{
  "title": "Line of Sight - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "What is the line of sight in trigonometry?", "options": {"A": "Line from eye to object", "B": "Horizontal line", "C": "Vertical line", "D": "Line along ground"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "Angle of elevation is between:", "options": {"A": "Line of sight and horizontal upward", "B": "Line of sight and vertical", "C": "Two lines of sight", "D": "Ground and object"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "Angle of depression is between:", "options": {"A": "Line of sight and horizontal downward", "B": "Line of sight and vertical", "C": "Two objects", "D": "Ground and observer"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "If line of sight is horizontal, angle of elevation is:", "options": {"A": "0°", "B": "45°", "C": "90°", "D": "180°"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "For same horizontal distance, as height increases, line of sight gets:", "options": {"A": "Steeper", "B": "Flatter", "C": "Same", "D": "Vertical"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "Two people looking at same object from different points have:", "options": {"A": "Same line of sight", "B": "Different lines of sight", "C": "Parallel lines", "D": "Perpendicular lines"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "Line of sight from top of tower to car on ground makes angle of:", "options": {"A": "Depression", "B": "Elevation", "C": "0°", "D": "90°"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "From ground to top of building, line of sight makes angle of:", "options": {"A": "Elevation", "B": "Depression", "C": "0°", "D": "90°"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "If observer and object at same height, line of sight is:", "options": {"A": "Horizontal", "B": "Vertical", "C": "45°", "D": "60°"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "For angle of elevation θ, line of sight is:", "options": {"A": "Above horizontal", "B": "Below horizontal", "C": "Horizontal", "D": "Vertical"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "Length of line of sight from 30m tower to point 40m away:", "options": {"A": "50 m", "B": "70 m", "C": "10 m", "D": "35 m"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "From 20m height to object 15m away horizontally, line of sight length:", "options": {"A": "25 m", "B": "35 m", "C": "5 m", "D": "20 m"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "Line of sight, horizontal line, and vertical distance form:", "options": {"A": "Right triangle", "B": "Square", "C": "Circle", "D": "Line"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "If angle of elevation=45°, line of sight makes equal angles with:", "options": {"A": "Horizontal and vertical", "B": "Two horizontals", "C": "Two verticals", "D": "None"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "For very distant object, line of sight is nearly:", "options": {"A": "Horizontal", "B": "Vertical", "C": "45°", "D": "90°"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "Observer at height h looking at object at ground distance d. Line of sight length:", "options": {"A": "√(h² + d²)", "B": "h + d", "C": "h - d", "D": "h × d"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Two observers at same point looking at different objects have:", "options": {"A": "Different lines of sight", "B": "Same line of sight", "C": "Parallel lines", "D": "Equal angles"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "If line of sight is vertical, angle of elevation is:", "options": {"A": "90°", "B": "0°", "C": "45°", "D": "180°"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "Line of sight from submarine to ship is:", "options": {"A": "Upward", "B": "Downward", "C": "Horizontal", "D": "Vertical"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "In line of sight problems, we typically use:", "options": {"A": "Trigonometric ratios", "B": "Algebra", "C": "Geometry", "D": "All"}, "correct_answer": "D"},

