of a circle subtends ∠AOB = 100° at center O. The angle subtended by AB at point C on minor arc is:", "options": {"A": "50°", "B": "100°", "C": "200°", "D": "260°"}, "correct_answer": "A"},
    
    {"id": 21, "difficulty": "medium", "question": "In a circle with center O, chord AB subtends ∠AOB = 110°. Find angle subtended by AB at point C on major arc.", "options": {"A": "55°", "B": "110°", "C": "125°", "D": "250°"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Two chords AB and CD are equal. If ∠AOB = 80° where O is center, find ∠COD.", "options": {"A": "40°", "B": "80°", "C": "160°", "D": "100°"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "In circle with center O, chord PQ subtends ∠POQ = 120°. R is a point on major arc PQ. Find ∠PRQ.", "options": {"A": "60°", "B": "120°", "C": "240°", "D": "30°"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "AB is a diameter of circle with center O. C is any point on circle. Find ∠ACB.", "options": {"A": "45°", "B": "60°", "C": "90°", "D": "120°"}, "correct_answer": "C"},
    {"id": 25, "difficulty": "medium", "question": "In a circle, chord XY subtends 50° at center. Point Z lies on minor arc XY. Find ∠XZY.", "options": {"A": "25°", "B": "50°", "C": "100°", "D": "130°"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Two chords AB and AC of a circle with center O are equal. If ∠BOC = 100°, find ∠BAC.", "options": {"A": "50°", "B": "100°", "C": "130°", "D": "80°"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "In circle with center O, chord RS subtends ∠ROS = 140°. T is point on minor arc RS. Find ∠RTS.", "options": {"A": "70°", "B": "110°", "C": "40°", "D": "20°"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "AB and CD are two equal chords of a circle with center O. If ∠AOB = 70°, find reflex ∠COD.", "options": {"A": "70°", "B": "110°", "C": "250°", "D": "290°"}, "correct_answer": "D"},
    {"id": 29, "difficulty": "medium", "question": "In a circle, chord LM subtends 65° at center. Point N lies on major arc LM. Find ∠LNM.", "options": {"A": "32.5°", "B": "65°", "C": "115°", "D": "295°"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "PQ is a diameter of circle. R is point on circle such that ∠PRQ = 90°. This illustrates:", "options": {"A": "angle in semicircle is 90°", "B": "angles in same segment equal", "C": "angle at center twice", "D": "none"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Two chords AB and CD subtend angles 40° and 80° at center respectively. Which chord is longer?", "options": {"A": "AB", "B": "CD", "C": "Equal", "D": "Cannot determine"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "In circle with center O, chord EF subtends ∠EOF = 150°. G is point on minor arc EF. Find ∠EGF.", "options": {"A": "75°", "B": "105°", "C": "150°", "D": "30°"}, "correct_answer": "B"},
    {"id": 33, "difficulty": "medium", "question": "Chords AB and AC are equal. If ∠BAC = 50°, find ∠BOC where O is center.", "options": {"A": "50°", "B": "100°", "C": "130°", "D": "25°"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "In circle, chord JK subtends 72° at center. Point L on major arc JK. Find ∠JLK.", "options": {"A": "36°", "B": "72°", "C": "108°", "D": "144°"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "AB is chord of circle with center O. ∠AOB = 90°. Point C on minor arc AB. Find ∠ACB.", "options": {"A": "45°", "B": "90°", "C": "135°", "D": "180°"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "Two equal chords subtend angles 65° and 115° at center. Is this possible?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only if chords are diameters"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "In circle, chord MN subtends ∠MON = 160° at center O. Point P on major arc MN. Find ∠MPN.", "options": {"A": "80°", "B": "100°", "C": "20°", "D": "160°"}, "correct_answer": "B"},
