{
  "title": "Angle of Elevation - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "The angle of elevation of the sun when a 10m pole casts a 10√3m shadow is:", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "From a point on ground, the angle of elevation of top of a tower is 45°. If the tower is 50m high, the distance of point from tower is:", "options": {"A": "25 m", "B": "50 m", "C": "75 m", "D": "100 m"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "The angle of elevation of top of a tower from a point 20m away is 60°. Height of tower is:", "options": {"A": "20√3 m", "B": "40/√3 m", "C": "20/√3 m", "D": "40√3 m"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "If the height of tower is equal to its shadow length, the sun's elevation angle is:", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "From a point 50m from tower, angle of elevation to top is 30°. Tower height:", "options": {"A": "50/√3 m", "B": "50√3 m", "C": "100/√3 m", "D": "100 m"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "A ladder makes angle 60° with ground when leaning against wall. If foot is 2m from wall, ladder length:", "options": {"A": "2 m", "B": "4 m", "C": "2√3 m", "D": "4/√3 m"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "Angle of elevation is always measured from:", "options": {"A": "Horizontal upward", "B": "Vertical upward", "C": "Horizontal downward", "D": "Any direction"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "If tanθ = 1 for elevation angle, θ =", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "0°"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "From 40m away, elevation to tower top is 30°. Tower height:", "options": {"A": "40/√3 m", "B": "40√3 m", "C": "80 m", "D": "20√3 m"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "A 6m pole casts 2√3m shadow. Sun's elevation:", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "75°"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "Elevation angle can be maximum:", "options": {"A": "90°", "B": "180°", "C": "45°", "D": "60°"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "If elevation angle doubles from 30° to 60°, height to distance ratio becomes:", "options": {"A": "Doubles", "B": "Triples", "C": "√3 times", "D": "3 times"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "From point, elevation to tower is 45°. Moving 20m closer makes it 60°. Initial distance:", "options": {"A": "10(3+√3) m", "B": "20(√3+1) m", "C": "30 m", "D": "40 m"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "A kite string makes 60° with ground. If string is 100m, kite height:", "options": {"A": "50 m", "B": "50√3 m", "C": "100/√3 m", "D": "100 m"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "If height = distance × √3, elevation angle:", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "75°"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "From ship, elevation to lighthouse top is 30°. If lighthouse is 60m, distance:", "options": {"A": "60√3 m", "B": "120 m", "C": "60/√3 m", "D": "30√3 m"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Elevation and depression are equal when:", "options": {"A": "Always equal", "B": "Observer at higher level", "C": "Same horizontal level", "D": "Never equal"}, "correct_answer": "C"},
    {"id": 18, "difficulty": "easy", "question": "A 1.5m tall boy sees top of pole at 45°. If 10m from pole, pole height:", "options": {"A": "10 m", "B": "11.5 m", "C": "12 m", "D": "13.5 m"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "If shadow length = √3 × height, sun's elevation:", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "75°"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "Elevation angle measured using:", "options": {"A": "Protractor", "B": "Sextant", "C": "Theodolite", "D": "All of these"}, "correct_answer": "D"},

