{
  "title": "Infinite GP (Sum to Infinity) - Grade 10 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "For an infinite GP to have a finite sum, the common ratio must be ______. 📉", "options": {"A": "greater than 1", "B": "less than -1", "C": "between -1 and 1 (|r|<1)", "D": "equal to 1"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "Sum to infinity formula for GP is ______. ∞", "options": {"A": "a/(1-r) where |r|<1", "B": "a/(r-1)", "C": "a(1-rⁿ)/(1-r)", "D": "a(rⁿ-1)/(r-1)"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "First term = 9, common ratio = 1/3. Sum to infinity? 🔢", "options": {"A": "12", "B": "13.5", "C": "27", "D": "18"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "If a = 16, r = 0.25, S∞ = ? 🧮", "options": {"A": "64", "B": "32", "C": "21.33", "D": "20"}, "correct_answer": "C"},
    {"id": 5, "difficulty": "easy", "question": "Infinite GP: 1, 1/2, 1/4, ... S∞ = ? ✨", "options": {"A": "1", "B": "2", "C": "0.5", "D": "1.5"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "Which can have a finite infinite sum? 🤔", "options": {"A": "2, 6, 18, ...", "B": "100, 50, 25, ...", "C": "1, -2, 4, -8, ...", "D": "5, 10, 20, ..."}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "If r = 0.5, a = 8, sum to infinity = ? ⚡", "options": {"A": "12", "B": "16", "C": "20", "D": "24"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "|r| ≥ 1 implies S∞ is ______. 🚫", "options": {"A": "finite", "B": "infinite/undefined", "C": "zero", "D": "negative"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "Sum of infinite GP: 0.9, 0.09, 0.009, ... = ? 🔄", "options": {"A": "0.99", "B": "1", "C": "0.999", "D": "0.9"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "If S∞ = 20 and a = 5, r = ? 📏", "options": {"A": "0.5", "B": "0.6", "C": "0.75", "D": "0.8"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "Infinite decreasing GP exists if ______. 📉", "options": {"A": "r > 1", "B": "0 < r < 1", "C": "r < -1", "D": "r = 1"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "First term 12, S∞ = 16. Find r. 🧐", "options": {"A": "0.2", "B": "0.25", "C": "0.3", "D": "0.5"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "Find S∞ for 27, 9, 3, ... 🌟", "options": {"A": "40", "B": "40.5", "C": "54", "D": "81"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "If a = 1, S∞ = 4, then r = ? 🔢", "options": {"A": "0.5", "B": "0.75", "C": "0.6", "D": "0.8"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "Which value of r allows finite S∞? ✅", "options": {"A": "2", "B": "-2", "C": "0.2", "D": "1.5"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "S∞ for GP with a=0.4, r=0.1 = ? 🔟", "options": {"A": "0.444...", "B": "0.5", "C": "0.4", "D": "0.45"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "If r = -0.5, a = 8, S∞ = ? ➖", "options": {"A": "16/3", "B": "16", "C": "5.33", "D": "6"}, "correct_answer": "C"},
    {"id": 18, "difficulty": "easy", "question": "Infinite GP sum formula is valid when |r| __ 1. ⚠️", "options": {"A": ">", "B": "<", "C": "=", "D": ">="}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "S∞ = 30, r = 2/3, a = ? 🔍", "options": {"A": "10", "B": "12", "C": "15", "D": "20"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "GP: 64, 32, 16, ... S∞ = ? 📐", "options": {"A": "128", "B": "96", "C": "112", "D": "108"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "If S∞ = 15 and a = 9, find r. 🤔", "options": {"A": "0.3", "B": "0.4", "C": "0.5", "D": "0.6"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "Sum of infinite series 5 - 5/2 + 5/4 - 5/8 + ... = ? 