    {"id": 21, "difficulty": "medium", "question": "From point A, line of sight to tower top makes 30° with horizontal. From point B 40m closer, makes 45°. Distance AB:", "options": {"A": "40 m", "B": "20(√3+1) m", "C": "20(3+√3) m", "D": "60 m"}, "correct_answer": "C"},
    {"id": 22, "difficulty": "medium", "question": "Two observers 100m apart have lines of sight to same balloon making 60° and 30° with horizontal. If observers same height, balloon height:", "options": {"A": "50√3 m", "B": "100/√3 m", "C": "50 m", "D": "100 m"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Line of sight from ship to lighthouse top makes 30° with horizontal. If lighthouse 60m, distance:", "options": {"A": "60√3 m", "B": "120 m", "C": "60/√3 m", "D": "30√3 m"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "From top of tower, line of sight to car makes 45° depression. If tower 50m, horizontal distance:", "options": {"A": "50 m", "B": "25 m", "C": "75 m", "D": "100 m"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Two lines of sight from same point to top and bottom of pole make 60° and 30° with horizontal. If point 20m from pole, pole height:", "options": {"A": "20√3 m", "B": "40/√3 m", "C": "20/√3 m", "D": "40√3 m"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Line of sight from airplane to runway makes 30° depression. If height 3000m, line of sight length:", "options": {"A": "6000 m", "B": "3000√3 m", "C": "3000/√3 m", "D": "1500√3 m"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Observer height 1.5m. Line of sight to top of 20m pole makes 45°. Distance from pole:", "options": {"A": "18.5 m", "B": "20 m", "C": "21.5 m", "D": "22 m"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "Two lines of sight from points A and B to top of tower make angles α and β with horizontal. If AB=d, height:", "options": {"A": "d sinα sinβ/sin(α-β)", "B": "d/(cotα - cotβ)", "C": "d tanα tanβ/(tanβ-tanα)", "D": "All"}, "correct_answer": "D"},
    {"id": 29, "difficulty": "medium", "question": "Line of sight from hilltop to town makes 30° depression. After descending 200m, depression becomes 45°. Initial height:", "options": {"A": "100(3+√3) m", "B": "200√3 m", "C": "300 m", "D": "400/√3 m"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "From point, lines of sight to top and bottom of building make 45° and 30° with horizontal. If point 40m away, building height:", "options": {"A": "40(√3-1)/√3 m", "B": "40(3-√3) m", "C": "20√3 m", "D": "40/√3 m"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Two observers at same height have lines of sight to balloon making 45° and 60°. If observers 50m apart, balloon height:", "options": {"A": "25(3+√3) m", "B": "50/(√3-1) m", "C": "25√3 m", "D": "50 m"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "Line of sight from top of 80m tower to object makes 45° depression. Object distance:", "options": {"A": "80 m", "B": "40 m", "C": "120 m", "D": "160 m"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "From ship, lines of sight to top and bottom of cliff make 30° and 45°. If cliff height 100m, distance:", "options": {"A": "100(√3+1) m", "B": "50√3 m", "C": "150 m", "D": "200 m"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Line of sight length from 24m tower to point on ground 10m away:", "options": {"A": "26 m", "B": "34 m", "C": "14 m", "D": "25 m"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "Two lines of sight from same point to two towers make equal angles. If towers heights h1,h2, distances ratio:", "options": {"A": "h1:h2", "B": "h2:h1", "C": "√h1:√h2", "D": "1:1"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "From point on ground, line of sight to cloud makes 60°. Reflection in water makes 30°. If point 150m above water, cloud height:", "options": {"A": "300 m", "B": "450 m", "C": "600 m", "D": "750 m"}, "correct_answer": "C"},
