{
  "title": "Cross Multiplication Method (Simultaneous Linear Equations) - Grade 9 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "The cross multiplication method is used to solve: 🧮", "options": {"A": "One equation", "B": "Two simultaneous linear equations", "C": "Quadratic equations", "D": "Cubic equations"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, cross multiplication formula gives x = ? 📐", "options": {"A": "(b₁c₂ - b₂c₁)/(a₁b₂ - a₂b₁)", "B": "(b₂c₁ - b₁c₂)/(a₁b₂ - a₂b₁)", "C": "(c₁a₂ - c₂a₁)/(a₁b₂ - a₂b₁)", "D": "(b₁c₂ - b₂c₁)/(a₂b₁ - a₁b₂)"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "In cross multiplication, y = ? 🔄", "options": {"A": "(c₁a₂ - c₂a₁)/(a₁b₂ - a₂b₁)", "B": "(a₁c₂ - a₂c₁)/(a₁b₂ - a₂b₁)", "C": "(c₂a₁ - c₁a₂)/(a₁b₂ - a₂b₁)", "D": "(a₂c₁ - a₁c₂)/(a₁b₂ - a₂b₁)"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "The denominator in cross multiplication formula is: 🏁", "options": {"A": "a₁b₁ - a₂b₂", "B": "a₁b₂ - a₂b₁", "C": "a₂b₂ - a₁b₁", "D": "a₁a₂ - b₁b₂"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "If a₁b₂ = a₂b₁, then lines are: 📏", "options": {"A": "Intersecting", "B": "Parallel", "C": "Coincident", "D": "Perpendicular"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "If a₁/a₂ = b₁/b₂ = c₁/c₂, then system has: ∞", "options": {"A": "Unique solution", "B": "No solution", "C": "Infinite solutions", "D": "Two solutions"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "Cross multiplication is also known as: 🔄", "options": {"A": "Substitution method", "B": "Elimination method", "C": "Cramer's rule", "D": "Formula method"}, "correct_answer": "D"},
    {"id": 8, "difficulty": "easy", "question": "For 2x + 3y = 5 and 4x + 6y = 10, cross multiplication gives denominator = ? 🧮", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "If denominator ≠ 0, system has: ✅", "options": {"A": "No solution", "B": "Unique solution", "C": "Infinite solutions", "D": "Cannot say"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "In cross multiplication, we write coefficients in how many rows? 📝", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "The formula x/(b₁c₂ - b₂c₁) = y/(c₁a₂ - c₂a₁) = 1/(a₁b₂ - a₂b₁) is valid when: 📐", "options": {"A": "a₁b₂ - a₂b₁ ≠ 0", "B": "Always", "C": "Only for positive numbers", "D": "Only for integers"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "If a₁b₂ - a₂b₁ = 0 but b₁c₂ - b₂c₁ ≠ 0, system is: 🚫", "options": {"A": "Consistent", "B": "Inconsistent", "C": "Dependent", "D": "Homogeneous"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "For equations 3x + 4y = 7 and 6x + 8y = 14, denominator = ? ➗", "options": {"A": "0", "B": "24", "C": "-24", "D": "12"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Cross multiplication directly gives solution without: 🔄", "options": {"A": "Adding equations", "B": "Subtracting equations", "C": "Making coefficients equal", "D": "All of these"}, "correct_answer": "D"},
    {"id": 15, "difficulty": "easy", "question": "If equations are x + y = 3 and x - y = 1, denominator = ? 🧮", "options": {"A": "0", "B": "-2", "C": "2", "D": "1"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "For homogeneous equations (c₁=c₂=0), cross multiplication gives: ⚖️", "options": {"A": "x=0, y=0", "B": "Infinite solutions", "C": "No solution", "D": "Unique non-zero solution"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "In cross multiplication diagram, arrows go from: ➡️", "options": {"A": "Top to bottom", "B": "Bottom to top", "C": "Crosswise", "D": "Randomly"}, "correct_answer": "C"},
    {"id": 18, "difficulty": "easy", "question": "If a₁=2, b₁=3, a₂=4, b₂=6, denominator = ? ➖", "options": {"A": "0", "B": "12", "C": "-12", "D": "24"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "When we cross multiply, we find values of x and y by equating to: 🏁", "options": {"A": "Zero", "B": "One", "C": "Reciprocal of denominator", "D": "Sum of coefficients"}, "correct_answer": "C"},
