{
  "title": "Inequalities in a Triangle - Grade 9 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "In triangle ABC, if AB = 5 cm and BC = 3 cm, which is greater: AC or 2 cm?", "options": {"A": "AC < 2 cm", "B": "AC = 2 cm", "C": "AC > 2 cm", "D": "Cannot be determined"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "According to triangle inequality theorem, can sides 3 cm, 4 cm, and 8 cm form a triangle?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Always"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "In any triangle, the sum of any two sides is always ______ the third side.", "options": {"A": "less than", "B": "equal to", "C": "greater than", "D": "half of"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "Which of these sets of sides cannot form a triangle?", "options": {"A": "2, 3, 4", "B": "5, 6, 10", "C": "3, 4, 7", "D": "6, 8, 9"}, "correct_answer": "C"},
    {"id": 5, "difficulty": "easy", "question": "In triangle ABC, if ∠A = 40° and ∠B = 60°, which side is the longest?", "options": {"A": "AB", "B": "BC", "C": "AC", "D": "All equal"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "The difference between any two sides of a triangle is always ______ the third side.", "options": {"A": "greater than", "B": "equal to", "C": "less than", "D": "double of"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "If a triangle has sides 5 cm and 7 cm, the third side must be between ______ cm.", "options": {"A": "2 and 12", "B": "0 and 12", "C": "3 and 11", "D": "2 and 11"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "In triangle PQR, if ∠P is the largest angle, which side is the longest?", "options": {"A": "PQ", "B": "QR", "C": "PR", "D": "Cannot determine"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "Can sides 6 cm, 8 cm, and 14 cm form a triangle?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Always"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "If two sides of a triangle are 4 cm and 9 cm, what is the possible range for the third side?", "options": {"A": "5 < x < 13", "B": "0 < x < 13", "C": "4 < x < 9", "D": "5 < x < 9"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "Which statement is always true for any triangle?", "options": {"A": "Longest side = shortest side", "B": "Longest side opposite largest angle", "C": "All sides equal", "D": "All angles equal"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "In triangle XYZ, if XY = 7 cm, YZ = 10 cm, then XZ must be ______.", "options": {"A": "greater than 3 cm", "B": "less than 17 cm", "C": "between 3 and 17 cm", "D": "exactly 10 cm"}, "correct_answer": "C"},
    {"id": 13, "difficulty": "easy", "question": "If ∠A > ∠B > ∠C in triangle ABC, which side is shortest?", "options": {"A": "AB", "B": "BC", "C": "AC", "D": "All equal"}, "correct_answer": "C"},
    {"id": 14, "difficulty": "easy", "question": "The sum of three sides of a triangle is 15 cm. If two sides are 4 cm and 6 cm, what is the third side?", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "Which set can form a triangle?", "options": {"A": "1, 2, 3", "B": "2, 3, 6", "C": "3, 4, 5", "D": "4, 5, 10"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "In a right triangle, the longest side is called the ______.", "options": {"A": "base", "B": "perpendicular", "C": "hypotenuse", "D": "median"}, "correct_answer": "C"},
    {"id": 17, "difficulty": "easy", "question": "If triangle has sides 8 cm and 15 cm, the third side cannot be ______.", "options": {"A": "10 cm", "B": "20 cm", "C": "7 cm", "D": "22 cm"}, "correct_answer": "D"},