    {"id": 37, "difficulty": "medium", "question": "Line of sight from airplane to two ships makes 45° and 30°. If height 3000m, distance between ships:", "options": {"A": "3000(√3-1) m", "B": "3000 m", "C": "6000 m", "D": "3000√3 m"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Observer moves toward tower. Line of sight angle changes from 30° to 45°. If initial distance 60m, height:", "options": {"A": "30(√3+1) m", "B": "60/(√3-1) m", "C": "30√3 m", "D": "60 m"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Two lines of sight from top of poles to each other's base make 45° each. If poles 20m apart, height difference:", "options": {"A": "20 m", "B": "10 m", "C": "30 m", "D": "40 m"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "Line of sight from top of building to car makes 30° depression. After car moves 50m, depression 45°. Building height:", "options": {"A": "25(√3+1) m", "B": "50/(√3-1) m", "C": "25√3 m", "D": "50 m"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "medium", "question": "From two points A and B, lines of sight to tower top make angles α and β. If AB=50m and α=30°, β=45°, height:", "options": {"A": "25(√3+1) m", "B": "50/(√3-1) m", "C": "25√3 m", "D": "50 m"}, "correct_answer": "B"},
    {"id": 42, "difficulty": "medium", "question": "Line of sight from balloon to point on ground makes 60° depression. After ascending 100m, depression 45°. Initial height:", "options": {"A": "50(3+√3) m", "B": "100(√3+1) m", "C": "150 m", "D": "200/√3 m"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "medium", "question": "Two observers at different heights have lines of sight to same object making same angle. Then:", "options": {"A": "They are equidistant", "B": "Heights proportional to distances", "C": "Product of heights constant", "D": "No relation"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "medium", "question": "Line of sight from top of 90m cliff to ship makes 30° depression. Ship distance:", "options": {"A": "90√3 m", "B": "180 m", "C": "90/√3 m", "D": "45√3 m"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "From point, lines of sight to top of two poles make complementary angles. If poles heights a,b, distances ratio:", "options": {"A": "a:b", "B": "b:a", "C": "√a:√b", "D": "a²:b²"}, "correct_answer": "B"},
    {"id": 46, "difficulty": "medium", "question": "Line of sight from airplane at 5000m to city makes 45° depression. Horizontal distance:", "options": {"A": "5000 m", "B": "2500 m", "C": "5000√2 m", "D": "2500√2 m"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "Two lines of sight from hilltop to two towns make 30° and 45°. If towns 5km apart, hill height:", "options": {"A": "2.5(√3+1) km", "B": "5/(√3+1) km", "C": "2.5√3 km", "D": "5(√3-1) km"}, "correct_answer": "B"},
    {"id": 48, "difficulty": "medium", "question": "Observer height 1.6m. Line of sight to top of 25m building makes 45°. Distance:", "options": {"A": "23.4 m", "B": "25 m", "C": "26.6 m", "D": "27 m"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "medium", "question": "Line of sight from top of lighthouse to horizon makes 0°. This means:", "options": {"A": "Earth curvature", "B": "Object at infinity", "C": "Height negligible", "D": "All"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "medium", "question": "Two lines of sight from ends of baseline to object form triangle. This is:", "options": {"A": "Triangulation", "B": "Parallax", "C": "Both", "D": "Neither"}, "correct_answer": "C"},
    {"id": 51, "difficulty": "medium", "question": "Line of sight from 120m tower to car makes 45°. After car moves, angle 30°. Distance moved:", "options": {"A": "120(√3-1) m", "B": "120 m", "C": "240 m", "D": "120√3 m"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "From point, lines of sight to top and bottom of flagpole make 60° and 45°. If flagpole 10m, point distance:", "options": {"A": "5(√3+1) m", "B": "10/(√3-1) m", "C": "5√3 m", "D": "10 m"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "medium", "question": "Two observers at sea level have lines of sight to ship making 30° and 45°. If observers 100m apart, ship distance from nearer:", "options": {"A": "50(√3+1) m", "B": "100/(√3-1) m", "C": "50√3 m", "D": "100 m"}, "correct_answer": "B"},
