{
  "title": "Algebraic Identities - Grade 8 ICSE",
  "total_questions": 60,
  "questions": [
    {
      "id": 1,
      "difficulty": "easy",
      "question": "Using identity, find: (x+3)² 📐",
      "options": {
        "A": "x² + 6x + 9",
        "B": "x² + 3x + 9",
        "C": "x² + 9",
        "D": "x² + 6x + 6"
      },
      "correct_answer": "A"
    },
    {
      "id": 2,
      "difficulty": "easy",
      "question": "What is (y - 5)²? ✨",
      "options": {
        "A": "y² - 10y + 25",
        "B": "y² - 5y + 25",
        "C": "y² - 25",
        "D": "y² + 10y + 25"
      },
      "correct_answer": "A"
    },
    {
      "id": 3,
      "difficulty": "easy",
      "question": "Expand using identity: (2 + a)² 🤔",
      "options": {
        "A": "4 + 4a + a²",
        "B": "4 + a²",
        "C": "4 + 2a + a²",
        "D": "2 + 4a + a²"
      },
      "correct_answer": "A"
    },
    {
      "id": 4,
      "difficulty": "easy",
      "question": "Find (p + 4)(p - 4) using identity. 🔍",
      "options": {
        "A": "p² - 16",
        "B": "p² - 8",
        "C": "p² - 4",
        "D": "p² + 16"
      },
      "correct_answer": "A"
    },
    {
      "id": 5,
      "difficulty": "easy",
      "question": "Simplify: (7 + m)(7 - m) 🧩",
      "options": {
        "A": "49 - m²",
        "B": "49 + m²",
        "C": "49 - 14m",
        "D": "7 - m²"
      },
      "correct_answer": "A"
    },
    {
      "id": 6,
      "difficulty": "easy",
      "question": "What is the expanded form of (3x + 1)²? 📏",
      "options": {
        "A": "9x² + 6x + 1",
        "B": "9x² + 3x + 1",
        "C": "9x² + 1",
        "D": "3x² + 6x + 1"
      },
      "correct_answer": "A"
    },
    {
      "id": 7,
      "difficulty": "easy",
      "question": "Which identity is used for (a + b)² = a² + 2ab + b²? 🏷️",
      "options": {
        "A": "Square of a sum",
        "B": "Square of a difference",
        "C": "Difference of squares",
        "D": "None of these"
      },
      "correct_answer": "A"
    },
    {
      "id": 8,
      "difficulty": "easy",
      "question": "Find: (0.5 + n)² 🎯",
      "options": {
        "A": "0.25 + n + n²",
        "B": "0.25 + n²",
        "C": "0.5 + n + n²",
        "D": "0.25 + 0.5n + n²"
      },
      "correct_answer": "A"
    },
    {
      "id": 9,
      "difficulty": "easy",
      "question": "What is (10 - c)²? 🧠",
      "options": {
        "A": "100 - 20c + c²",
        "B": "100 - c²",
        "C": "10 - 20c + c²",
        "D": "100 - 10c + c²"
      },
      "correct_answer": "A"
    },
    {
      "id": 10,
      "difficulty": "easy",
      "question": "Using identity, find (1/2 + y)² 🍰",
      "options": {
        "A": "0.25 + y + y²",
        "B": "0.5 + y + y²",
        "C": "0.25 + 0.5y + y²",
        "D": "0.25 + y²"
      },
      "correct_answer": "A"
    },
    {
      "id": 11,
      "difficulty": "easy",
      "question": "Expand: (x + 1/x)² (Assume x ≠ 0) ⚖️",
      "options": {
        "A": "x² + 2 + 1/x²",
        "B": "x² + 1/x²",
        "C": "x² + 1 + 1/x²",
        "D": "x + 2 + 1/x"
      },
      "correct_answer": "A"
    },
    {
      "id": 12,
      "difficulty": "easy",
      "question": "Find (2k - 3)². 🪀",
      "options": {
        "A": "4k² - 12k + 9",
        "B": "4k² - 6k + 9",
        "C": "2k² - 12k + 9",
        "D": "4k² - 9"
      },
      "correct_answer": "A"
    },
    {
      "id": 13,
      "difficulty": "easy",
      "question": "What is the result of (a + b)(a - b)? 🔄",
      "options": {
        "A": "a² - b²",
        "B": "a² + b²",
        "C": "a² - 2ab + b²",
        "D": "a² + 2ab + b²"
      },
      "correct_answer": "A"
    },
    {
      "id": 14,
      "difficulty": "easy",
      "question": "Simplify: (5p)² - (4q)² using identity. 🧮",
      "options": {
        "A": "25p² - 16q²",
        "B": "(5p - 4q)²",
        "C": "25p² + 16q²",
        "D": "(5p + 4q)²"
      },