    {"id": 18, "difficulty": "easy", "question": "In triangle, side opposite to ______ angle is the longest.", "options": {"A": "smallest", "B": "largest", "C": "right", "D": "acute"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "Can 5 cm, 12 cm, 13 cm form a triangle?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "In ΔABC, if AB = 8 cm, AC = 6 cm, then BC must be less than ______.", "options": {"A": "14 cm", "B": "2 cm", "C": "10 cm", "D": "8 cm"}, "correct_answer": "A"},
    
    {"id": 21, "difficulty": "medium", "question": "In triangle ABC, AB = 7 cm, BC = 9 cm, and AC = 5 cm. Which angle is the largest?", "options": {"A": "∠A", "B": "∠B", "C": "∠C", "D": "Cannot determine"}, "correct_answer": "C"},
    {"id": 22, "difficulty": "medium", "question": "In ΔPQR, PQ = 8 cm, QR = 15 cm, and PR = 17 cm. Which angle is 90°?", "options": {"A": "∠P", "B": "∠Q", "C": "∠R", "D": "No right angle"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "If in triangle ABC, ∠A = 50°, ∠B = 70°, arrange sides in ascending order.", "options": {"A": "AB < BC < AC", "B": "BC < AB < AC", "C": "AC < AB < BC", "D": "AB < AC < BC"}, "correct_answer": "D"},
    {"id": 24, "difficulty": "medium", "question": "Two sides of a triangle are 12 cm and 8 cm. If the perimeter is 30 cm, find the third side.", "options": {"A": "10 cm", "B": "12 cm", "C": "14 cm", "D": "16 cm"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "In ΔXYZ, ∠X = 60°, ∠Y = 50°. Which side is the shortest?", "options": {"A": "XY", "B": "YZ", "C": "XZ", "D": "Cannot determine"}, "correct_answer": "B"},
    {"id": 26, "difficulty": "medium", "question": "If sides of a triangle are in ratio 3:4:5 and perimeter is 48 cm, find the longest side.", "options": {"A": "12 cm", "B": "16 cm", "C": "20 cm", "D": "24 cm"}, "correct_answer": "C"},
    {"id": 27, "difficulty": "medium", "question": "In ΔABC, AB = 6 cm, BC = 8 cm, and AC = 10 cm. What type of triangle is this?", "options": {"A": "Acute", "B": "Obtuse", "C": "Right", "D": "Equilateral"}, "correct_answer": "C"},
    {"id": 28, "difficulty": "medium", "question": "In triangle PQR, PQ = 5 cm, QR = 12 cm. If PR is an integer, how many possible values for PR exist?", "options": {"A": "8", "B": "9", "C": "10", "D": "11"}, "correct_answer": "B"},
    {"id": 29, "difficulty": "medium", "question": "In ΔABC, AB = 5 cm, BC = 7 cm, and ∠B = 110°. Which side is longest?", "options": {"A": "AB", "B": "BC", "C": "AC", "D": "Cannot determine without calculation"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "If two sides of triangle are 7.5 cm and 3.5 cm, what is the maximum possible length of third side if it's an integer?", "options": {"A": "10 cm", "B": "11 cm", "C": "12 cm", "D": "13 cm"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "In triangle, ∠A = 45°, ∠B = 55°. Arrange sides BC, AC, AB in descending order.", "options": {"A": "BC > AC > AB", "B": "AC > BC > AB", "C": "AB > AC > BC", "D": "BC > AB > AC"}, "correct_answer": "D"},
    {"id": 32, "difficulty": "medium", "question": "In isosceles triangle ABC with AB = AC = 10 cm, BC = 4 cm. Which angle is largest?", "options": {"A": "∠A", "B": "∠B", "C": "∠C", "D": "All equal"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "If perimeter of triangle is 24 cm and two sides are 9 cm and 7 cm, find the range of possible values for third side.", "options": {"A": "6 < x < 8", "B": "7 < x < 9", "C": "8 < x < 10", "D": "5 < x < 7"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "In ΔPQR, PQ = 8 cm, QR = 6 cm, and ∠Q = 90°. Find PR.", "options": {"A": "10 cm", "B": "12 cm", "C": "14 cm", "D": "16 cm"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "Triangle has sides 13 cm, 14 cm, 15 cm. Which angle is largest?", "options": {"A": "Opposite 13 cm", "B": "Opposite 14 cm", "C": "Opposite 15 cm", "D": "Cannot determine"}, "correct_answer": "C"},