    {"id": 54, "difficulty": "medium", "question": "Line of sight from top of 75m tower to object on ground makes θ. After tower height increased to 100m, angle becomes φ. If tanθ=3/4, tanφ:", "options": {"A": "1", "B": "4/3", "C": "3/2", "D": "2"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "From two points on line through object, lines of sight make angles α,β. Distance between points:", "options": {"A": "h(cotβ - cotα)", "B": "h(tanα - tanβ)", "C": "h(sinα - sinβ)", "D": "h(cosβ - cosα)"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "Line of sight from balloon to two points on ground makes 45° and 30°. If points 200m apart, balloon height:", "options": {"A": "100(√3+1) m", "B": "200/(√3-1) m", "C": "100√3 m", "D": "200 m"}, "correct_answer": "B"},
    {"id": 57, "difficulty": "medium", "question": "Observer moves from point A to B. Line of sight angle changes from 45° to 60°. If AB=20m, height:", "options": {"A": "10(3+√3) m", "B": "20(√3+1) m", "C": "10√3 m", "D": "20 m"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "Two lines of sight from same height to top of tower make 30° and 60°. If points 80m apart, height:", "options": {"A": "20√3 m", "B": "40√3 m", "C": "60√3 m", "D": "80√3 m"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "Line of sight from top of 50m tower to horizon. Earth radius 6370km. Distance to horizon:", "options": {"A": "√(2×6370×0.05) km", "B": "√(6370×0.05) km", "C": "2√(6370×0.05) km", "D": "√(6370×0.1) km"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "From point A, line of sight to top of tower makes angle α. From B vertically above A, makes angle β. If AB=h, height:", "options": {"A": "h/(cotα - cotβ)", "B": "h sinα sinβ/sin(β-α)", "C": "h tanα tanβ/(tanβ-tanα)", "D": "All"}, "correct_answer": "D"},
    {"id": 61, "difficulty": "medium", "question": "Two lines of sight from hilltop to two villages make 30° and 45°. If villages 3km apart, hill height:", "options": {"A": "1.5(√3+1) km", "B": "3/(√3+1) km", "C": "1.5√3 km", "D": "3(√3-1) km"}, "correct_answer": "B"},
    {"id": 62, "difficulty": "medium", "question": "Line of sight from 40m tower to car makes 45°. After car moves 20m, angle 30°. Initial distance:", "options": {"A": "10(3+√3) m", "B": "20(√3+1) m", "C": "10√3 m", "D": "20 m"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "medium", "question": "From two ships, lines of sight to lighthouse top make 45° and 60°. If ships 500m apart, lighthouse height:", "options": {"A": "250(√3+1) m", "B": "500/(√3-1) m", "C": "250√3 m", "D": "500 m"}, "correct_answer": "B"},
    {"id": 64, "difficulty": "medium", "question": "Observer at height h1 sees object at angle α. Object at height h2 sees observer at angle β. Then:", "options": {"A": "h1/h2 = tanα/tanβ", "B": "h1/h2 = sinα/sinβ", "C": "h1/h2 = cotα/cotβ", "D": "No relation"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "Line of sight from airplane to airport makes 30°. After flying 100km, angle 45°. Initial height:", "options": {"A": "50(√3+1) km", "B": "100/(√3-1) km", "C": "50√3 km", "D": "100 km"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "medium", "question": "Two lines of sight from ends of 100m base to top of mountain make 45° and 30°. Mountain height:", "options": {"A": "50(√3+1) m", "B": "100/(√3-1) m", "C": "50√3 m", "D": "100 m"}, "correct_answer": "B"},
    {"id": 67, "difficulty": "medium", "question": "From point, line of sight to top of tree makes 60°. Moving 10m back makes 30°. Tree height:", "options": {"A": "5√3 m", "B": "10√3 m", "C": "15 m", "D": "20 m"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "medium", "question": "Line of sight from top of 80m building to bottom of 60m building makes 45°. Distance between:", "options": {"A": "20 m", "B": "40 m", "C": "60 m", "D": "80 m"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "medium", "question": "Two observers at different points see same cloud. Lines of sight make 30° and 45°. If observers 200m apart, cloud height:", "options": {"A": "100(√3+1) m", "B": "200/(√3-1) m", "C": "100√3 m", "D": "200 m"}, "correct_answer": "B"},
    {"id": 70, "difficulty": "medium", "question": "Line of sight from top of 150m tower to car makes θ. After car moves 50m, angle 45°. tanθ:", "options": {"A": "2", "B": "3", "C": "1.5", "D": "2.5"}, "correct_answer": "B"},