    {"id": 21, "difficulty": "medium", "question": "Two poles 6m and 11m tall. Line joining tops makes 45° with horizontal. Distance between poles:", "options": {"A": "5 m", "B": "10 m", "C": "15 m", "D": "20 m"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "From point on ground, elevations to top and bottom of flag on tower are 60° and 45°. If flag is 10m, tower height:", "options": {"A": "5(3+√1) m", "B": "10(√3+1) m", "C": "15 m", "D": "20/√3 m"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "A man 1.8m tall observes top of tower at 30°. Moving 30m closer makes it 60°. Tower height:", "options": {"A": "15.8 m", "B": "16.8 m", "C": "17.8 m", "D": "18.8 m"}, "correct_answer": "B"},
    {"id": 24, "difficulty": "medium", "question": "From two points opposite sides of tower, elevations are 45° and 60°. If points are 100m apart, tower height:", "options": {"A": "100/(√3+1) m", "B": "50√3 m", "C": "75 m", "D": "100√3/(√3+1) m"}, "correct_answer": "D"},
    {"id": 25, "difficulty": "medium", "question": "A ladder reaches window at height h making angle α. Pushed to make angle β, now reaches height 2h. cotα + cotβ = ?", "options": {"A": "1", "B": "2", "C": "√2", "D": "√3"}, "correct_answer": "B"},
    {"id": 26, "difficulty": "medium", "question": "From top of 75m tower, elevation to top of taller tower is 30° and depression to foot is 45°. Height of taller tower:", "options": {"A": "75(√3+1) m", "B": "150 m", "C": "100√3 m", "D": "125 m"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Two towers equal height. From point between them, elevations are 30° and 60°. If point is 40m from nearer tower, height:", "options": {"A": "20√3 m", "B": "30√3 m", "C": "40√3 m", "D": "50√3 m"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "A balloon rises vertically. From point, elevation changes from 30° to 60° in 10min. If speed constant, initial height:", "options": {"A": "5√3 × speed", "B": "10√3 × speed", "C": "15 × speed", "D": "20 × speed"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "From point A, elevation to tower is 30°. From B 40m closer, elevation is 45°. Height:", "options": {"A": "20(√3+1) m", "B": "40/(√3-1) m", "C": "20√3 m", "D": "40 m"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "Two ships see lighthouse at elevations 45° and 30°. If ships are 100m apart, lighthouse height:", "options": {"A": "50/(√3-1) m", "B": "100/(√3+1) m", "C": "50√3 m", "D": "100 m"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "From point on ground, elevation to cloud is 60° and to reflection in water is 30°. If point is 150m above water, cloud height:", "options": {"A": "300 m", "B": "450 m", "C": "600 m", "D": "750 m"}, "correct_answer": "C"},
    {"id": 32, "difficulty": "medium", "question": "A man 2m tall sees top of pole at 45°. Moving 10m closer makes it 60°. Pole height:", "options": {"A": "12 + 10√3 m", "B": "10(√3+1) m", "C": "15.32 m", "D": "17.32 m"}, "correct_answer": "D"},
    {"id": 33, "difficulty": "medium", "question": "From top of tower 50m high, elevations to two cars on same side are 30° and 45°. Distance between cars:", "options": {"A": "50(√3-1) m", "B": "50√3 m", "C": "100 m", "D": "50(3-√3) m"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Two vertical poles. From midpoint, elevations are complementary. If poles heights h1, h2, then:", "options": {"A": "h1h2 = d²/4", "B": "h1+h2 = d", "C": "h1/h2 = d", "D": "h1 = h2"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "A tower subtends angle θ at point. Moving d closer makes it 2θ. If original distance = x, then:", "options": {"A": "x = d(2cosθ-1)", "B": "x = d/(2cosθ-1)", "C": "x = 2d cosθ", "D": "x = d cos2θ"}, "correct_answer": "B"},
    {"id": 36, "difficulty": "medium", "question": "From point A, elevation is α. After moving a distance d toward tower, elevation becomes β. Height = ?", "options": {"A": "d sinα sinβ/sin(β-α)", "B": "d tanα tanβ/(tanβ-tanα)", "C": "d/(cotα - cotβ)", "D": "All of these"}, "correct_answer": "D"},