      "correct_answer": "A"
    },
    {
      "id": 15,
      "difficulty": "easy",
      "question": "Find (0.1 + 0.2)² using identity. ➕",
      "options": { "A": "0.09", "B": "0.03", "C": "0.3", "D": "0.9" },
      "correct_answer": "A"
    },
    {
      "id": 16,
      "difficulty": "easy",
      "question": "Expand: (√2 + 1)² 📐",
      "options": {
        "A": "3 + 2√2",
        "B": "2 + 2√2",
        "C": "3 + √2",
        "D": "2 + √2"
      },
      "correct_answer": "A"
    },
    {
      "id": 17,
      "difficulty": "easy",
      "question": "What is 103² using identity (100+3)²? 🚀",
      "options": { "A": "10609", "B": "10009", "C": "10603", "D": "11209" },
      "correct_answer": "A"
    },
    {
      "id": 18,
      "difficulty": "easy",
      "question": "Find (x + 7)(x + 7). It's a... 🔂",
      "options": {
        "A": "x² + 14x + 49",
        "B": "x² + 7x + 49",
        "C": "x² + 49",
        "D": "2x + 14"
      },
      "correct_answer": "A"
    },
    {
      "id": 19,
      "difficulty": "easy",
      "question": "Which is the square of (t - 1)? 🎪",
      "options": {
        "A": "t² - 2t + 1",
        "B": "t² - t + 1",
        "C": "t² - 1",
        "D": "t² + 2t + 1"
      },
      "correct_answer": "A"
    },
    {
      "id": 20,
      "difficulty": "easy",
      "question": "If (a+b)² = a² + 2ab + b², what is (m+n)²? ✅",
      "options": {
        "A": "m² + 2mn + n²",
        "B": "m² + mn + n²",
        "C": "m² + n²",
        "D": "2m + 2n"
      },
      "correct_answer": "A"
    },
    {
      "id": 21,
      "difficulty": "medium",
      "question": "Find 98² using a suitable identity. 🧠",
      "options": { "A": "9604", "B": "9608", "C": "9804", "D": "10004" },
      "correct_answer": "A"
    },
    {
      "id": 22,
      "difficulty": "medium",
      "question": "Simplify: (2x + 3y)² - (2x - 3y)². What's the result? ⚡",
      "options": { "A": "24xy", "B": "12xy", "C": "8x² + 18y²", "D": "0" },
      "correct_answer": "A"
    },
    {
      "id": 23,
      "difficulty": "medium",
      "question": "If x + 1/x = 5, find x² + 1/x². 🧩",
      "options": { "A": "23", "B": "25", "C": "5", "D": "27" },
      "correct_answer": "A"
    },
    {
      "id": 24,
      "difficulty": "medium",
      "question": "Expand: (3a + 4b - 5c)² 🚧",
      "options": {
        "A": "9a²+16b²+25c²+24ab-30ac-40bc",
        "B": "9a²+16b²+25c²+24ab-40ac-30bc",
        "C": "9a²+16b²+25c²+12ab-30ac-40bc",
        "D": "3a²+4b²+5c²+24ab-30ac-40bc"
      },
      "correct_answer": "A"
    },
    {
      "id": 25,
      "difficulty": "medium",
      "question": "Find product using identity: 10.5 × 9.5 🛒",
      "options": { "A": "99.75", "B": "100.25", "C": "99.5", "D": "100.0" },
      "correct_answer": "A"
    },
    {
      "id": 26,
      "difficulty": "medium",
      "question": "Simplify: (√5 + √3)(√5 - √3) 🌱",
      "options": {
        "A": "2",
        "B": "8",
        "C": "√15",
        "D": "√2"
      },
      "correct_answer": "A"
    },
    {
      "id": 27,
      "difficulty": "medium",
      "question": "If a² + b² = 29 and ab = 10, find (a + b)². 🔢",
      "options": {
        "A": "49",
        "B": "9",
        "C": "29",
        "D": "39"
      },
      "correct_answer": "A"
    },
    {
      "id": 28,
      "difficulty": "medium",
      "question": "Find value of 501 × 499 using identity. 🧮",
      "options": { "A": "249999", "B": "250001", "C": "249001", "D": "250999" },
      "correct_answer": "A"
    },
    {
      "id": 29,
      "difficulty": "medium",
      "question": "Simplify: (x + y)² + (x - y)² ➕",
      "options": {
        "A": "2(x² + y²)",
        "B": "2(x² - y²)",
        "C": "4xy",
        "D": "2x² + y²"
      },
      "correct_answer": "A"
    },
    {
      "id": 30,
      "difficulty": "medium",
      "question": "If p - q = 7 and pq = 18, find p² + q². ⚖️",
      "options": { "A": "85", "B": "67", "C": "49", "D": "36" },