    {"id": 38, "difficulty": "medium", "question": "AB and BC are two chords with AB = BC. If ∠AOB = 85° and O is center, find ∠BOC.", "options": {"A": "85°", "B": "95°", "C": "170°", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Chord PQ subtends 42° at point R on circle. Find angle subtended at center.", "options": {"A": "21°", "B": "42°", "C": "84°", "D": "168°"}, "correct_answer": "C"},
    {"id": 40, "difficulty": "medium", "question": "In circle with center O, chord ST subtends ∠SOT = 130°. Point U on minor arc ST. Find ∠SUT.", "options": {"A": "65°", "B": "115°", "C": "50°", "D": "25°"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "medium", "question": "AB is diameter, C is point on circle. If ∠ABC = 35°, find ∠BAC.", "options": {"A": "35°", "B": "55°", "C": "90°", "D": "125°"}, "correct_answer": "B"},
    {"id": 42, "difficulty": "medium", "question": "Two chords XY and XZ are equal. If ∠YXZ = 60°, find ∠YOX where O is center.", "options": {"A": "60°", "B": "120°", "C": "30°", "D": "240°"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "medium", "question": "Chord AB subtends 55° at point C on major arc. Find ∠AOB at center.", "options": {"A": "27.5°", "B": "55°", "C": "110°", "D": "125°"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "medium", "question": "In circle, chord DE subtends 38° at point F on minor arc. Find angle subtended at center.", "options": {"A": "19°", "B": "38°", "C": "76°", "D": "142°"}, "correct_answer": "C"},
    {"id": 45, "difficulty": "medium", "question": "AB and AC are chords with AB = AC. If ∠BOC = 120° where O is center, find ∠BAC.", "options": {"A": "60°", "B": "120°", "C": "30°", "D": "240°"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "medium", "question": "Chord PQ subtends 68° at center. Point R on major arc. Find ∠PRQ.", "options": {"A": "34°", "B": "68°", "C": "112°", "D": "146°"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "Two chords KL and KM are equal. If ∠LKM = 45°, find reflex ∠LOM where O is center.", "options": {"A": "45°", "B": "90°", "C": "270°", "D": "315°"}, "correct_answer": "C"},
    {"id": 48, "difficulty": "medium", "question": "In circle, chord UV subtends 124° at center. Point W on minor arc UV. Find ∠UWV.", "options": {"A": "62°", "B": "118°", "C": "56°", "D": "28°"}, "correct_answer": "B"},
    {"id": 49, "difficulty": "medium", "question": "AB is diameter, C on circle. If ∠CAB = 25°, find ∠CBA.", "options": {"A": "25°", "B": "65°", "C": "90°", "D": "115°"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "medium", "question": "Two equal chords subtend angles x and 180-x at center. Is this possible?", "options": {"A": "Yes, always", "B": "No, never", "C": "Only if x=90", "D": "Only if chords are diameters"}, "correct_answer": "B"},
    {"id": 51, "difficulty": "medium", "question": "Chord CD subtends 46° at point E on major arc. Find ∠COD at center.", "options": {"A": "23°", "B": "46°", "C": "92°", "D": "134°"}, "correct_answer": "C"},
    {"id": 52, "difficulty": "medium", "question": "In circle, chord FG subtends 152° at center. Point H on major arc FG. Find ∠FHG.", "options": {"A": "76°", "B": "104°", "C": "28°", "D": "14°"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "AB and AC are chords with AB = AC. If ∠BAC = 70°, find ∠BOC.", "options": {"A": "70°", "B": "140°", "C": "110°", "D": "40°"}, "correct_answer": "B"},