    {"id": 36, "difficulty": "medium", "question": "If in triangle, side a = 8 cm, side b = 15 cm, and side c = 17 cm, prove it's a right triangle. Which side is hypotenuse?", "options": {"A": "8 cm", "B": "15 cm", "C": "17 cm", "D": "Not a right triangle"}, "correct_answer": "C"},
    {"id": 37, "difficulty": "medium", "question": "In ΔABC, AB = 5√2 cm, BC = 5√2 cm, AC = 10 cm. Find largest angle.", "options": {"A": "∠A", "B": "∠B", "C": "∠C", "D": "All equal"}, "correct_answer": "C"},
    {"id": 38, "difficulty": "medium", "question": "Two sides of triangle are 5 cm and 8 cm. If third side is x cm, which inequality is correct?", "options": {"A": "3 < x < 13", "B": "0 < x < 13", "C": "5 < x < 8", "D": "3 ≤ x ≤ 13"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "In triangle, if one angle is 120°, then side opposite to it is ______ than sum of other two sides.", "options": {"A": "greater", "B": "less", "C": "equal", "D": "double"}, "correct_answer": "B"},
    {"id": 40, "difficulty": "medium", "question": "Triangle has vertices A(0,0), B(4,0), C(0,3). Which side is longest?", "options": {"A": "AB", "B": "BC", "C": "AC", "D": "All equal"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "medium", "question": "If in ΔPQR, ∠P = 70°, ∠Q = 50°, and PQ = 6 cm, find the shortest side.", "options": {"A": "PQ", "B": "QR", "C": "PR", "D": "Cannot determine"}, "correct_answer": "C"},
    {"id": 42, "difficulty": "medium", "question": "Sides of triangle are x, x+2, x+4. If perimeter is 24 cm, find longest side.", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 43, "difficulty": "medium", "question": "In right triangle, hypotenuse is 13 cm, one leg is 5 cm. Find the other leg.", "options": {"A": "10 cm", "B": "11 cm", "C": "12 cm", "D": "14 cm"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "medium", "question": "In ΔABC, AB = 7 cm, BC = 10 cm, CA = 12 cm. Which angle is smallest?", "options": {"A": "∠A", "B": "∠B", "C": "∠C", "D": "Cannot determine"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "medium", "question": "Can triangle have sides 4.5 cm, 5.5 cm, and 10.5 cm?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Depends on angles"}, "correct_answer": "B"},
    {"id": 46, "difficulty": "medium", "question": "In triangle, if one side is 10 cm and another is 6 cm, what is minimum possible integer length of third side?", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "medium", "question": "In ΔPQR, PQ = 9 cm, QR = 12 cm, PR = 15 cm. Arrange angles in ascending order.", "options": {"A": "∠P < ∠Q < ∠R", "B": "∠Q < ∠P < ∠R", "C": "∠R < ∠Q < ∠P", "D": "∠R < ∠P < ∠Q"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "medium", "question": "Two sides of triangle are 11 cm and 8 cm. If third side is even number, how many possibilities?", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "C"},
    {"id": 49, "difficulty": "medium", "question": "In triangle ABC, ∠A = 40°, AB = 8 cm, AC = 10 cm. Which side is shortest?", "options": {"A": "AB", "B": "BC", "C": "AC", "D": "Cannot determine"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "medium", "question": "In isosceles triangle, equal sides are 13 cm each, base is 10 cm. Which angle is largest?", "options": {"A": "Vertex angle", "B": "Base angles", "C": "All equal", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "If triangle sides are 2x, 3x, 4x and perimeter is 36 cm, find longest side.", "options": {"A": "8 cm", "B": "12 cm", "C": "16 cm", "D": "20 cm"}, "correct_answer": "C"},