      "correct_answer": "A"
    },
    {
      "id": 31,
      "difficulty": "medium",
      "question": "Find (2.5)² - (1.5)² using identity. 🎯",
      "options": { "A": "4", "B": "6.25", "C": "2.25", "D": "1" },
      "correct_answer": "A"
    },
    {
      "id": 32,
      "difficulty": "medium",
      "question": "Expand: (a/2 + 2/a)² (a ≠ 0) 📦",
      "options": {
        "A": "a²/4 + 2 + 4/a²",
        "B": "a²/2 + 2 + 4/a²",
        "C": "a²/4 + 4/a²",
        "D": "a/2 + 2 + 2/a"
      },
      "correct_answer": "A"
    },
    {
      "id": 33,
      "difficulty": "medium",
      "question": "Simplify: (3m + 4n)² + (3m - 4n)². It equals... 🧲",
      "options": {
        "A": "18m² + 32n²",
        "B": "9m² + 16n²",
        "C": "48mn",
        "D": "6m² + 8n²"
      },
      "correct_answer": "A"
    },
    {
      "id": 34,
      "difficulty": "medium",
      "question": "Find (999)² using identity. 🚀",
      "options": {
        "A": "998001",
        "B": "999999",
        "C": "1000001",
        "D": "998999"
      },
      "correct_answer": "A"
    },
    {
      "id": 35,
      "difficulty": "medium",
      "question": "If (x + y) = 12 and xy = 27, find x² + y². 🌟",
      "options": { "A": "90", "B": "144", "C": "118", "D": "54" },
      "correct_answer": "A"
    },
    {
      "id": 36,
      "difficulty": "medium",
      "question": "Simplify: (a + b + c)² + (a - b - c)². Long one! 🐍",
      "options": {
        "A": "2a² + 2b² + 2c² + 4bc",
        "B": "2a² + 2b² + 2c²",
        "C": "4a²",
        "D": "2(a² + b² + c²)"
      },
      "correct_answer": "A"
    },
    {
      "id": 37,
      "difficulty": "medium",
      "question": "Find (√7 + 2)² - (√7 - 2)². Quick! ⚡",
      "options": { "A": "8√7", "B": "4√7", "C": "8", "D": "4" },
      "correct_answer": "A"
    },
    {
      "id": 38,
      "difficulty": "medium",
      "question": "Using (a+b)², find (205)². 💡",
      "options": { "A": "42025", "B": "42005", "C": "41025", "D": "40025" },
      "correct_answer": "A"
    },
    {
      "id": 39,
      "difficulty": "medium",
      "question": "If a + b = 9 and a² + b² = 41, find ab. 🔍",
      "options": { "A": "20", "B": "40", "C": "81", "D": "√20" },
      "correct_answer": "A"
    },
    {
      "id": 40,
      "difficulty": "medium",
      "question": "Simplify: (p + 2q)² - (p - 2q)². The answer is? 🎲",
      "options": { "A": "8pq", "B": "4pq", "C": "4p²", "D": "8q²" },
      "correct_answer": "A"
    },
    {
      "id": 41,
      "difficulty": "hard",
      "question": "If x² + y² = 74 and xy = 35, find the value of |(x+y)(x-y)|. 🎯",
      "options": {
        "A": "24",
        "B": "12",
        "C": "74",
        "D": "35"
      },
      "correct_answer": "A"
    },
    {
      "id": 42,
      "difficulty": "hard",
      "question": "Simplify: (a + b + c)² - (a - b - c)². Find the final expression. 🌪️",
      "options": {
        "A": "4a(b + c)",
        "B": "4ab + 4ac",
        "C": "2a² + 4bc",
        "D": "4bc"
      },
      "correct_answer": "A"
    },
    {
      "id": 43,
      "difficulty": "hard",
      "question": "If a + b + c = 15 and a² + b² + c² = 83, find ab + bc + ca. 🔥",
      "options": { "A": "71", "B": "225", "C": "83", "D": "142" },
      "correct_answer": "A"
    },
    {
      "id": 44,
      "difficulty": "hard",
      "question": "Prove using identities: (a - b)² = (a + b)² - 4ab. If a-b=3 and a+b=7, find ab. 🧠",
      "options": {
        "A": "10",
        "B": "9",
        "C": "40",
        "D": "21/4"
      },
      "correct_answer": "A"
    },
    {
      "id": 45,
      "difficulty": "hard",
      "question": "Find the square of 1001 using identity. Also find (1001+999)(1001-999). What's the product? ⚡",
      "options": { "A": "4000", "B": "1002001", "C": "2000", "D": "998001" },
      "correct_answer": "A"
    },
    {
      "id": 46,
      "difficulty": "hard",