   {"id": 54, "difficulty": "medium", "question": "Chord IJ subtends 58° at center. Point K on minor arc IJ. Find ∠IKJ.", "options": {"A": "29°", "B": "58°", "C": "122°", "D": "151°"}, "correct_answer": "D"}, {"id": 55, "difficulty": "medium", "question": "Two chords LM and LN are equal. If ∠MLN = 55°, find ∠LOM where O is center.", "options": {"A": "55°", "B": "110°", "C": "125°", "D": "70°"}, "correct_answer": "B"},
    {"id": 56, "difficulty": "medium", "question": "In circle, chord OP subtends 134° at center. Point Q on major arc OP. Find ∠OQP.", "options": {"A": "67°", "B": "113°", "C": "46°", "D": "23°"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "AB is diameter, C on circle. If ∠ACB = 90°, find ∠CAB + ∠CBA.", "options": {"A": "90°", "B": "180°", "C": "270°", "D": "360°"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "Two equal chords subtend 75° and 105° at center. This is impossible because:", "options": {"A": "equal chords subtend equal angles at center", "B": "angles should sum to 180°", "C": "angles should be complementary", "D": "none of these"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "Chord QR subtends 49° at point S on minor arc. Find ∠QOR at center.", "options": {"A": "24.5°", "B": "49°", "C": "98°", "D": "131°"}, "correct_answer": "C"},
    {"id": 60, "difficulty": "medium", "question": "In circle, chord TU subtends 168° at center. Point V on minor arc TU. Find ∠TVU.", "options": {"A": "84°", "B": "96°", "C": "12°", "D": "6°"}, "correct_answer": "B"},
    {"id": 61, "difficulty": "medium", "question": "AB and AC are equal chords. If ∠BOC = 130°, find ∠BAC.", "options": {"A": "65°", "B": "130°", "C": "50°", "D": "25°"}, "correct_answer": "A"},
    {"id": 62, "difficulty": "medium", "question": "Chord WX subtends 62° at center. Point Y on major arc WX. Find ∠WYX.", "options": {"A": "31°", "B": "62°", "C": "118°", "D": "149°"}, "correct_answer": "A"},
   {"id": 63, "difficulty": "medium", "question": "Two chords AB and BC are equal. If ∠AOC = 150° where O is center, find ∠ABC.", "options": {"A": "75°", "B": "150°", "C": "105°", "D": "30°"}, "correct_answer": "A"}, {"id": 64, "difficulty": "medium", "question": "In circle, chord ZZ' subtends 142° at center. Point Y on minor arc ZZ'. Find ∠YZ'Z.", "options": {"A": "71°", "B": "109°", "C": "38°", "D": "19°"}, "correct_answer": "B"},
    {"id": 65, "difficulty": "medium", "question": "AB is diameter, C on circle. If arc AC = 70°, find ∠ABC.", "options": {"A": "35°", "B": "70°", "C": "110°", "D": "20°"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "medium", "question": "Two equal chords subtend angles that are supplementary at center. Is this possible?", "options": {"A": "Yes, always", "B": "No, never", "C": "Only if chords are diameters", "D": "Only if center angle is 90°"}, "correct_answer": "B"},
    {"id": 67, "difficulty": "medium", "question": "Chord AA' subtends 53° at point B on major arc. Find ∠AOA' at center.", "options": {"A": "26.5°", "B": "53°", "C": "106°", "D": "127°"}, "correct_answer": "C"},
    {"id": 68, "difficulty": "medium", "question": "In circle, chord BC subtends 176° at center. Point D on major arc BC. Find ∠BDC.", "options": {"A": "88°", "B": "92°", "C": "4°", "D": "2°"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "medium", "question": "Chords PQ and PR are equal. If ∠QPR = 40°, find ∠QOR where O is center.", "options": {"A": "40°", "B": "80°", "C": "140°", "D": "100°"}, "correct_answer": "B"},
