{
  "title": "Algebraic Identities - Grade 9 CBSE",
  "total_questions": 60,
  "questions": [
    {
      "id": 1,
      "difficulty": "easy",
      "question": "What is (a + b)² equal to?",
      "options": {
        "A": "a² + b² + 2ab",
        "B": "a² + b² - 2ab",
        "C": "a² - b²",
        "D": "a² + b²"
      },
      "correct_answer": "A"
    },
    {
      "id": 2,
      "difficulty": "easy",
      "question": "Simplify: (x + 3)²",
      "options": {
        "A": "x² + 9",
        "B": "x² + 6x + 9",
        "C": "x² + 3x + 9",
        "D": "x² + 9x + 9"
      },
      "correct_answer": "B"
    },
    {
      "id": 3,
      "difficulty": "easy",
      "question": "What is (a - b)² equal to?",
      "options": {
        "A": "a² + b² - 2ab",
        "B": "a² + b² + 2ab",
        "C": "a² - b²",
        "D": "a² + b²"
      },
      "correct_answer": "A"
    },
    {
      "id": 4,
      "difficulty": "easy",
      "question": "Expand: (2x - 1)²",
      "options": {
        "A": "4x² - 4x + 1",
        "B": "4x² + 4x + 1",
        "C": "4x² - 1",
        "D": "4x² + 1"
      },
      "correct_answer": "A"
    },
    {
      "id": 5,
      "difficulty": "easy",
      "question": "What is a² - b² equal to?",
      "options": {
        "A": "(a + b)(a - b)",
        "B": "(a - b)²",
        "C": "(a + b)²",
        "D": "a² + b²"
      },
      "correct_answer": "C"
    },
    {
      "id": 6,
      "difficulty": "easy",
      "question": "Simplify: (x + 4)(x - 4)",
      "options": {
        "A": "x² - 16",
        "B": "x² + 16",
        "C": "x² - 8",
        "D": "x² + 8"
      },
      "correct_answer": "A"
    },
    {
      "id": 7,
      "difficulty": "easy",
      "question": "Expand: (x + 5)²",
      "options": {
        "A": "x² + 25",
        "B": "x² + 10x + 25",
        "C": "x² + 5x + 25",
        "D": "x² + 25x + 25"
      },
      "correct_answer": "B"
    },
    {
      "id": 8,
      "difficulty": "easy",
      "question": "What is (a + b)(a - b) equal to?",
      "options": {
        "A": "a² - b²",
        "B": "a² + b²",
        "C": "a² + 2ab + b²",
        "D": "a² - 2ab + b²"
      },
      "correct_answer": "D"
    },
    {
      "id": 9,
      "difficulty": "easy",
      "question": "Simplify: (3x + 2)²",
      "options": {
        "A": "9x² + 12x + 4",
        "B": "9x² + 6x + 4",
        "C": "9x² + 4",
        "D": "9x² + 12x + 2"
      },
      "correct_answer": "A"
    },
    {
      "id": 10,
      "difficulty": "easy",
      "question": "What is (x + y)² - (x - y)² equal to?",
      "options": {
        "A": "2xy",
        "B": "4xy",
        "C": "2x² + 2y²",
        "D": "x² + y²"
      },
      "correct_answer": "B"
    },
    {
      "id": 11,
      "difficulty": "easy",
      "question": "Expand: (2a - 3b)²",
      "options": {
        "A": "4a² - 12ab + 9b²",
        "B": "4a² + 12ab + 9b²",
        "C": "4a² - 9b²",
        "D": "4a² + 9b²"
      },
      "correct_answer": "A"
    },
    {
      "id": 12,
      "difficulty": "easy",
      "question": "Simplify: (p + 7)(p - 7)",
      "options": {
        "A": "p² - 49",
        "B": "p² + 49",
        "C": "p² - 14",
        "D": "p² + 14"
      },
      "correct_answer": "C"
    },
    {
      "id": 13,
      "difficulty": "easy",
      "question": "What is (a + b + c)² equal to?",
      "options": {
        "A": "a² + b² + c² + 2ab + 2bc + 2ca",
        "B": "a² + b² + c² + ab + bc + ca",
        "C": "a² + b² + c²",
        "D": "a² + b² + c² + 2abc"
      },
      "correct_answer": "A"
    },
    {
      "id": 14,
      "difficulty": "easy",
      "question": "Simplify: (x + 2)² - (x - 2)²",
      "options": {
        "A": "8x",
        "B": "4x",
        "C": "8",