    {"id": 37, "difficulty": "medium", "question": "Two towers heights 20m and 30m. From top of shorter, elevation to top of taller is 45°. Distance:", "options": {"A": "10 m", "B": "20 m", "C": "30 m", "D": "40 m"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "From point on ground 40m from tower, elevation to top is 30°. Flag on tower makes elevation 45° from same point. Flag 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": 39, "difficulty": "medium", "question": "A man sees top of tower at 60°. After walking 20m away, sees at 30°. Initial distance:", "options": {"A": "10 m", "B": "15 m", "C": "20 m", "D": "25 m"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "From two points A and B on ground, elevations to tower top are α and β. If AB = d, height = ?", "options": {"A": "d sinα sinβ/sin(α-β)", "B": "d/(cotα - cotβ)", "C": "d tanα tanβ/(tanα-tanβ)", "D": "All equivalent"}, "correct_answer": "D"},
    {"id": 41, "difficulty": "medium", "question": "A ladder length L makes angle α with wall. When foot pulled x away, makes angle β. L = ?", "options": {"A": "x/(cosβ - cosα)", "B": "x sin(α+β)/sin(α-β)", "C": "x/(sinα - sinβ)", "D": "x tan(α+β)"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "medium", "question": "Two men on same side of tower see top at 30° and 60°. If they are 40m apart, tower height:", "options": {"A": "20√3 m", "B": "30√3 m", "C": "40√3 m", "D": "50√3 m"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "medium", "question": "From point A, elevation is 30°. From B vertically above A, elevation is 45°. If AB=20m, height:", "options": {"A": "10(3+√3) m", "B": "20(√3+1) m", "C": "30 m", "D": "40/√3 m"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "medium", "question": "A pole broken by wind, top touches ground at 30° 10m from foot. Original height:", "options": {"A": "10√3 m", "B": "20/√3 m", "C": "15 m", "D": "17.32 m"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "From top of tower 100m, elevations to two hills are 45° and 30°. If hills are same height and tower between them, distance between hills:", "options": {"A": "100(√3+1) m", "B": "200 m", "C": "100√3 m", "D": "150√3 m"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "medium", "question": "A man 1.5m tall sees cloud at 30°. Reflection in lake shows 60°. Cloud height:", "options": {"A": "3 m", "B": "4.5 m", "C": "6 m", "D": "7.5 m"}, "correct_answer": "C"},
    {"id": 47, "difficulty": "medium", "question": "Two towers heights a and b. From midpoint, elevations complementary. Distance between towers:", "options": {"A": "2√ab", "B": "√ab", "C": "√(a²+b²)", "D": "a+b"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "medium", "question": "From point, elevation to tower is θ. After moving half distance closer, elevation becomes φ. tanφ = ?", "options": {"A": "2tanθ/(2-tanθ)", "B": "2tanθ", "C": "3tanθ/2", "D": "tanθ(1+sinθ)"}, "correct_answer": "B"},
    {"id": 49, "difficulty": "medium", "question": "A balloon rises vertically. From point, elevation changes from 45° to 60° in 1min. If speed 10√3 m/min, initial height:", "options": {"A": "10(3+√3) m", "B": "20√3 m", "C": "30 m", "D": "40 m"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "medium", "question": "From top of tower 150m, angles of depression of two cars on same side are 45° and 30°. Distance between cars:", "options": {"A": "150(√3-1) m", "B": "150 m", "C": "300 m", "D": "150√3 m"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "Two poles 15m and 30m. From midpoint, elevation to tops are complementary. Distance between poles:", "options": {"A": "15√2 m", "B": "30 m", "C": "15√6 m", "D": "45 m"}, "correct_answer": "B"},
    {"id": 52, "difficulty": "medium", "question": "A tower subtends maximum angle at point on ground at distance:", "options": {"A": "Height", "B": "Height/√2", "C": "Height/√3", "D": "Height × √2"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "From point A, elevation is α. From B, elevation is β. If AB=d and A,B,tower foot collinear, height:", "options": {"A": "d sinα sinβ/sin(β-α)", "B": "d/(cotα - cotβ)", "C": "d tanα tanβ/(tanβ-tanα)", "D": "All"}, "correct_answer": "D"},