    {"id": 52, "difficulty": "medium", "question": "In ΔXYZ, XY = 7 cm, YZ = 24 cm, XZ = 25 cm. Which angle is 90°?", "options": {"A": "∠X", "B": "∠Y", "C": "∠Z", "D": "No right angle"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "medium", "question": "In triangle, if ∠A = 55°, ∠B = 65°, and side AB = 12 cm, find approximate length of longest side.", "options": {"A": "11.5 cm", "B": "12.8 cm", "C": "13.2 cm", "D": "14.0 cm"}, "correct_answer": "C"},
    {"id": 54, "difficulty": "medium", "question": "Two sides of triangle are 7 cm and 10 cm. If triangle is obtuse, which inequality must third side satisfy?", "options": {"A": "x < 3 or x > 17", "B": "3 < x < √149", "C": "√149 < x < 17", "D": "x > 17 only"}, "correct_answer": "C"},
    {"id": 55, "difficulty": "medium", "question": "In ΔABC, AB = 8 cm, BC = 15 cm. If AC is prime number, how many possible triangles?", "options": {"A": "4", "B": "5", "C": "6", "D": "7"}, "correct_answer": "B"},
    {"id": 56, "difficulty": "medium", "question": "Triangle has sides 8 cm, 15 cm, 17 cm. Classify by angles.", "options": {"A": "Acute", "B": "Right", "C": "Obtuse", "D": "Equiangular"}, "correct_answer": "B"},
    {"id": 57, "difficulty": "medium", "question": "In ΔPQR, PQ = 5 cm, QR = 7 cm, ∠Q = 100°. Which side is longest?", "options": {"A": "PQ", "B": "QR", "C": "PR", "D": "Cannot determine"}, "correct_answer": "C"},
    {"id": 58, "difficulty": "medium", "question": "If median to side BC in ΔABC is 5 cm, and AB = 8 cm, AC = 6 cm, which inequality holds?", "options": {"A": "BC < 14", "B": "BC > 2", "C": "2 < BC < 14", "D": "BC = 10"}, "correct_answer": "C"},
    {"id": 59, "difficulty": "medium", "question": "In triangle, sides are 9 cm, 12 cm, and x cm. If triangle is acute, what is possible x?", "options": {"A": "5", "B": "15", "C": "17", "D": "20"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "medium", "question": "Two sides of triangle are 6.5 cm and 3.5 cm. If perimeter is 18 cm, find third side.", "options": {"A": "7 cm", "B": "8 cm", "C": "9 cm", "D": "10 cm"}, "correct_answer": "B"},
    {"id": 61, "difficulty": "medium", "question": "In ΔABC, AB = 14 cm, BC = 8 cm. If AC is even integer, how many possible values?", "options": {"A": "7", "B": "8", "C": "9", "D": "10"}, "correct_answer": "B"},
    {"id": 62, "difficulty": "medium", "question": "Triangle has angles 30°, 60°, 90°. If shortest side is 4 cm, find longest side.", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "B"},
    {"id": 63, "difficulty": "medium", "question": "In quadrilateral ABCD, AB=5, BC=6, CD=7, DA=8. Which diagonal is longer: AC or BD?", "options": {"A": "AC", "B": "BD", "C": "Equal", "D": "Cannot determine"}, "correct_answer": "B"},
    {"id": 64, "difficulty": "medium", "question": "In ΔPQR, PQ = 10 cm, QR = 17 cm, PR = 21 cm. Which angle is largest?", "options": {"A": "∠P", "B": "∠Q", "C": "∠R", "D": "All equal"}, "correct_answer": "C"},
    {"id": 65, "difficulty": "medium", "question": "If triangle sides are a, b, c with a+b=15, b+c=18, c+a=17, find longest side.", "options": {"A": "a=7", "B": "b=8", "C": "c=10", "D": "c=9"}, "correct_answer": "C"},
    {"id": 66, "difficulty": "medium", "question": "In triangle, altitude from vertex to base is 6 cm. If adjacent sides are 10 cm and 8 cm, find range for base.", "options": {"A": "2 < base < 18", "B": "4 < base < 16", "C": "6 < base < 14", "D": "8 < base < 12"}, "correct_answer": "B"},