        "D": "4"
      },
      "correct_answer": "A"
    },
    {
      "id": 15,
      "difficulty": "easy",
      "question": "Expand: (3x + 4y)²",
      "options": {
        "A": "9x² + 24xy + 16y²",
        "B": "9x² + 12xy + 16y²",
        "C": "9x² + 16y²",
        "D": "9x² + 24xy + 4y²"
      },
      "correct_answer": "B"
    },
    {
      "id": 16,
      "difficulty": "easy",
      "question": "What is (x + 1)³ equal to?",
      "options": {
        "A": "x³ + 3x² + 3x + 1",
        "B": "x³ + 1",
        "C": "x³ + 3x + 1",
        "D": "x³ + x² + x + 1"
      },
      "correct_answer": "A"
    },
    {
      "id": 17,
      "difficulty": "easy",
      "question": "Simplify: (2x - 3)(2x + 3)",
      "options": {
        "A": "4x² - 9",
        "B": "4x² + 9",
        "C": "4x² - 6",
        "D": "4x² + 6"
      },
      "correct_answer": "C"
    },
    {
      "id": 18,
      "difficulty": "easy",
      "question": "Expand: (a - 6)²",
      "options": {
        "A": "a² - 12a + 36",
        "B": "a² + 12a + 36",
        "C": "a² - 36",
        "D": "a² + 36"
      },
      "correct_answer": "A"
    },
    {
      "id": 19,
      "difficulty": "easy",
      "question": "What is (x + y)³ equal to?",
      "options": {
        "A": "x³ + y³ + 3xy(x + y)",
        "B": "x³ + y³",
        "C": "x³ + y³ + 3x²y + 3xy²",
        "D": "x³ + 3x²y + 3xy² + y³"
      },
      "correct_answer": "D"
    },
    {
      "id": 20,
      "difficulty": "easy",
      "question": "Simplify: (5m + 2n)(5m - 2n)",
      "options": {
        "A": "25m² - 4n²",
        "B": "25m² + 4n²",
        "C": "25m² - 10mn + 4n²",
        "D": "25m² + 10mn - 4n²"
      },
      "correct_answer": "B"
    },
    {
      "id": 21,
      "difficulty": "medium",
      "question": "Simplify: (2x + 3y)² + (2x - 3y)²",
      "options": {
        "A": "8x² + 18y²",
        "B": "4x² + 9y²",
        "C": "8x² + 9y²",
        "D": "4x² + 18y²"
      },
      "correct_answer": "A"
    },
    {
      "id": 22,
      "difficulty": "medium",
      "question": "If a + b = 7 and ab = 12, find a² + b²",
      "options": {
        "A": "25",
        "B": "37",
        "C": "49",
        "D": "13"
      },
      "correct_answer": "B"
    },
    {
      "id": 23,
      "difficulty": "medium",
      "question": "Simplify: (x + 2y + 3z)²",
      "options": {
        "A": "x² + 4y² + 9z² + 4xy + 12yz + 6xz",
        "B": "x² + 4y² + 9z² + 4xy + 6yz + 12xz",
        "C": "x² + 4y² + 9z² + 2xy + 6yz + 3xz",
        "D": "x² + 4y² + 9z²"
      },
      "correct_answer": "A"
    },
    {
      "id": 24,
      "difficulty": "medium",
      "question": "If x - y = 5 and xy = 24, find x² + y²",
      "options": {
        "A": "73",
        "B": "49",
        "C": "25",
        "D": "1"
      },
      "correct_answer": "A"
    },
    {
      "id": 25,
      "difficulty": "medium",
      "question": "Simplify: (3a - 4b)³",
      "options": {
        "A": "27a³ - 108a²b + 144ab² - 64b³",
        "B": "27a³ - 64b³",
        "C": "27a³ - 36a²b + 48ab² - 64b³",
        "D": "27a³ - 108ab + 64b³"
      },
      "correct_answer": "A"
    },
    {
      "id": 26,
      "difficulty": "medium",
      "question": "If x + 1/x = 5, find x² + 1/x²",
      "options": {
        "A": "23",
        "B": "25",
        "C": "27",
        "D": "29"
      },
      "correct_answer": "A"
    },
    {
      "id": 27,
      "difficulty": "medium",
      "question": "Simplify: (2x + 5y)² - (2x - 5y)²",
      "options": {
        "A": "40xy",
        "B": "20xy",
        "C": "4x² + 25y²",
        "D": "8x² + 50y²"
      },
      "correct_answer": "A"
    },
    {
      "id": 28,
      "difficulty": "medium",
      "question": "If a² + b² = 29 and ab = 10, find a + b",