      "question": "Simplify: (2x + 3y + 4z)² - (2x - 3y - 4z)². Find the coefficient of 'yz' in the result. 🏗️",
      "options": {
        "A": "48x",
        "B": "24x",
        "C": "0",
        "D": "96z"
      },
      "correct_answer": "C"
    },
    {
      "id": 47,
      "difficulty": "hard",
      "question": "If x + 1/x = √13, find x² + 1/x² and x⁴ + 1/x⁴. Give x⁴ + 1/x⁴. 🚀",
      "options": {
        "A": "119",
        "B": "121",
        "C": "11",
        "D": "13"
      },
      "correct_answer": "A"
    },
    {
      "id": 48,
      "difficulty": "hard",
      "question": "Using (a+b+c)², if a²+b²+c² = 50 and ab+bc+ca=47, find |a+b+c|. 🌟",
      "options": {
        "A": "12",
        "B": "-12",
        "C": "97",
        "D": "144"
      },
      "correct_answer": "A"
    },
    {
      "id": 49,
      "difficulty": "hard",
      "question": "Simplify: [(a+b)² - (a-b)²] / (4ab). What's the result? ➗",
      "options": { "A": "1", "B": "0", "C": "2", "D": "4" },
      "correct_answer": "A"
    },
    {
      "id": 50,
      "difficulty": "hard",
      "question": "Find 997 × 1003 using two different identities and verify. Give the product. ✅",
      "options": {
        "A": "999991",
        "B": "1000009",
        "C": "999999",
        "D": "1000001"
      },
      "correct_answer": "A"
    },
    {
      "id": 51,
      "difficulty": "hard",
      "question": "If (p+q)² = 100 and pq = 21, find p² + q² and (p-q)². Give (p-q)². 🧩",
      "options": { "A": "16", "B": "100", "C": "58", "D": "42" },
      "correct_answer": "A"
    },
    {
      "id": 52,
      "difficulty": "hard",
      "question": "Expand: (x² + y² + z²)² - (x² - y² - z²)². Find the simplified form. 🌌",
      "options": {
        "A": "4x²y² - 4x²z²",
        "B": "4x²(y²+z²)",
        "C": "2x²(y²+z²)",
        "D": "4x⁴"
      },
      "correct_answer": "B"
    },
    {
      "id": 53,
      "difficulty": "hard",
      "question": "If a + b = 5 and a³ + b³ = 35, find a² + b². Hint: Use (a+b)³. 🎲",
      "options": {
        "A": "11",
        "B": "13",
        "C": "25",
        "D": "10"
      },
      "correct_answer": "B"
    },
    {
      "id": 54,
      "difficulty": "hard",
      "question": "Simplify: (a + 2b + 3c)² + (a - 2b - 3c)². Long expression! 🐉",
      "options": {
        "A": "2a² + 8b² + 18c² + 24bc",
        "B": "2a² + 4b² + 9c²",
        "C": "2(a²+4b²+9c²)",
        "D": "4a²"
      },
      "correct_answer": "A"
    },
    {
      "id": 55,
      "difficulty": "hard",
      "question": "Find 1234² - 1233² using identity. What is the result? 🚅",
      "options": { "A": "2467", "B": "1", "C": "1234", "D": "2466" },
      "correct_answer": "A"
    },
    {
      "id": 56,
      "difficulty": "hard",
      "question": "If (m + 1/m) = 4, find m² + 1/m² and m⁴ + 1/m⁴. Give m⁴ + 1/m⁴. 📈",
      "options": { "A": "194", "B": "16", "C": "14", "D": "196" },
      "correct_answer": "A"
    },
    {
      "id": 57,
      "difficulty": "hard",
      "question": "Simplify: [(x+y)² - x² - y²] / xy. Quick division! ➗",
      "options": { "A": "2", "B": "1", "C": "0", "D": "(x+y)/xy" },
      "correct_answer": "A"
    },
    {
      "id": 58,
      "difficulty": "hard",
      "question": "Prove (a+b)² - (a-b)² = 4ab. If this equals 80 and a²+b²=50, find |a+b|. 🔑",
      "options": {
        "A": "√130",
        "B": "10",
        "C": "√90",
        "D": "13"
      },
      "correct_answer": "C"
    },
    {
      "id": 59,
      "difficulty": "hard",
      "question": "Find (0.99)² using (1 - 0.01)². Also find 1 - (0.99)². Give the second result. 🎯",
      "options": { "A": "0.0199", "B": "0.9801", "C": "0.99", "D": "0.01" },
      "correct_answer": "A"
    },
    {
      "id": 60,
      "difficulty": "hard",
      "question": "If √a + √b = 5 and √ab = 6, find a + b. Use (√a+√b)² identity. 🌳",
      "options": { "A": "13", "B": "25", "C": "37", "D": "1" },
      "correct_answer": "A"
    }
  ]
}