🔄", "options": {"A": "10/3", "B": "3.33", "C": "5", "D": "20/3"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "The sum 0.6 + 0.06 + 0.006 + ... ∞ = ? ✨", "options": {"A": "2/3", "B": "0.666", "C": "0.6", "D": "0.06"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "If each term of infinite GP is squared, new S∞ = ? 📈", "options": {"A": "a²/(1-r²)", "B": "(a/(1-r))²", "C": "a/(1-r²)", "D": "a²/(1-r)²"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "First term = 0.5, S∞ = 2/3, find r. 🧮", "options": {"A": "0.2", "B": "0.25", "C": "0.3", "D": "0.4"}, "correct_answer": "B"},
    {"id": 26, "difficulty": "medium", "question": "Sum to infinity of 1, -1/2, 1/4, -1/8, ... = ? ➖", "options": {"A": "2/3", "B": "3/4", "C": "4/5", "D": "5/6"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "If S∞ = 25, second term = 5, find a. 🔎", "options": {"A": "10", "B": "12.5", "C": "15", "D": "20"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "Infinite GP with r = 0.2, sum is 125. Find 3rd term. 📊", "options": {"A": "2", "B": "4", "C": "8", "D": "16"}, "correct_answer": "B"},
    {"id": 29, "difficulty": "medium", "question": "If S∞ for a GP is 8 and a=6, r=? 🧐", "options": {"A": "1/4", "B": "1/3", "C": "1/5", "D": "1/6"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "0.12 + 0.0012 + 0.000012 + ... ∞ = ? 🔢", "options": {"A": "0.1212...", "B": "4/33", "C": "0.12", "D": "0.012"}, "correct_answer": "B"},
    {"id": 31, "difficulty": "medium", "question": "For infinite GP, if a = 20, S∞ = 25, find r. ⚖️", "options": {"A": "0.1", "B": "0.2", "C": "0.25", "D": "0.3"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "If sum to infinity of GP is twice the first term, r = ? 📐", "options": {"A": "1/4", "B": "1/3", "C": "1/2", "D": "2/3"}, "correct_answer": "C"},
    {"id": 33, "difficulty": "medium", "question": "S∞ of GP with a = √2, r = 1/√2 = ? ∞", "options": {"A": "√2 + 2", "B": "2+√2", "C": "√2/(√2-1)", "D": "2√2"}, "correct_answer": "C"},
    {"id": 34, "difficulty": "medium", "question": "If S∞ = 3a (a≠0), r = ? 🤯", "options": {"A": "1/3", "B": "2/3", "C": "3/4", "D": "1/2"}, "correct_answer": "B"},
    {"id": 35, "difficulty": "medium", "question": "Sum of infinite GP 1, 2/5, 4/25, ... = ? 📉", "options": {"A": "5/3", "B": "5/4", "C": "5/2", "D": "4/3"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "If S∞ = 16, first two terms sum = 12, find a. 🔧", "options": {"A": "6", "B": "8", "C": "9", "D": "10"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "0.4777... as infinite GP sum = ? (Hint: 0.4 + 0.0777...) 🔁", "options": {"A": "43/90", "B": "47/99", "C": "7/15", "D": "4/9"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Three numbers in GP have infinite sum 9/2. If first is 3, find r. 📏", "options": {"A": "1/4", "B": "1/3", "C": "1/2", "D": "2/3"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "If every term of infinite GP is 3 times sum of all later terms, r=? 🔥", "options": {"A": "1/4", "B": "1/3", "C": "1/2", "D": "2/3"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "S∞ for GP: x, x², x³, ... |x|<1, = ? 📚", "options": {"A": "x/(1-x)", "B": "1/(1-x)", "C": "x/(1-x²)", "D": "1/(1-x²)"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "hard", "question": "If S∞ of GP is S, and sum of squares of terms is T, find a in terms of S,T. 🧠", "options": {"A": "S²/√(S²-T)", "B": "S²/(S²-T)", "C": "√(S²T/(S²-T))", "D": "S²/(S+T)"}, "correct_answer": "B"},