    {"id": 71, "difficulty": "medium", "question": "From two points A and B, lines of sight to top of pole make equal angles. If AB=d and pole height=h, distances from pole:", "options": {"A": "Equal", "B": "Sum=h", "C": "Product=h²", "D": "No relation"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "Line of sight from balloon to point A makes 45°. After ascending 100m, to same point makes 30°. Initial height:", "options": {"A": "50(3+√3) m", "B": "100(√3+1) m", "C": "150 m", "D": "200/√3 m"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "Two lines of sight from same point to two towers make complementary angles. If towers heights equal, distances:", "options": {"A": "Equal", "B": "Inversely proportional", "C": "Sum constant", "D": "Product constant"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "medium", "question": "From point on ground, line of sight to top of 50m tower makes 45°. After moving, makes 30°. Distance moved:", "options": {"A": "50(√3-1) m", "B": "50 m", "C": "100 m", "D": "50√3 m"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "Line of sight from ship to top of 100m cliff makes 30°. Distance:", "options": {"A": "100√3 m", "B": "200 m", "C": "100/√3 m", "D": "50√3 m"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "medium", "question": "Two observers at heights h1,h2 see each other at angles α,β. Then:", "options": {"A": "h1/h2 = tanα/tanβ", "B": "h1/h2 = sinα/sinβ", "C": "h1/h2 = cotα/cotβ", "D": "h1+h2 constant"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "Line of sight from top of 60m tower to bottom of 40m building makes 45°. Distance between:", "options": {"A": "20 m", "B": "40 m", "C": "60 m", "D": "80 m"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "From two points on opposite banks, lines of sight to top of tree make 45° and 60°. If river width 50m, tree height:", "options": {"A": "25(√3+1) m", "B": "50/(√3-1) m", "C": "25√3 m", "D": "50 m"}, "correct_answer": "B"},
    {"id": 79, "difficulty": "medium", "question": "Line of sight from airplane at 6000m to two ships makes 45° and 30°. Distance between ships:", "options": {"A": "6000(√3-1) m", "B": "6000 m", "C": "12000 m", "D": "6000√3 m"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "medium", "question": "Observer moves toward tower. Line of sight angle changes from 30° to 45°. If height 50m, distance moved:", "options": {"A": "50(√3-1) m", "B": "50 m", "C": "100 m", "D": "50√3 m"}, "correct_answer": "A"},

    {"id": 81, "difficulty": "hard", "question": "Three collinear points A,B,C with AB=BC. From these, lines of sight to tower top make angles α,β,γ with tanα=1/√3, tanβ=1, tanγ=√3. Tower height if AB=100m:", "options": {"A": "100 m", "B": "100√3 m", "C": "200 m", "D": "150 m"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "From point P, line of sight to top of tower makes angle α. From Q vertically above P, makes angle β. If PQ=h, height:", "options": {"A": "h/(cotα - cotβ)", "B": "h sinα sinβ/sin(β-α)", "C": "h tanα tanβ/(tanβ-tanα)", "D": "All equivalent"}, "correct_answer": "D"},
    {"id": 83, "difficulty": "hard", "question": "Two towers heights h1,h2. From point on line joining bases, lines of sight to tops make complementary angles. Then point divides distance in ratio:", "options": {"A": "h1²:h2²", "B": "h1:h2", "C": "√h1:√h2", "D": "h2:h1"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "From three non-collinear points, lines of sight to top of tower make equal angles. Then:", "options": {"A": "Points lie on circle", "B": "Tower at center", "C": "All equidistant", "D": "No such tower"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "Line of sight from top of tower of height H to points A and B make angles α and β. If AB=d and A,B,tower foot collinear, H=", "options": {"A": "d sinα sinβ/sin(α-β)", "B": "d/(cotα - cotβ)", "C": "d tanα tanβ/(tanα-tanβ)", "D": "All"}, "correct_answer": "D"},
