{
  "title": "Heights & Distances - Grade 10 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "What trigonometric ratio is 'Opposite/Hypotenuse'? 📐", "options": {"A": "Sine", "B": "Cosine", "C": "Tangent", "D": "Cotangent"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "To find the height of a tree 🌳 using trigonometry, we need the distance from tree and?", "options": {"A": "Angle of elevation", "B": "Angle of rotation", "C": "Angle of descent", "D": "Angle of twist"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "If a ladder 🪜 makes 60° with ground and is 10m long, height it reaches = 10 × ?", "options": {"A": "cos 60°", "B": "sin 60°", "C": "tan 60°", "D": "cot 60°"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "tan 45° = ?", "options": {"A": "0", "B": "1/√2", "C": "1", "D": "√3"}, "correct_answer": "C"},
    {"id": 5, "difficulty": "easy", "question": "From a point 20m away, angle to top of pole is 45°. Height of pole? 📏", "options": {"A": "10 m", "B": "20 m", "C": "20√2 m", "D": "40 m"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "sin 30° = ?", "options": {"A": "√3/2", "B": "1/√2", "C": "1/2", "D": "1"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "A 6m shadow is cast by a pole when sun's elevation is 45°. Pole height? 🌞", "options": {"A": "3 m", "B": "6 m", "C": "6√2 m", "D": "12 m"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "cos 60° = ?", "options": {"A": "1/2", "B": "√3/2", "C": "1/√2", "D": "1"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "If height = distance × tan θ, θ is angle of?", "options": {"A": "Depression", "B": "Elevation", "C": "Inclination", "D": "Rotation"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "A kite 🪁 is flying at height h with string length 2h. What is sin of elevation angle?", "options": {"A": "1/2", "B": "1/√2", "C": "√3/2", "D": "1"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "tan θ = Height / ?", "options": {"A": "Hypotenuse", "B": "Distance", "C": "Shadow length", "D": "String length"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "√3 is value of tan ?°", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 13, "difficulty": "easy", "question": "A person 1.8m tall casts 1.8m shadow. Sun's elevation? 👤", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "If distance = height, then tan θ = ?", "options": {"A": "0", "B": "1", "C": "√3", "D": "1/√3"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "From top of 30m tower, depression of car is 45°. Car distance? 🚗", "options": {"A": "15 m", "B": "30 m", "C": "30√2 m", "D": "60 m"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "1/√2 = sin ?°", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
    {"id": 17, "difficulty": "easy", "question": "Height of equilateral triangle of side 2cm? 🔺", "options": {"A": "1 cm", "B": "√3 cm", "C": "2 cm", "D": "√2 cm"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "A flagstaff on building subtends equal angles at two points. This uses concept of?", "options": {"A": "Similar triangles", "B": "Congruence", "C": "Circle theorems", "D": "Symmetry"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "tan 30° = ?", "options": {"A": "1/√3", "B": "√3", "C": "1", "D": "0"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "To find height of mountain 🏔️, we measure angle of elevation from two points. This method is?", "options": {"A": "Single observation", "B": "Double observation", "C": "Triangulation", "D": "Approximation"}, "correct_answer": "C"},

    {"id": 21, "difficulty": "medium", "question": "A tree 🌲 broken by wind forms right triangle. Top touches ground 10m from foot making 30° angle. Original height?", "options": {"A": "10√3 m", "B": "20/√3 m", "C": "10(√3+1) m", "D": "15 m"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "From top of lighthouse, 100m above sea, depression of ship is 30°. Distance of ship? 🚢", "options": {"A": "100 m", "B": "100√3 m", "C": "200 m", "D": "100/√3 m"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "A 1.6m tall girl 👧 sees top of tower at 30° elevation from 20√3 m away. Tower height?", "options": {"A": "21.6 m", "B": "20 m", "C": "22.4 m", "D": "18.4 m"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "Two poles of heights 6m & 11m. Line joining tops makes 45° with horizontal. Distance between poles? 🚩🚩", "options": {"A": "5 m", "B": "10 m", "C": "15 m", "D": "17 m"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "A vertical tower's shadow is √3 times its height. Sun's elevation?", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "From a point P on ground, top of building is 60°. From top, depression of P is 30°. If building is 50m, distance of P? 🏢", "options": {"A": "50/√3 m", "B": "50√3 m", "C": "100/√3 m", "D": "100 m"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "A balloon 🎈 is tied to ground by string. Wind moves it, string makes 60° with ground. If string is 60m, balloon height?", "options": {"A": "30 m", "B": "30√3 m", "C": "60√3 m", "D": "120 m"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "The angle of elevation of top of hill from foot is 60°. After walking 200m towards hill, elevation is 75°. Hill height? ⛰️", "options": {"A": "100√3 m", "B": "200(√3+1) m", "C": "273.2 m", "D": "173.2 m"}, "correct_answer": "C"},