    {"id": 42, "difficulty": "hard", "question": "Sum of infinite GP is 4, sum of cubes of terms is 192/7. Find r. ⚡", "options": {"A": "1/2", "B": "1/3", "C": "1/4", "D": "1/5"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "If S∞ of GP is 6, and sum of reciprocals of terms is 3/2, find a. 🔄", "options": {"A": "2", "B": "3", "C": "4", "D": "6"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "hard", "question": "GP with a=1, S∞ = r/(1-r). Find r. 🤔", "options": {"A": "1/2", "B": "golden ratio φ", "C": "√2 - 1", "D": "1/√2"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "hard", "question": "Sum of infinite GP 1 + 2r + 3r² + 4r³ + ... for |r|<1 = ? 📈", "options": {"A": "1/(1-r)", "B": "1/(1-r)²", "C": "r/(1-r)²", "D": "1/(1+r)²"}, "correct_answer": "B"},
    {"id": 46, "difficulty": "hard", "question": "If S∞ = 3, second term is a/4, find a. 📊", "options": {"A": "2", "B": "2.25", "C": "2.5", "D": "3"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "hard", "question": "Infinite GP with r positive, S∞ = 4, sum of first 2 terms = 3. Find r. 🧩", "options": {"A": "1/2", "B": "1/3", "C": "2/3", "D": "3/4"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "hard", "question": "If first term of infinite GP is 1 and every term equals sum of all later terms, r=? 🏆", "options": {"A": "1/3", "B": "1/2", "C": "2/3", "D": "1/4"}, "correct_answer": "B"},
    {"id": 49, "difficulty": "hard", "question": "S∞ for GP: 1/(1+√x), 1/(1-x), 1/(1-√x), ... (0<x<1) = ? 🌟", "options": {"A": "1/(1-x)", "B": "√x/(1-x)", "C": "1/(1-√x)", "D": "1/(1+√x)"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "Sum to infinity of sinθ + sin²θ + sin³θ + ... (0<θ<π) = ? 📐", "options": {"A": "sinθ/(1-sinθ)", "B": "cosθ/(1-sinθ)", "C": "tanθ", "D": "cosecθ - 1"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "If S∞ = 5, and product of all terms = 1, find a. (a>0) 🔥", "options": {"A": "2", "B": "2.5", "C": "3", "D": "1.5"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "hard", "question": "Infinite GP with r negative, S∞ = 5, first term = 6. Find r. ➖", "options": {"A": "-1/5", "B": "-1/4", "C": "-1/3", "D": "-1/2"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "S∞ = 12, sum of first 4 terms = 9.375. Find a. 🧮", "options": {"A": "6", "B": "7.5", "C": "8", "D": "9"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "Sum of infinite series 1 + (1+a)r + (1+a+a²)r² + ... (|r|<1) = ? 📚", "options": {"A": "1/[(1-r)(1-ar)]", "B": "1/(1-r)²", "C": "1/(1-ar)", "D": "1/[(1-r)(1-r²)]"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "Infinite GP with common ratio = 1/3. If sum of first 2 terms = 4, S∞ = ? 🚀", "options": {"A": "6", "B": "6.75", "C": "7", "D": "9"}, "correct_answer": "B"},
    {"id": 56, "difficulty": "hard", "question": "S∞ for GP where each term is half the sum of all later terms = ? ⚖️", "options": {"A": "2a", "B": "3a", "C": "4a", "D": "a/(1-r)"}, "correct_answer": "B"},
    {"id": 57, "difficulty": "hard", "question": "If S∞ = k times first term (k>1), r expressed in terms of k? 🔢", "options": {"A": "r = (k-1)/k", "B": "r = 1 - 1/k", "C": "r = 1/k", "D": "r = k/(k+1)"}, "correct_answer": "B"},
    {"id": 58, "difficulty": "hard", "question": "Sum of infinite GP is 9, sum of squares of terms is 27. Find a. 📉", "options": {"A": "3", "B": "4.5", "C": "6", "D": "7.5"}, "correct_answer": "C"},
    {"id": 59, "difficulty": "hard", "question": "If S∞ = m and sum of cubes of terms = n, find r in terms of m,n. 🔷", "options": {"A": "√[1 - (a³/n)]", "B": "∛[n/(m³)]", "C": "√[1 - m³/n]", "D": "∛[1 - m³/n]"}, "correct_answer": "C"},
    {"id": 60, "difficulty": "hard", "question": "Infinite GP with a=2, S∞ = 3. Find sum of infinite GP formed by taking every odd term (1st,3rd,5th,...). 🔥", "options": {"A": "4.5", "B": "5", "C": "6", "D": "9/2"}, "correct_answer": "A"}
  ]
}