   {"id": 70, "difficulty": "medium", "question": "Chord EF subtends 66° at center. Point G on minor arc EF. Find ∠EGF.", "options": {"A": "33°", "B": "66°", "C": "114°", "D": "147°"}, "correct_answer": "D"}, {"id": 71, "difficulty": "medium", "question": "AB is diameter, C on circle. If ∠BAC = 30°, find ∠ACB.", "options": {"A": "30°", "B": "60°", "C": "90°", "D": "120°"}, "correct_answer": "B"},
    {"id": 72, "difficulty": "medium", "question": "Two equal chords cannot subtend angles 80° and 100° at center because:", "options": {"A": "they would be unequal", "B": "sum should be 180°", "C": "they should be equal", "D": "both A and C"}, "correct_answer": "C"},
    {"id": 73, "difficulty": "medium", "question": "Chord GH subtends 47° at point I on minor arc. Find ∠GOH at center.", "options": {"A": "23.5°", "B": "47°", "C": "94°", "D": "133°"}, "correct_answer": "C"},
    {"id": 74, "difficulty": "medium", "question": "In circle, chord JK subtends 158° at center. Point L on minor arc JK. Find ∠JLK.", "options": {"A": "79°", "B": "101°", "C": "22°", "D": "11°"}, "correct_answer": "B"},
    {"id": 75, "difficulty": "medium", "question": "Chords MN and MP are equal. If ∠NMP = 65°, find ∠NOP where O is center.", "options": {"A": "65°", "B": "130°", "C": "115°", "D": "50°"}, "correct_answer": "B"},
    {"id": 76, "difficulty": "medium", "question": "Chord QR subtends 72° at center. Point S on major arc QR. Find ∠QSR.", "options": {"A": "36°", "B": "72°", "C": "108°", "D": "144°"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "AB is diameter, C on circle. If ∠CBA = 40°, find ∠CAB.", "options": {"A": "40°", "B": "50°", "C": "90°", "D": "140°"}, "correct_answer": "B"},
    {"id": 78, "difficulty": "medium", "question": "Two equal chords always subtend equal angles at center. True or False?", "options": {"A": "True", "B": "False", "C": "Sometimes true", "D": "Never true"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "medium", "question": "Chord ST subtends 51° at point U on major arc. Find ∠SOT at center.", "options": {"A": "25.5°", "B": "51°", "C": "102°", "D": "129°"}, "correct_answer": "C"},
    {"id": 80, "difficulty": "medium", "question": "In circle, chord UV subtends 148° at center. Point W on major arc UV. Find ∠UWV.", "options": {"A": "74°", "B": "106°", "C": "32°", "D": "16°"}, "correct_answer": "A"},
    
   {"id": 81, "difficulty": "hard", "question": "In circle with center O, chords AB and CD intersect at P. If ∠APC = 50° and arc AC = 80°, find ∠BOD.", "options": {"A": "50°", "B": "80°", "C": "100°", "D": "130°"}, "correct_answer": "C"}, {"id": 82, "difficulty": "hard", "question": "Two chords AB and CD intersect at point P inside circle. If ∠APD = 100°, arc AD = 120°, and arc BC = 80°, find ∠APC.", "options": {"A": "60°", "B": "80°", "C": "100°", "D": "120°"}, "correct_answer": "B"},
    {"id": 83, "difficulty": "hard", "question": "In circle, chord AB subtends 50° at point C on major arc. Chord AD subtends 70° at point C. Find ∠BAD.", "options": {"A": "10°", "B": "20°", "C": "30°", "D": "40°"}, "correct_answer": "C"},
    {"id": 84, "difficulty": "hard", "question": "AB is diameter. Chord AC makes angle 30° with AB. Chord BC makes angle 40° with AB. Find ∠ACB.", "options": {"A": "50°", "B": "70°", "C": "90°", "D": "110°"}, "correct_answer": "C"},
    {"id": 85, "difficulty": "hard", "question": "In circle with center O, chords PQ and RS intersect at T. If ∠PTR = 110°, arc PR = 100°, find arc QS.", "options": {"A": "60°", "B": "70°", "C": "120°", "D": "140°"}, "correct_answer": "C"},