    {"id": 29, "difficulty": "medium", "question": "A person observes cloud ☁️ above lake at 30°. Its reflection in lake has depression 60°. If eye is 1.5m above lake, cloud height?", "options": {"A": "3 m", "B": "4.5 m", "C": "6 m", "D": "7.5 m"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "From a window 10m high, top of a pole has elevation 30° & bottom has depression 45°. Pole height? 📊", "options": {"A": "10(√3+1) m", "B": "10(√3-1) m", "C": "10/√3 m", "D": "20 m"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "A vertical tree is broken by wind. Top touches ground 15m from foot making 30° angle. Broken height from ground?", "options": {"A": "5√3 m", "B": "10√3 m", "C": "15/√3 m", "D": "30 m"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "Two ships ⛵ are observed from top of 75m cliff. Their depressions are 30° & 45°. Distance between ships if on same side?", "options": {"A": "75(√3-1) m", "B": "75 m", "C": "75√3 m", "D": "150 m"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "A tower's top from point A is 45°, from point B in line with A is 30°. If AB=40m, tower height? 🗼", "options": {"A": "20(√3+1) m", "B": "40(√3-1) m", "C": "54.64 m", "D": "20√3 m"}, "correct_answer": "C"},
    {"id": 34, "difficulty": "medium", "question": "A pole 5√3m high is fixed on a building. From ground, top of pole is 60° & top of building is 30°. Building height? 🏗️", "options": {"A": "5 m", "B": "5√3 m", "C": "10 m", "D": "15 m"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "A person sees top of tower at 45°. After walking 15m, angle becomes 30°. Tower height? 👀", "options": {"A": "15(√3+1)/2 m", "B": "15(√3-1) m", "C": "20.49 m", "D": "35.49 m"}, "correct_answer": "C"},
    {"id": 36, "difficulty": "medium", "question": "From an airplane ✈️ 1500m high, depression of a bridge is 30°. After 15s, depression is 60°. Plane speed? (km/hr)", "options": {"A": "720", "B": "360", "C": "180", "D": "540"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "A flagstaff on tower subtends same angle α at two points on ground. If points are d apart, flagstaff height? 🚩", "options": {"A": "d tan α", "B": "d/(2 tan α)", "C": "(d/2) cot α", "D": "d sin α"}, "correct_answer": "B"},
    {"id": 38, "difficulty": "medium", "question": "The shadow of a tower is 30m when sun's elevation is 30°. When elevation is 60°, shadow length?", "options": {"A": "10 m", "B": "10√3 m", "C": "15 m", "D": "20 m"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "A vertical pole subtends equal angles at distances a & b from its foot. Pole height? 📍", "options": {"A": "√(ab)", "B": "(a+b)/2", "C": "ab/(a+b)", "D": "2ab/(a+b)"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "From top of 7m building, top of cable tower has elevation 60° & bottom has depression 45°. Tower height? 🏙️", "options": {"A": "7(√3+1) m", "B": "7√3 m", "C": "7(√3-1) m", "D": "14 m"}, "correct_answer": "A"},

    {"id": 41, "difficulty": "hard", "question": "A tower stands on bank of river. From opposite bank, elevation is 60°. Move 20m back, elevation is 30°. River width? 🌉", "options": {"A": "10 m", "B": "20 m", "C": "30 m", "D": "40 m"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "A man on cliff observes boat at 30° depression. After 10 mins, depression is 60°. If boat moves at 2 m/s towards cliff, cliff height? ⛰️", "options": {"A": "600√3 m", "B": "1200/√3 m", "C": "400√3 m", "D": "800 m"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "hard", "question": "Two vertical towers are 60m apart. From midpoint, their tops have elevations tan⁻¹(3/4) & tan⁻¹(5/12). Height difference? 🗼", "options": {"A": "15 m", "B": "20 m", "C": "25 m", "D": "30 m"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "hard", "question": "A pole stands inside circular garden. From point on circumference, elevation is α. Walking a distance d along circumference, elevation is β. Pole height? 🌳", "options": {"A": "d/(cot α - cot β)", "B": "d/(tan β - tan α)", "C": "d sin α sin β/sin(β-α)", "D": "d cos α cos β/cos(β-α)"}, "correct_answer": "C"},