    {"id": 54, "difficulty": "medium", "question": "A man sees top of tower at 30°. Moving 20m toward tower, elevation becomes 45°. Height above eye level (1.6m):", "options": {"A": "10(√3+1)+1.6", "B": "10(3+√3)+1.6", "C": "20(√3-1)+1.6", "D": "27.32"}, "correct_answer": "D"},
    {"id": 55, "difficulty": "medium", "question": "Two vertical poles. From point on line joining bases, elevations to tops have equal tangents. Then:", "options": {"A": "Point is midpoint", "B": "Poles equal height", "C": "Point divides distance inversely as heights", "D": "Always true"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "medium", "question": "From point on ground, elevation to tower is 60°. Moving 40m away makes it 30°. Height:", "options": {"A": "20√3 m", "B": "30√3 m", "C": "40√3 m", "D": "50√3 m"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "A ladder length L reaches height h. If foot pulled x making angle θ, new height:", "options": {"A": "√(L² - (L sinα - x)²)", "B": "L sinθ", "C": "h - x cotα", "D": "All"}, "correct_answer": "D"},
    {"id": 58, "difficulty": "medium", "question": "From two points A,B on line through tower foot, elevations are α,β. If AB=d, height=", "options": {"A": "d tanα tanβ/(tanβ-tanα)", "B": "d/(cotα-cotβ)", "C": "d sinα sinβ/sin(β-α)", "D": "All"}, "correct_answer": "D"},
    {"id": 59, "difficulty": "medium", "question": "A tower stands on bank of river. From opposite bank 60m away, elevation=30°. From point 20m back, elevation=?", "options": {"A": "tan⁻¹(3/4)", "B": "45°", "C": "60°", "D": "tan⁻¹(2/3)"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "Two towers heights h1,h2. From midpoint, elevations complementary. Then:", "options": {"A": "h1h2 = d²/4", "B": "√h1 + √h2 = d", "C": "h1+h2 = d", "D": "h1/h2 = d²"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "medium", "question": "From point, elevation=30°. Moving 20m toward tower, elevation=45°. Height:", "options": {"A": "10(√3+1) m", "B": "20/(√3-1) m", "C": "27.32 m", "D": "All represent same"}, "correct_answer": "D"},
    {"id": 62, "difficulty": "medium", "question": "A man 1.8m tall sees top of tower at 45°. Moving 10m away, sees at 30°. Tower height:", "options": {"A": "1.8+10(√3+1)", "B": "1.8+5(3+√3)", "C": "23.66", "D": "24.8"}, "correct_answer": "C"},
    {"id": 63, "difficulty": "medium", "question": "Two ships see lighthouse at 45° and 30°. If ships 200m apart, height:", "options": {"A": "200/(√3+1) m", "B": "100(√3-1) m", "C": "100/(√3-1) m", "D": "200√3/(√3+1) m"}, "correct_answer": "C"},
    {"id": 64, "difficulty": "medium", "question": "From top of tower 50m, elevation to hill top=45°, depression to foot=30°. Hill height:", "options": {"A": "50(√3+1) m", "B": "100 m", "C": "75√3 m", "D": "125 m"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "A pole broken by storm, top touches ground 10m from foot at 30°. Broken height:", "options": {"A": "20/√3 m", "B": "10√3 m", "C": "15 m", "D": "17.32 m"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "medium", "question": "From point A, elevation=α. From B d meters closer, elevation=β. Height=", "options": {"A": "d sinα sinβ/sin(β-α)", "B": "d/(cotα-cotβ)", "C": "d tanα tanβ/(tanβ-tanα)", "D": "All"}, "correct_answer": "D"},
    {"id": 67, "difficulty": "medium", "question": "Two vertical poles. From point, elevations have equal sines. Then:", "options": {"A": "Point equidistant", "B": "Poles equal height", "C": "Point on perpendicular bisector", "D": "Can't say"}, "correct_answer": "B"},
    {"id": 68, "difficulty": "medium", "question": "From two points on line through tower, elevations α,β. Distance between points:", "options": {"A": "h(cotα - cotβ)", "B": "h(tanβ - tanα)", "C": "h(sinβ - sinα)", "D": "h(cosα - cosβ)"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "medium", "question": "A tower subtends angle θ at point distant d from foot. Height=", "options": {"A": "d tanθ", "B": "d sinθ", "C": "d cotθ", "D": "d secθ"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "medium", "question": "Two men see top of tower at 30° and 45°. If 40m apart and between tower, height:", "options": {"A": "40/(√3+1) m", "B": "40√3/(√3+1) m", "C": "20(3-√3) m", "D": "40(√3-1) m"}, "correct_answer": "B"},