    {"id": 67, "difficulty": "medium", "question": "Triangle has vertices (1,1), (4,1), (1,5). Find longest side length.", "options": {"A": "3 units", "B": "4 units", "C": "5 units", "D": "6 units"}, "correct_answer": "C"},
    {"id": 68, "difficulty": "medium", "question": "In ΔABC, ∠A = 70°, AB = 10 cm, AC = 12 cm. Which is correct about BC?", "options": {"A": "BC < 10 cm", "B": "10 < BC < 12", "C": "12 < BC < 22", "D": "BC > 22 cm"}, "correct_answer": "C"},
    {"id": 69, "difficulty": "medium", "question": "Two sides of triangle are 9 cm and 13 cm. If triangle is isosceles, possible third side?", "options": {"A": "9 cm only", "B": "13 cm only", "C": "9 or 13 cm", "D": "22 cm"}, "correct_answer": "C"},
    {"id": 70, "difficulty": "medium", "question": "In right triangle, legs are 9 cm and 12 cm. Find hypotenuse.", "options": {"A": "13 cm", "B": "15 cm", "C": "17 cm", "D": "20 cm"}, "correct_answer": "B"},
    {"id": 71, "difficulty": "medium", "question": "If triangle sides are 5n, 12n, 13n, what type of triangle?", "options": {"A": "Acute", "B": "Right", "C": "Obtuse", "D": "Equilateral"}, "correct_answer": "B"},
    {"id": 72, "difficulty": "medium", "question": "In ΔPQR, PQ=√50, QR=√72, PR=√98. Arrange angles in ascending order.", "options": {"A": "∠P<∠Q<∠R", "B": "∠Q<∠P<∠R", "C": "∠R<∠P<∠Q", "D": "∠R<∠Q<∠P"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "Triangle has sides 10, 17, 21. What type of triangle?", "options": {"A": "Acute", "B": "Right", "C": "Obtuse", "D": "Cannot determine"}, "correct_answer": "C"},
    {"id": 74, "difficulty": "medium", "question": "In ΔABC, AB=8, BC=15. If ∠B=60°, find approximate AC.", "options": {"A": "12.5", "B": "13.5", "C": "14.5", "D": "15.5"}, "correct_answer": "B"},
    {"id": 75, "difficulty": "medium", "question": "If in triangle, side a=10, b=24, and triangle is right, find c.", "options": {"A": "26", "B": "28", "C": "25 or 26", "D": "26 or 28"}, "correct_answer": "C"},
    {"id": 76, "difficulty": "medium", "question": "In ΔXYZ, XY=13, YZ=20, XZ=21. Which angle is smallest?", "options": {"A": "∠X", "B": "∠Y", "C": "∠Z", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "Triangle perimeter is 30 cm. If sides are integers and triangle is scalene, maximum possible longest side?", "options": {"A": "13", "B": "14", "C": "15", "D": "16"}, "correct_answer": "B"},
    {"id": 78, "difficulty": "medium", "question": "In ΔABC, median AD=6 cm. If AB=10, AC=8, find range for BC.", "options": {"A": "2<BC<18", "B": "4<BC<16", "C": "6<BC<14", "D": "8<BC<12"}, "correct_answer": "B"},
    {"id": 79, "difficulty": "medium", "question": "Two sides of triangle are 7 cm and 10 cm. If area is maximum, what is third side?", "options": {"A": "√149 cm", "B": "12 cm", "C": "13 cm", "D": "17 cm"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "medium", "question": "In triangle, if angles are in ratio 2:3:4, arrange sides in ascending order.", "options": {"A": "a<b<c", "B": "b<c<a", "C": "c<a<b", "D": "a<c<b"}, "correct_answer": "A"},
    
    {"id": 81, "difficulty": "hard", "question": "In ΔABC, AD is median. If AB=10, AC=14, and AD=6, find BC.", "options": {"A": "8√2", "B": "10", "C": "12", "D": "2√31"}, "correct_answer": "D"},
    {"id": 82, "difficulty": "hard", "question": "Triangle has sides a,b,c with a²+b²=2c². Which statement is always true?", "options": {"A": "c is longest side", "B": "Triangle is right", "C": "∠C=90°", "D": "Triangle is acute"}, "correct_answer": "D"},
