{
  "title": "Section formula - Grade 10 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "📍 Point dividing (2,3) and (4,5) in ratio 1:1 is: 🎯", "options": {"A": "(3,4)", "B": "(2,4)", "C": "(3,3)", "D": "(4,5)"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "📐 Section formula for internal division m:n: ✏️", "options": {"A": "((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))", "B": "((mx₁+nx₂)/(m+n), (my₁+ny₂)/(m+n))", "C": "((x₁+x₂)/2, (y₁+y₂)/2)", "D": "((x₁-x₂)/2, (y₁-y₂)/2)"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "🎯 Point dividing (0,0) and (6,8) in ratio 2:1: ⭐", "options": {"A": "(4, 16/3)", "B": "(2, 8/3)", "C": "(6,8)", "D": "(3,4)"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "📏 For midpoint, ratio is: 🔢", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "🔢 Point dividing (1,2) and (4,5) in ratio 2:1 internally: 🎨", "options": {"A": "(3,4)", "B": "(2,3)", "C": "(4,5)", "D": "(5,6)"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "📍 For external division, ratio m:n is: 🤔", "options": {"A": "m and n opposite sign", "B": "m and n same sign", "C": "m=n", "D": "m=0"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "📐 Point dividing (3,4) and (6,8) in ratio 1:2 internally: 🌈", "options": {"A": "(4, 16/3)", "B": "(5, 20/3)", "C": "(3,4)", "D": "(6,8)"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "🎯 External division formula when m:n: ⚡", "options": {"A": "((mx₂-nx₁)/(m-n), (my₂-ny₁)/(m-n))", "B": "((mx₁-nx₂)/(m-n), (my₁-ny₂)/(m-n))", "C": "same as internal", "D": "different sign"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "📏 Point dividing (2,3) and (5,7) externally in ratio 2:1: ✏️", "options": {"A": "(8,11)", "B": "(7,10)", "C": "(6,9)", "D": "(4,5)"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "🔢 If ratio is k:1, coordinates simplify to: 📐", "options": {"A": "((kx₂+x₁)/(k+1), (ky₂+y₁)/(k+1))", "B": "((x₁+kx₂)/(k+1), (y₁+ky₂)/(k+1))", "C": "((kx₁+x₂)/(k+1), (ky₁+y₂)/(k+1))", "D": "((x₁+x₂)/(k+1), (y₁+y₂)/(k+1))"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "📍 Point on segment closer to (x₁,y₁) if ratio: 🎯", "options": {"A": "m<n", "B": "m>n", "C": "m=n", "D": "m=0"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "📐 For external division, point lies: 🔍", "options": {"A": "Outside segment", "B": "Inside segment", "C": "At endpoint", "D": "At midpoint"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "🎯 Ratio in which (3,4) divides (1,2) and (5,6): ⭐", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "📏 Point dividing (0,0) and (10,0) in ratio 3:2: ✏️", "options": {"A": "(6,0)", "B": "(4,0)", "C": "(5,0)", "D": "(8,0)"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "🔢 If point is midpoint, then m:n = ? 🎨", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "equal to coordinates"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "📍 External division with ratio 2:-1 is same as internal with ratio: 🤔", "options": {"A": "2:1", "B": "1:2", "C": "-2:1", "D": "1:-2"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "📐 Point dividing (a,b) and (c,d) in ratio 1:1 is: 🌈", "options": {"A": "((a+c)/2, (b+d)/2)", "B": "(a+c, b+d)", "C": "((a-c)/2, (b-d)/2)", "D": "(ac, bd)"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "🎯 For ratio m:0, point is: ⚡", "options": {"A": "First point", "B": "Second point", "C": "Midpoint", "D": "Outside"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "📏 Point dividing (2,3) and (5,7) in ratio equal to distances from axes? 🔢", "options": {"A": "Depends", "B": "1:1", "C": "2:1", "D": "3:2"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "🔢 Trisection points divide segment in ratio: 📐", "options": {"A": "1:2 and 2:1", "B": "1:1", "C": "2:3", "D": "3:2"}, "correct_answer": "A"},
    
    {"id": 21, "difficulty": "medium", "question": "📍 Find ratio in which (4,5) divides (2,3) and (6,7): 🎯", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "📐 Point dividing (1,2) and (5,6) in ratio 3:2 internally: ✏️", "options": {"A": "(17/5, 22/5)", "B": "(13/5, 18/5)", "C": "(11/5, 14/5)", "D": "(19/5, 24/5)"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "🎯 Find point on line through (1,2) and (4,5) that is twice as far from first as second: ⭐", "options": {"A": "(-2,-1)", "B": "(7,8)", "C": "(3,4)", "D": "(5,6)"}, "correct_answer": "B"},