    {"id": 45, "difficulty": "hard", "question": "From top of tower, depression of a car is θ. After car travels distance d towards tower, depression is φ. Tower height? 🚗", "options": {"A": "d/(cot θ - cot φ)", "B": "d/(tan φ - tan θ)", "C": "d sin θ sin φ/sin(φ-θ)", "D": "d cos θ cos φ/cos(φ-θ)"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "hard", "question": "Two vertical poles of heights a & b. Line joining tops makes angle α with horizontal. Distance between poles?", "options": {"A": "(b-a) cot α", "B": "(b-a) tan α", "C": "(a+b) cot α", "D": "(a+b) tan α"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "hard", "question": "A balloon 🎈 rising vertically is observed from point on ground. After time t1, elevation is α; after t2, elevation is β. Uniform speed of balloon?", "options": {"A": "d(cot α - cot β)/(t2-t1)", "B": "d(tan β - tan α)/(t2-t1)", "C": "d/((t2-t1)(cot α - cot β))", "D": "Cannot be determined"}, "correct_answer": "D"},
    {"id": 48, "difficulty": "hard", "question": "A tower subtends angle θ at point A. After walking a distance a towards tower, it subtends angle φ. Tower height? 🏯", "options": {"A": "a sin θ sin φ/sin(φ-θ)", "B": "a cos θ cos φ/cos(φ-θ)", "C": "a tan θ tan φ/(tan φ - tan θ)", "D": "a/(cot θ - cot φ)"}, "correct_answer": "D"},
    {"id": 49, "difficulty": "hard", "question": "From a point on ground, top of minar has elevation θ. Move distance d towards minar, elevation becomes 45°. Move further d, elevation becomes (90°-θ). Minar height? 🕌", "options": {"A": "d√(tan θ)", "B": "d√(cot θ)", "C": "d√2", "D": "d(tan θ + cot θ)"}, "correct_answer": "C"},
    {"id": 50, "difficulty": "hard", "question": "A man sees top of tower at elevation α. After walking distance d towards tower on slope inclined at β to horizontal, elevation is γ. Tower height? 🧗", "options": {"A": "d sin β(sin(γ-α))/(sin(γ-β) sin(α-β))", "B": "d cos β(tan γ - tan α)", "C": "d/(cot α - cot γ)", "D": "Complex expression"}, "correct_answer": "D"},
    {"id": 51, "difficulty": "hard", "question": "Two towers of equal height are opposite each other on banks of river of width w. From point on other bank in line with first tower, elevations are α & β. Height? 🌊", "options": {"A": "w tan α tan β/(tan α + tan β)", "B": "w/(cot α + cot β)", "C": "w sin α sin β/sin(α+β)", "D": "Both A & B"}, "correct_answer": "D"},
    {"id": 52, "difficulty": "hard", "question": "A vertical tower stands on horizontal plane. From point A, elevation is α. From point B, due south of A, elevation is β. From point C, due east of A, elevation is γ. Tower height? 🧭", "options": {"A": "AB/(cot α - cot β)", "B": "AC/(cot α - cot γ)", "C": "√((AB²)/(cot²α - cot²β))", "D": "Requires 3D geometry"}, "correct_answer": "D"},
    {"id": 53, "difficulty": "hard", "question": "A pole is divided by a mark into two parts that subtend equal angles at point on ground. If lower part is a & upper is b, distance of point? 📍", "options": {"A": "√(ab)", "B": "√(a(a+b))", "C": "√(b(a+b))", "D": "(a+b)/2"}, "correct_answer": "B"},
    {"id": 54, "difficulty": "hard", "question": "From top of tower of height h, depression of foot of another tower is α. From bottom, elevation of top of second tower is β. Height of second tower? 🗼🗼", "options": {"A": "h(1+ tan β cot α)", "B": "h(1+ cot β tan α)", "C": "h(tan β/tan α)", "D": "h(sin(α+β)/sin α)"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "A person observes two objects on same side at depressions α & β (α>β). After time t, depressions become γ & δ. If objects move at constant speeds v1 & v2, height of observer? 👁️", "options": {"A": "(v1 cot α - v2 cot β)t/(cot β - cot α)", "B": "(v2 - v1)t/(tan γ - tan δ)", "C": "Cannot be found", "D": "(v1+v2)t/(cot α - cot β)"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "hard", "question": "A tower subtends greatest angle α at point P on ground. Distance of P from tower? 📏", "options": {"A": "h cot α", "B": "h tan α", "C": "h/√(tan α)", "D": "h√(cot α)"}, "correct_answer": "D"},
    {"id": 57, "difficulty": "hard", "question": "From a point between two towers, their tops have elevations α & β. If towers are heights h1 & h2 and point distances x from first, find x. 📊", "options": {"A": "(h1 cot α)/(h2 cot β)", "B": "(h2 tan β - h1 tan α)/(tan α - tan β)", "C": "(h1 - h2)/(cot α - cot β)", "D": "(h1 cot β - h2 cot α)/(cot α - cot β)"}, "correct_answer": "C"},
    {"id": 58, "difficulty": "hard", "question": "A vertical tower PQ. From A, elevation of P is α. From B, vertically below A, elevation of P is β. If AB = a, tower height? 🔼", "options": {"A": "a sin α sin β/sin(α-β)", "B": "a cos α cos β/cos(α-β)", "C": "a tan α tan β/(tan α - tan β)", "D": "a/(cot α - cot β)"}, "correct_answer": "D"},
    {"id": 59, "difficulty": "hard", "question": "A flagstaff on tower subtends equal angles at two points on ground in line with tower. If points are a & b from tower, flagstaff height? 🚩", "options": {"A": "√(ab)", "B": "√((a²+b²)/2)", "C": "√(b² - a²)", "D": "(b-a)/2"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "Two vertical towers have heights h1 & h2. Line joining tops makes angle θ with horizontal. Distance between towers?", "options": {"A": "(h2-h1) cot θ", "B": "(h2-h1) tan θ", "C": "(h1+h2) cot θ", "D": "(h1+h2) tan θ"}, "correct_answer": "A"}
  ]
}