    {"id": 86, "difficulty": "hard", "question": "Two observers at heights h1,h2 see each other at angles α,β. Distance between them:", "options": {"A": "√(h1² + h2² - 2h1h2 cos(α+β))", "B": "(h1 cotα + h2 cotβ)", "C": "√(h1² cot²α + h2² cot²β)", "D": "Complex expression"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "From point A, line of sight to top of tower makes angle α. After moving distance d at angle θ to original direction, makes angle β. Height:", "options": {"A": "d sinα sinβ sinθ/sin(β-α)", "B": "d/(cotα - cotβ cosθ)", "C": "Complex", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "Three towers at vertices of equilateral triangle. From centroid, lines of sight to tops make equal angles. If side=s, angle=", "options": {"A": "tan⁻¹(√3h/s)", "B": "tan⁻¹(2h/√3s)", "C": "tan⁻¹(h√3/s)", "D": "tan⁻¹(h/s√3)"}, "correct_answer": "B"},
    {"id": 89, "difficulty": "hard", "question": "Two lines of sight from points A and B to top of tower make angles α,β. From point C on AB extended, makes angle γ. If AC:CB=m:n, relation:", "options": {"A": "m cotβ + n cotα = (m+n) cotγ", "B": "m tanα + n tanβ = (m+n) tanγ", "C": "Both equivalent", "D": "Neither"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "Line of sight from top of tower to horizon. Earth radius R. Height h. Distance to horizon:", "options": {"A": "√(2Rh)", "B": "√(Rh)", "C": "√(2Rh+h²)", "D": "√(R²+h²)-R"}, "correct_answer": "C"},
    {"id": 91, "difficulty": "hard", "question": "From four points forming square, lines of sight to top of tower at center make equal angles. If side=a, height:", "options": {"A": "a/√2 tanθ", "B": "a tanθ/2", "C": "a tanθ/√2", "D": "a√2 tanθ"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "hard", "question": "Two towers heights H1,H2. From point where line joining tops subtends maximum angle, distances to bases:", "options": {"A": "√(H1H2)", "B": "H1+H2", "C": "|H1-H2|", "D": "√(H1²+H2²)"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "From point P, lines of sight to tops of two towers make angles α,β. If towers heights h1,h2 and P,h1,h2 collinear, distance between towers:", "options": {"A": "h1 cotα - h2 cotβ", "B": "h1 tanβ - h2 tanα", "C": "|h1 cotα - h2 cotβ|", "D": "|h1 tanα - h2 tanβ|"}, "correct_answer": "C"},
    {"id": 94, "difficulty": "hard", "question": "Three collinear points A,B,C. Lines of sight to top of tower make angles 30°,45°,60°. If AB=100m, BC=", "options": {"A": "100(√3-1) m", "B": "100 m", "C": "100√3 m", "D": "200 m"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "Line of sight from top of tower of height h to point on ground makes angle θ. Maximum area of triangle formed by tower, point, and ground:", "options": {"A": "h²/2", "B": "h²", "C": "h²/4", "D": "2h²"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "Two observers at sea level have lines of sight to ship making angles α,β. If observers distance=d, ship distance from line joining observers:", "options": {"A": "d/(cotα + cotβ)", "B": "d tanα tanβ/(tanα+tanβ)", "C": "d sinα sinβ/sin(α+β)", "D": "All"}, "correct_answer": "D"},
    {"id": 97, "difficulty": "hard", "question": "From point on ground, line of sight to top of tower makes angle α. After moving distance d toward tower at angle θ to original line, makes angle β. Height:", "options": {"A": "d sinα sinβ sinθ/sin(β-α)", "B": "d/(cotα - cotβ cosθ)", "C": "Complex", "D": "Multiple solutions"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "Four towers at vertices of rectangle. From center, lines of sight to tops make angles θ1,θ2,θ3,θ4. Relation:", "options": {"A": "tan²θ1 + tan²θ3 = tan²θ2 + tan²θ4", "B": "cot²θ1 + cot²θ3 = cot²θ2 + cot²θ4", "C": "Both if heights equal", "D": "No relation"}, "correct_answer": "B"},
    {"id": 99, "difficulty": "hard", "question": "Line of sight from top of tower to three collinear points makes angles α,β,γ. If distances from foot to points are in AP, then:", "options": {"A": "cotα, cotβ, cotγ in AP", "B": "tanα, tanβ, tanγ in AP", "C": "sinα, sinβ, sinγ in AP", "D": "cosα, cosβ, cosγ in AP"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "Two towers heights a,b. From point where line joining tops subtends angle θ, distance to line joining bases:", "options": {"A": "√(ab) cot(θ/2)", "B": "√(ab) tan(θ/2)", "C": "(a+b) cotθ", "D": "(a-b) tanθ"}, "correct_answer": "A"}
  ]
}