    {"id": 71, "difficulty": "medium", "question": "From point A, elevation=45°. From B 20m away, elevation=30°. Height if A,B,tower foot collinear:", "options": {"A": "10(√3+1) m", "B": "20/(√3-1) m", "C": "27.32 m", "D": "All"}, "correct_answer": "D"},
    {"id": 72, "difficulty": "medium", "question": "A ladder against wall slips. Initially angle=60°, finally=30°. If foot moved 2m, ladder length:", "options": {"A": "2√3 m", "B": "4 m", "C": "3 m", "D": "5 m"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "From point on ground, elevation=θ. After moving a toward tower, elevation=2θ. Height=", "options": {"A": "a sin2θ sinθ/sinθ", "B": "a tanθ tan2θ/(tan2θ-tanθ)", "C": "a/(cotθ - cot2θ)", "D": "All"}, "correct_answer": "D"},
    {"id": 74, "difficulty": "medium", "question": "Two towers equal height. From point between, elevations 30°,60°. Distance to nearer tower=40m. Height=", "options": {"A": "40√3 m", "B": "20√3 m", "C": "30√3 m", "D": "50√3 m"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "A man sees top of tower at 60°. Moving 40m away, sees at 30°. Height above eye (1.5m):", "options": {"A": "20√3+1.5", "B": "34.64+1.5", "C": "36.14", "D": "35.5"}, "correct_answer": "C"},
    {"id": 76, "difficulty": "medium", "question": "From top of tower 100m, elevations to two cars 45°,30°. If cars on opposite sides, distance between:", "options": {"A": "100(√3+1) m", "B": "200 m", "C": "100√3 m", "D": "150√3 m"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "Two poles 8m,15m. From point, elevations equal. Distance from point to poles ratio:", "options": {"A": "8:15", "B": "15:8", "C": "64:225", "D": "√8:√15"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "From point, elevation=α. After moving d away, elevation=β. Height=", "options": {"A": "d sinα sinβ/sin(α-β)", "B": "d/(cotβ - cotα)", "C": "d tanα tanβ/(tanα-tanβ)", "D": "All"}, "correct_answer": "D"},
    {"id": 79, "difficulty": "medium", "question": "A balloon rises vertically. Elevation changes 30° to 60° in 2min at speed 10√3 m/min. Initial height:", "options": {"A": "10(3+√3) m", "B": "20√3 m", "C": "30 m", "D": "40 m"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "medium", "question": "Two towers heights a,b. From point, elevations complementary. Distance from point to towers ratio:", "options": {"A": "√a:√b", "B": "a:b", "C": "b:a", "D": "a²:b²"}, "correct_answer": "B"},

    {"id": 81, "difficulty": "hard", "question": "From point A on ground, elevation to top of tower is α. From point B vertically above A, elevation is β. If AB = h, tower height = ?", "options": {"A": "h sinα sinβ/sin(β-α)", "B": "h/(cotα - cotβ)", "C": "h tanα/(tanβ - tanα)", "D": "h(1 + tanα/tan(β-α))"}, "correct_answer": "B"},
    {"id": 82, "difficulty": "hard", "question": "Three vertical towers at vertices of equilateral triangle. From centroid, elevation to each top is 30°. If side = 60m, tower height:", "options": {"A": "20√3 m", "B": "30 m", "C": "40/√3 m", "D": "50√3 m"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "A,B,C collinear with tower. From A, elevation=30°, from B=45°, from C=60°. If AB=BC, ratio of distances:", "options": {"A": "√3:1:1/√3", "B": "3:√3:1", "C": "√3+1:1:√3-1", "D": "2:√2:1"}, "correct_answer": "C"},
    {"id": 84, "difficulty": "hard", "question": "Two towers heights h1,h2. From point on line joining bases, elevations complementary. Prove point divides distance in ratio h1²:h2².", "options": {"A": "True", "B": "False, it's h1:h2", "C": "False, it's √h1:√h2", "D": "False, it's h2:h1"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "From point P, elevation=θ. After moving distance d along line making angle α with line to tower, elevation becomes φ. Height=", "options": {"A": "d sinθ sinφ sinα/sin(φ-θ)", "B": "d/(cotθ - cotφ cosα)", "C": "Complex expression", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "A tower subtends maximum angle α at point on