    {"id": 20, "difficulty": "easy", "question": "If x = (b₁c₂ - b₂c₁)/(a₁b₂ - a₂b₁), then y = ? 🔄", "options": {"A": "(c₁a₂ - c₂a₁)/(a₁b₂ - a₂b₁)", "B": "(a₁c₂ - a₂c₁)/(a₁b₂ - a₂b₁)", "C": "(b₂c₁ - b₁c₂)/(a₁b₂ - a₂b₁)", "D": "None"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "Solve by cross multiplication: 3x + 2y = 7, 4x - 3y = -2. Find x. 📐", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Solve: 5x + 3y = 11, 2x + 7y = 12. Find y. 🧮", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "For equations 7x - 3y = 5, 14x - 6y = 10, cross multiplication gives: 🔄", "options": {"A": "x=0, y=0", "B": "No solution", "C": "Infinite solutions", "D": "x=1, y=2"}, "correct_answer": "C"},
    {"id": 24, "difficulty": "medium", "question": "If 2x + 5y = 12 and 4x + 10y = k are inconsistent, k = ? 🚫", "options": {"A": "12", "B": "24", "C": "Any except 24", "D": "Any number"}, "correct_answer": "C"},
    {"id": 25, "difficulty": "medium", "question": "Solve: 0.5x + 0.2y = 1.3, 0.3x + 0.4y = 1.4. Find x. 🔢", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 26, "difficulty": "medium", "question": "By cross multiplication, solve: x/2 + y/3 = 1, x/3 + y/2 = 1. Find x+y. ➕", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "Equations ax + by = c and bx + ay = d have unique solution if: 📐", "options": {"A": "a = b", "B": "a ≠ b", "C": "a² ≠ b²", "D": "a² = b²"}, "correct_answer": "C"},
    {"id": 28, "difficulty": "medium", "question": "If denominator in cross multiplication is zero and numerator for x also zero, then system is: 🔄", "options": {"A": "Inconsistent", "B": "Consistent with unique solution", "C": "Consistent with infinite solutions", "D": "Homogeneous"}, "correct_answer": "C"},
    {"id": 29, "difficulty": "medium", "question": "Solve: (x+y)/2 = 5, (x-y)/3 = 1. Find y. 🧮", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "For what k will 3x + 4y = 7 and 6x + 8y = k have infinite solutions? ∞", "options": {"A": "7", "B": "14", "C": "21", "D": "Any number"}, "correct_answer": "B"},
    {"id": 31, "difficulty": "medium", "question": "Solve: 3(x+2) - 4(y-1) = 14, 5(x-1) + 2(y+3) = 18. Find x. 🔢", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "If cross multiplication gives x/2 = y/(-3) = 1/(-1), then x,y = ? 📐", "options": {"A": "2,-3", "B": "-2,3", "C": "-2,-3", "D": "2,3"}, "correct_answer": "B"},
    {"id": 33, "difficulty": "medium", "question": "Solve: 2/x + 3/y = 13, 5/x - 4/y = -2. Let 1/x = u, 1/y = v. Find y. 🔄", "options": {"A": "1/2", "B": "1/3", "C": "1/4", "D": "1/5"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "The pair 4x + 6y = 9, 2x + 3y = 6 is: 📏", "options": {"A": "Consistent", "B": "Inconsistent", "C": "Dependent", "D": "Homogeneous"}, "correct_answer": "B"},
    {"id": 35, "difficulty": "medium", "question": "Solve by cross multiplication: 3x + 4y = 25, 5x - 2y = 9. Find y. 🧮", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "B"},
    {"id": 36, "difficulty": "medium", "question": "If a₁b₂ - a₂b₁ = 12, b₁c₂ - b₂c₁ = 24, then x = ? 📐", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "For equations 2x + py = 8, 4x + 2y = q to have infinite solutions, p,q = ? ∞", "options": {"A": "4,16", "B": "2,8", "C": "1,4", "D": "3,12"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Solve: (2x+1)/3 + (y-2)/4 = 2, (3x-1)/2 - (y+1)/3 = 1. Find x-y. ➖", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "If cross multiplication yields x/(-10) = y/15 = 1/5, then x,y = ? 🔄", "options": {"A": "-2,3", "B": "2,-3", "C": "-2,-3", "D": "2,3"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "For 5x + ky = 12 and 10x + 6y = 24 to be consistent with infinite solutions, k = ? 🔢", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "hard", "question": "Solve by cross multiplication: (a-b)x + (a+b)y = a² - 2ab - b², (a+b)(x+y) = a² + b². Find x. 📐", "options": {"A": "a+b", "B": "a-b", "C": "2a", "D": "2b"}, "correct_answer": "B"},