    {"id": 86, "difficulty": "hard", "question": "Two chords AB and CD intersect externally at P. If ∠APC = 30°, arc AC = 100°, find arc BD.", "options": {"A": "40°", "B": "80°", "C": "140°", "D": "160°"}, "correct_answer": "D"},
    {"id": 87, "difficulty": "hard", "question": "In circle, chords XY and XZ are equal. If ∠XYZ = 40°, find ∠XZY.", "options": {"A": "40°", "B": "50°", "C": "70°", "D": "80°"}, "correct_answer": "C"},
    {"id": 88, "difficulty": "hard", "question": "AB and CD are two chords intersecting at P inside circle. If ∠APC = 80°, arc AC = 110°, find ∠BPD.", "options": {"A": "70°", "B": "80°", "C": "90°", "D": "100°"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "In circle, chord EF subtends 60° at point G on circle. Chord EH subtends 80° at point G. Find ∠FEH.", "options": {"A": "10°", "B": "20°", "C": "30°", "D": "40°"}, "correct_answer": "B"},
    {"id": 90, "difficulty": "hard", "question": "PQ is diameter. Chord PR makes angle 25° with PQ. Chord QR makes angle 35° with QP. Find ∠PRQ.", "options": {"A": "60°", "B": "75°", "C": "90°", "D": "120°"}, "correct_answer": "C"},
    {"id": 91, "difficulty": "hard", "question": "Two chords LM and LN are drawn from point L on circle. If ∠MLN = 50° and arc MN = 120°, find ∠LON where O is center.", "options": {"A": "60°", "B": "120°", "C": "130°", "D": "140°"}, "correct_answer": "B"},
    {"id": 92, "difficulty": "hard", "question": "Chords AB and CD intersect at P inside circle. If ∠APC = 70°, arc AD = 130°, and arc BC = 90°, find ∠APD.", "options": {"A": "70°", "B": "90°", "C": "110°", "D": "130°"}, "correct_answer": "C"},
    {"id": 93, "difficulty": "hard", "question": "In circle, chord ST subtends 55° at point U. Chord SV subtends 75° at point U. Find ∠TSV.", "options": {"A": "10°", "B": "20°", "C": "30°", "D": "40°"}, "correct_answer": "B"},
    {"id": 94, "difficulty": "hard", "question": "AB is diameter. Chord AC makes angle 20° with diameter. Chord BC makes angle 30° with diameter extended. Find ∠ACB.", "options": {"A": "50°", "B": "70°", "C": "90°", "D": "110°"}, "correct_answer": "C"},
    {"id": 95, "difficulty": "hard", "question": "Two chords PQ and RS intersect at T. If ∠PTR = 100°, arc PR = 120°, find arc QS.", "options": {"A": "80°", "B": "100°", "C": "120°", "D": "140°"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "Chords XY and XZ are equal. If ∠YXZ = 55°, find reflex ∠YOX where O is center.", "options": {"A": "110°", "B": "180°", "C": "250°", "D": "290°"}, "correct_answer": "C"},
    {"id": 97, "difficulty": "hard", "question": "In circle, chords AB and CD intersect at P. If ∠APC = 60°, arc AC = 80°, and arc BD = 100°, find ∠BPC.", "options": {"A": "40°", "B": "60°", "C": "80°", "D": "100°"}, "correct_answer": "D"},
    {"id": 98, "difficulty": "hard", "question": "Chord EF subtends 65° at point G. Chord EG subtends 85° at point F. Find ∠FEG.", "options": {"A": "15°", "B": "20°", "C": "25°", "D": "30°"}, "correct_answer": "B"},
    {"id": 99, "difficulty": "hard", "question": "AB is diameter. Chord AC makes 40° with AB. Chord AD makes 50° with AB on opposite side. Find ∠CAD.", "options": {"A": "10°", "B": "40°", "C": "50°", "D": "90°"}, "correct_answer": "D"},
    {"id": 100, "difficulty": "hard", "question": "Two chords LM and LN are such that ∠MLN = 60°. If arc MN = 140°, find angle subtended by chord LN at center.", "options": {"A": "80°", "B": "100°", "C": "120°", "D": "140°"}, "correct_answer": "B"}
  ]
}