    {"id": 83, "difficulty": "hard", "question": "In ΔABC, ∠B=60°, AB=8, BC=10. Find AC using cosine rule.", "options": {"A": "2√21", "B": "2√19", "C": "√84", "D": "√76"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "If in triangle, sinA:sinB:sinC=3:4:5, find ratio of sides.", "options": {"A": "3:4:5", "B": "5:4:3", "C": "4:3:5", "D": "3:5:4"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "Triangle sides are 13, 14, 15. Find length of altitude to side 14.", "options": {"A": "10", "B": "11", "C": "12", "D": "13"}, "correct_answer": "C"},
    {"id": 86, "difficulty": "hard", "question": "In ΔABC, AB=AC=10, BC=12. Find length of median from A to BC.", "options": {"A": "6", "B": "7", "C": "8", "D": "9"}, "correct_answer": "C"},
    {"id": 87, "difficulty": "hard", "question": "If triangle sides are x²+x+1, 2x+1, x²-1 (x>1), prove triangle is obtuse for x>2. Find condition.", "options": {"A": "x>2", "B": "x>3", "C": "x>4", "D": "x>5"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "In ΔABC, ∠A=120°, AB=5, AC=8. Find BC.", "options": {"A": "√129", "B": "7", "C": "√109", "D": "13"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "Triangle sides satisfy a³+b³+c³=3abc. What type of triangle?", "options": {"A": "Equilateral", "B": "Right", "C": "Isosceles", "D": "Scalene"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "In ΔPQR, PQ=2, QR=3, PR=4. Find ∠PQR using cosine formula.", "options": {"A": "cos⁻¹(1/8)", "B": "cos⁻¹(-1/4)", "C": "cos⁻¹(1/4)", "D": "cos⁻¹(-1/8)"}, "correct_answer": "B"},
    {"id": 91, "difficulty": "hard", "question": "In triangle ABC, AB=7, BC=8, CA=9. Find approximate ∠B.", "options": {"A": "48°", "B": "58°", "C": "68°", "D": "78°"}, "correct_answer": "C"},
    {"id": 92, "difficulty": "hard", "question": "Triangle has vertices (0,0), (6,0), (3,4). Find circumradius.", "options": {"A": "3.125", "B": "3.25", "C": "3.5", "D": "3.75"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "If in ΔABC, a=13, b=14, c=15, find area using Heron's formula.", "options": {"A": "84", "B": "91", "C": "96", "D": "102"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "In triangle, sides are 5, 7, 8. Find angle opposite to side 7.", "options": {"A": "cos⁻¹(1/5)", "B": "cos⁻¹(1/7)", "C": "cos⁻¹(1/8)", "D": "cos⁻¹(2/5)"}, "correct_answer": "D"},
    {"id": 95, "difficulty": "hard", "question": "Triangle sides are √3, √4, √5. Which angle is largest?", "options": {"A": "Opposite √3", "B": "Opposite √4", "C": "Opposite √5", "D": "All equal"}, "correct_answer": "C"},
    {"id": 96, "difficulty": "hard", "question": "In ΔABC, if sinA=3/5 and sinB=5/13, find sinC.", "options": {"A": "33/65", "B": "56/65", "C": "63/65", "D": "16/65"}, "correct_answer": "C"},
    {"id": 97, "difficulty": "hard", "question": "Triangle sides satisfy a/(b+c)=b/(c+a)=c/(a+b)=k. Find k if triangle is not equilateral.", "options": {"A": "1/2", "B": "1/3", "C": "1/4", "D": "No such k"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "In ΔABC, altitudes from B and C are 6 and 8 respectively. If BC=10, find ∠A.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "D"},
    {"id": 99, "difficulty": "hard", "question": "Triangle has sides x, x+1, x+2. For what x is triangle obtuse?", "options": {"A": "x>1", "B": "x>2", "C": "x>3", "D": "x>4"}, "correct_answer": "B"},
    {"id": 100, "difficulty": "hard", "question": "In ΔABC, AB=AC, ∠A=100°. Which side is shortest?", "options": {"A": "AB", "B": "BC", "C": "AC", "D": "All equal"}, "correct_answer": "B"}
  ]
}