    {"id": 42, "difficulty": "hard", "question": "If x = (b₁c₂ - b₂c₁)/(a₁b₂ - a₂b₁) and denominator is D, then y = (c₁a₂ - c₂a₁)/D. For equations 3x+4y=7, 6x+8y=14, D=? 🧮", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "Solve: 2(3u-v) = 5uv, 2(u+3v) = 5uv. Find u+v (u,v ≠ 0). 🔄", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "For what p will system 3x + 5y = 7, 6x + 10y = p have no solution? 🚫", "options": {"A": "14", "B": "Any except 14", "C": "7", "D": "Any number"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "hard", "question": "Solve: 3/(x+y) + 2/(x-y) = 3, 9/(x+y) - 4/(x-y) = 2. Find x. 🔢", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "C"},
    {"id": 46, "difficulty": "hard", "question": "If cross multiplication gives x/(2) = y/(-3) = 1/(k) and x=1, y=-2, then k = ? 📐", "options": {"A": "2", "B": "1/2", "C": "-1/2", "D": "-2"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "hard", "question": "Solve: ax + by = a² + b², bx - ay = 0. Find x. 🧮", "options": {"A": "a", "B": "b", "C": "a+b", "D": "a-b"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "hard", "question": "Equations (k+1)x + 3y = 5, 4x + 6y = 10 have infinite solutions when k = ? ∞", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "B"},
    {"id": 49, "difficulty": "hard", "question": "Solve: (7x-2y)/xy = 5, (8x+7y)/xy = 15. Find x+y. ➕", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "If x = (b₁c₂ - b₂c₁)/(a₁b₂ - a₂b₁) and equations are 2x+3y=7, 4x+6y=14, then x = ? 🔄", "options": {"A": "0", "B": "1", "C": "Any real", "D": "No solution"}, "correct_answer": "C"},
    {"id": 51, "difficulty": "hard", "question": "Solve by cross multiplication: (x/a) + (y/b) = a+b, (x/a²) + (y/b²) = 2. Find x. 📐", "options": {"A": "a²", "B": "b²", "C": "ab", "D": "a²b²"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "hard", "question": "For equations 3x + 2y = 5, 6x + 4y = 10, cross multiplication yields denominator 0 and numerator for x also 0. This means: 🔄", "options": {"A": "No solution", "B": "Unique solution", "C": "Infinite solutions", "D": "Error"}, "correct_answer": "C"},
    {"id": 53, "difficulty": "hard", "question": "Solve: 2x + 3y = 7, 6x - ky = 21. For unique solution, k ≠ ? 📏", "options": {"A": "-9", "B": "9", "C": "6", "D": "-6"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "If (x+2)/3 = (y-1)/4 = (x+y)/5, find x and y using cross multiplication like method. 🔢", "options": {"A": "2,3", "B": "3,4", "C": "4,5", "D": "5,6"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "Solve: (x+y-8)/2 = (x+2y-14)/3 = (3x-y)/4. Find x. 🧮", "options": {"A": "2", "B": "4", "C": "6", "D": "8"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "hard", "question": "For equations a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0, if a₁b₂ - a₂b₁ = 0 but c₁b₂ - c₂b₁ ≠ 0, then lines are: 📏", "options": {"A": "Parallel", "B": "Coincident", "C": "Intersecting", "D": "Perpendicular"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "Solve: 0.2x + 0.3y = 1.3, 0.4x + 0.5y = 2.3. Find y. 🔄", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "C"},
    {"id": 58, "difficulty": "hard", "question": "If cross multiplication diagram has coefficients a₁,b₁,c₁ in first row, a₂,b₂,c₂ in second, third row repeats first. Denominator = ? 📝", "options": {"A": "a₁b₂ - a₂b₁", "B": "b₁c₂ - b₂c₁", "C": "c₁a₂ - c₂a₁", "D": "a₁c₂ - a₂c₁"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "hard", "question": "Solve: (x/3) + (y/4) = 5, (x/2) - (y/5) = 1. Find x*y. ✖️", "options": {"A": "36", "B": "48", "C": "54", "D": "60"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "If system 4x + 6y = 10, 2x + 3y = k has no solution, then k ≠ ? 🚫", "options": {"A": "5", "B": "10", "C": "15", "D": "20"}, "correct_answer": "A"}
  ]
}