    {"id": 24, "difficulty": "medium", "question": "📏 Triangle vertices (0,0), (6,0), (0,8). Centroid coordinates? 🔍", "options": {"A": "(2, 8/3)", "B": "(3,4)", "C": "(4,3)", "D": "(6,8)"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "🔢 Point dividing (2,3) and (5,7) externally in ratio 3:2: 🎨", "options": {"A": "(11,15)", "B": "(10,14)", "C": "(9,13)", "D": "(8,12)"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "📍 Ratio in which y-axis divides (2,3) and (-4,5): 🤔", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "📐 Coordinates of point dividing (x₁,y₁) and (x₂,y₂) in ratio equal to their distances from origin: 🌈", "options": {"A": "Not simple", "B": "Midpoint", "C": "Weighted average", "D": "Geometric mean"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "🎯 Point on line through (1,2) and (4,5) at distance from first = 2×distance from second. Ratio? ⚡", "options": {"A": "2:1 externally", "B": "1:2 internally", "C": "2:1 internally", "D": "1:2 externally"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "📏 Find ratio in which line 2x+3y=5 divides (1,2) and (4,5): ✏️", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:2"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "🔢 Triangle vertices A(1,2), B(3,4), C(5,6). Point dividing median through A in ratio 2:1 is: 📐", "options": {"A": "(3,4)", "B": "(11/3, 14/3)", "C": "(10/3, 13/3)", "D": "(4,5)"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "📍 Point dividing join of (1,2) and (4,5) in ratio k:1. If point lies on line x+y=7, find k: 🎯", "options": {"A": "2", "B": "1", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "📐 Coordinates of points of trisection of (1,2) and (7,8): 🔢", "options": {"A": "(3,4) and (5,6)", "B": "(2,3) and (4,5)", "C": "(4,5) and (6,7)", "D": "(5,6) and (9,10)"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "🎯 Ratio in which point (11,15) divides (2,3) and (5,7): ⭐", "options": {"A": "3:2 externally", "B": "2:3 internally", "C": "3:2 internally", "D": "2:3 externally"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "📏 If point (3,4) divides (1,2) and (5,6) in ratio m:n, then m/n = ? 🔍", "options": {"A": "1", "B": "2", "C": "1/2", "D": "3"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "🔢 Point dividing (a,b) and (c,d) in ratio equal to difference of coordinates: 🎨", "options": {"A": "Special case", "B": "Midpoint", "C": "General formula", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "📍 Line through (1,2) and (4,5) extended to point where x-coordinate is 10. Find ratio: 🤔", "options": {"A": "3:2 externally", "B": "2:3 internally", "C": "3:1 externally", "D": "1:3 internally"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "📐 Centroid of triangle divides median in ratio: 🌈", "options": {"A": "2:1", "B": "1:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "🎯 Point on x-axis dividing (2,3) and (-4,5). Ratio? ⚡", "options": {"A": "3:5", "B": "5:3", "C": "2:3", "D": "3:2"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "📏 Coordinates of centroid using section formula for medians: ✏️", "options": {"A": "((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)", "B": "((x₁+x₂)/2, (y₁+y₂)/2)", "C": "((x₁+x₂+x₃)/2, (y₁+y₂+y₃)/2)", "D": "((x₁x₂x₃)/3, (y₁y₂y₃)/3)"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "🔢 Point dividing (sinθ, cosθ) and (cosθ, sinθ) in ratio 1:1: 📐", "options": {"A": "((sinθ+cosθ)/2, (cosθ+sinθ)/2)", "B": "(sinθ cosθ, cosθ sinθ)", "C": "(1,1)", "D": "(0,0)"}, "correct_answer": "A"},
    
    {"id": 41, "difficulty": "hard", "question": "📍 Points A(1,2), B(3,4), C(5,6). Find point dividing line joining A to midpoint of BC in ratio 2:3: 🎯", "options": {"A": "(17/5, 24/5)", "B": "(13/5, 18/5)", "C": "(11/5, 16/5)", "D": "(19/5, 26/5)"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "📐 Ratio in which line ax+by+c=0 divides (x₁,y₁) and (x₂,y₂) is given by: ✏️", "options": {"A": "-(ax₁+by₁+c)/(ax₂+by₂+c)", "B": "(ax₁+by₁+c)/(ax₂+by₂+c)", "C": "√[(ax₁+by₁+c)/(ax₂+by₂+c)]", "D": "(ax₁+by₁+c)²/(ax₂+by₂+c)²"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "🎯 Points A(2,3), B(4,5), C(6,7). Point D divides BC in ratio 2:1. Find ratio in which AD divides BC: ⭐", "options": {"A": "Doesn't