ground. If height=h, distance of this point=", "options": {"A": "h cotα", "B": "h/√(1+sinα)", "C": "h √(1+sinα)/sinα", "D": "h/√(cot²α-1)"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "Two towers heights a,b. From point where line joining tops subtends maximum angle, distance to bases ratio:", "options": {"A": "√a:√b", "B": "a:b", "C": "a²:b²", "D": "√a³:√b³"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "From three collinear points A,B,C, elevations to tower top are α,β,γ. If AB:BC=m:n, then:", "options": {"A": "(cotα-cotβ):(cotβ-cotγ)=m:n", "B": "(tanβ-tanα):(tanγ-tanβ)=m:n", "C": "Both A and B", "D": "Neither"}, "correct_answer": "C"},
    {"id": 89, "difficulty": "hard", "question": "A tower stands on plane. From three points A,B,C at same distance, elevations are 30°,45°,60°. If ABC equilateral, side length if height=100m:", "options": {"A": "100√6 m", "B": "200/√3 m", "C": "150√2 m", "D": "100√3 m"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "Two towers different heights. From point on line joining bases, tangents of elevations are equal. Point divides distance in ratio:", "options": {"A": "Heights ratio", "B": "Square of heights ratio", "C": "Inverse ratio of heights", "D": "Square root ratio"}, "correct_answer": "C"},
    {"id": 91, "difficulty": "hard", "question": "From point A, elevation=α. From B, elevation=β. From C, elevation=γ. If A,B,C collinear and AB:BC=p:q, then:", "options": {"A": "(cotα-cotβ):(cotβ-cotγ)=p:q", "B": "p(cotβ-cotγ)=q(cotα-cotβ)", "C": "Both equivalent", "D": "Neither"}, "correct_answer": "C"},
    {"id": 92, "difficulty": "hard", "question": "A tower of height h subtends maximum angle θ at point on ground. Then sinθ=", "options": {"A": "h/√(h²+d²)", "B": "√(1-(d/h)²)", "C": "h/(h+d)", "D": "1/√(1+(d/h)²)"}, "correct_answer": "D"},
    {"id": 93, "difficulty": "hard", "question": "Two towers heights h1,h2 distance d apart. From point on line joining bases, elevations complementary. Distance from point to tower1:", "options": {"A": "d√h1/(√h1+√h2)", "B": "d h1/(h1+h2)", "C": "d/2", "D": "d h1²/(h1²+h2²)"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "From point P, tower subtends angle α. After moving distance d toward tower, subtends angle β. Height=", "options": {"A": "d sinα sinβ/sin(β-α)", "B": "d/(cotα - cotβ)", "C": "d tan(α/2)tan(β/2)/[tan(β/2)-tan(α/2)]", "D": "All equivalent"}, "correct_answer": "D"},
    {"id": 95, "difficulty": "hard", "question": "Three towers same height at vertices of right triangle. From right angle vertex, elevation to each top=45°. If sides 30m,40m, height:", "options": {"A": "20√2 m", "B": "25 m", "C": "30 m", "D": "35 m"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "A tower on hill. From foot of hill, elevation to top=45°. From point 100m up hill at 30° incline, elevation=60°. Hill height:", "options": {"A": "50(3+√3) m", "B": "100(√3+1) m", "C": "150 m", "D": "200/√3 m"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "Two vertical towers. From point on ground, elevations have tangents in ratio 2:3. If heights ratio 4:9, point divides distance in ratio:", "options": {"A": "2:3", "B": "3:2", "C": "4:9", "D": "8:27"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "From point A, elevation=α. After moving distance d at angle θ to line to tower, elevation=β. Height=", "options": {"A": "d sinα sinβ sinθ/sin(β-α)", "B": "d/(cotα - cotβ cosθ)", "C": "Complex trig expression", "D": "Cannot find"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "A tower subtends angles α,β at two points distance d apart on line through foot. If points on same side, height=", "options": {"A": "d sinα sinβ/sin(α-β)", "B": "d/(cotβ - cotα)", "C": "d tanα tanβ/(tanα-tanβ)", "D": "All"}, "correct_answer": "D"},
    {"id": 100, "difficulty": "hard", "question": "Three collinear points A,B,C with AB=BC. From these, elevations to tower are 30°,45°,60°. If height=100m, AB=", "options": {"A": "100(√3-1) m", "B": "100 m", "C": "50√3 m", "D": "200/√3 m"}, "correct_answer": "A"}
  ]
}