divide", "B": "1:2", "C": "2:1", "D": "Internally 1:1"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "📏 Triangle vertices (0,0), (6,0), (0,8). Find point on median from (0,0) dividing it in ratio 3:2 from vertex: 🔍", "options": {"A": "(18/5, 12/5)", "B": "(12/5, 18/5)", "C": "(6/5, 8/5)", "D": "(24/5, 16/5)"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "hard", "question": "🔢 Points A(1,1), B(2,3), C(4,5). Find ratio in which point dividing AB in ratio 2:3 divides AC: 🎨", "options": {"A": "3:7", "B": "7:3", "C": "2:5", "D": "5:2"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "hard", "question": "📍 Line joining (1,2) to (5,6) is divided by line x+y=7. Find ratio: 🤔", "options": {"A": "1:2", "B": "2:1", "C": "1:1", "D": "3:1"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "hard", "question": "📐 If point P divides AB in ratio m:n and Q divides AB in ratio p:q, then ratio in which point dividing PQ divides AB: 🌈", "options": {"A": "Complex", "B": "m+n:p+q", "C": "mp:nq", "D": "m/p:n/q"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "hard", "question": "🎯 Harmonic conjugate of point dividing AB in ratio m:n is point dividing in ratio: ⚡", "options": {"A": "-m:n", "B": "m:-n", "C": "n:m", "D": "-m:-n"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "📏 Coordinates of incentre of triangle using section formula on angle bisectors: ✏️", "options": {"A": "((ax₁+bx₂+cx₃)/(a+b+c), (ay₁+by₂+cy₃)/(a+b+c))", "B": "((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)", "C": "((x₁x₂x₃)/(a+b+c), (y₁y₂y₃)/(a+b+c))", "D": "((a+b+c)/3, (a+b+c)/3)"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "🔢 Point dividing (2,3) and (5,7) in ratio such that its distance from (1,1) is √13. Possible ratios: 📐", "options": {"A": "1:2 or 2:1", "B": "1:1 only", "C": "3:2 only", "D": "2:3 only"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "📍 Locus of point dividing (a cosθ, b sinθ) and (a cosφ, b sinφ) in ratio m:n: 🎯", "options": {"A": "Conic", "B": "Line", "C": "Circle", "D": "Point"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "hard", "question": "📐 Points A(1,2), B(3,4), C(5,6). Find point dividing line joining A to centroid of triangle ABC in ratio 3:4: 🔢", "options": {"A": "(20/7, 26/7)", "B": "(17/7, 23/7)", "C": "(23/7, 29/7)", "D": "(26/7, 32/7)"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "🎯 If P divides AB in ratio 2:3 and Q divides AB in ratio 3:4, find ratio in which midpoint of PQ divides AB: ⭐", "options": {"A": "29:41", "B": "41:29", "C": "1:1", "D": "2:3"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "📏 Coordinates of point dividing line joining (x₁,y₁) and (x₂,y₂) in ratio equal to distance from two fixed points: 🔍", "options": {"A": "Complex locus", "B": "Midpoint", "C": "Weighted", "D": "Simple formula"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "🔢 Triangle vertices A(0,0), B(6,0), C(0,8). Points D,E,F on BC,CA,AB dividing sides in ratio 1:2. Area ratio triangle DEF : ABC: 🎨", "options": {"A": "2:9", "B": "1:3", "C": "1:4", "D": "2:7"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "hard", "question": "📍 Point moving such that its distances from (1,2) and (3,4) are in ratio 2:3. Locus? 🤔", "options": {"A": "Circle", "B": "Line", "C": "Parabola", "D": "Ellipse"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "📐 In triangle ABC, points D,E on AB,AC dividing in ratio 2:1 and 1:2. Find ratio in which DE divides BC: 🌈", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "🎯 Coordinates of point dividing line joining (at₁², 2at₁) and (at₂², 2at₂) in ratio t₁:t₂: ⚡", "options": {"A": "(a(t₁t₂), 2a√(t₁t₂))", "B": "(a(t₁+t₂), 2a(t₁+t₂))", "C": "(a(t₁²+t₂²), 2a(t₁+t₂))", "D": "(a(t₁t₂)², 2a(t₁t₂))"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "hard", "question": "📏 If point P divides AB in ratio m:n and Q divides AB in same ratio but externally, then PQ is divided by A and B in ratio: ✏️", "options": {"A": "m:n externally", "B": "n:m internally", "C": "m²:n²", "D": "m+n:m-n"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "🔢 Harmonic mean of distances from point dividing AB in ratio m:n to A and B is: 📐", "options": {"A": "2AB/(m+n)", "B": "AB/(m+n)", "C": "2mnAB/(m+n)", "D": "mnAB/(m+n)"}, "correct_answer": "A"}
  ]
}