      "options": {
        "A": "7",
        "B": "9",
        "C": "11",
        "D": "13"
      },
      "correct_answer": "A"
    },
    {
      "id": 29,
      "difficulty": "medium",
      "question": "Simplify: (x + y + z)(x + y - z)",
      "options": {
        "A": "x² + y² - z² + 2xy",
        "B": "x² + y² - z²",
        "C": "x² + y² + z²",
        "D": "x² - y² - z²"
      },
      "correct_answer": "A"
    },
    {
      "id": 30,
      "difficulty": "medium",
      "question": "If x² + y² = 34 and xy = 15, find x - y when x > y",
      "options": {
        "A": "2",
        "B": "4",
        "C": "6",
        "D": "8"
      },
      "correct_answer": "A"
    },
    {
      "id": 31,
      "difficulty": "medium",
      "question": "Simplify: (a + b + c)² + (a - b - c)²",
      "options": {
        "A": "2a² + 2b² + 2c² + 4bc",
        "B": "2a² + 2b² + 2c²",
        "C": "a² + b² + c²",
        "D": "4a² + 2b² + 2c²"
      },
      "correct_answer": "A"
    },
    {
      "id": 32,
      "difficulty": "medium",
      "question": "If x - 1/x = 3, find x³ - 1/x³",
      "options": {
        "A": "36",
        "B": "27",
        "C": "18",
        "D": "9"
      },
      "correct_answer": "A"
    },
    {
      "id": 33,
      "difficulty": "medium",
      "question": "Simplify: (2x + 3y + 4z)² - (2x - 3y - 4z)²",
      "options": {
        "A": "8x(3y + 4z)",
        "B": "24xy + 32xz",
        "C": "8x(6y + 8z)",
        "D": "48xy + 64xz"
      },
      "correct_answer": "C"
    },
    {
      "id": 34,
      "difficulty": "medium",
      "question": "If a + b + c = 0, find a²/bc + b²/ca + c²/ab",
      "options": {
        "A": "3",
        "B": "0",
        "C": "1",
        "D": "-3"
      },
      "correct_answer": "A"
    },
    {
      "id": 35,
      "difficulty": "medium",
      "question": "Simplify: (x + 2)³ - (x - 2)³",
      "options": {
        "A": "12x² + 16",
        "B": "12x² + 8",
        "C": "6x² + 8",
        "D": "2x³ + 24x"
      },
      "correct_answer": "A"
    },
    {
      "id": 36,
      "difficulty": "medium",
      "question": "If a + b + c = 15 and ab + bc + ca = 71, find a² + b² + c²",
      "options": {
        "A": "83",
        "B": "225",
        "C": "106",
        "D": "49"
      },
      "correct_answer": "A"
    },
    {
      "id": 37,
      "difficulty": "medium",
      "question": "Simplify: (p + q)³ - (p - q)³",
      "options": {
        "A": "2q(3p² + q²)",
        "B": "2p(3q² + p²)",
        "C": "6p²q + 2q³",
        "D": "6pq² + 2p³"
      },
      "correct_answer": "A"
    },
    {
      "id": 38,
      "difficulty": "medium",
      "question": "If x² + 1/x² = 7, find x³ + 1/x³",
      "options": {
        "A": "18",
        "B": "27",
        "C": "36",
        "D": "45"
      },
      "correct_answer": "A"
    },
    {
      "id": 39,
      "difficulty": "medium",
      "question": "Simplify: (a + 2b + 3c)² - (a - 2b - 3c)²",
      "options": {
        "A": "4a(2b + 3c)",
        "B": "8ab + 12ac",
        "C": "4a(4b + 6c)",
        "D": "16ab + 24ac"
      },
      "correct_answer": "C"
    },
    {
      "id": 40,
      "difficulty": "medium",
      "question": "If x + y + z = 0, prove that x³ + y³ + z³ = 3xyz. This is an example of which identity?",
      "options": {
        "A": "(x + y + z)³ identity",
        "B": "x³ + y³ + z³ - 3xyz identity",
        "C": "(x + y)³ identity",
        "D": "x³ - y³ identity"
      },
      "correct_answer": "B"
    },
    {
      "id": 41,
      "difficulty": "hard",
      "question": "If a + b + c = 6, a² + b² + c² = 14, and ab + bc + ca = 11, find a³ + b³ + c³ - 3abc",
      "options": {
        "A": "18",
        "B": "24",
        "C": "36",
        "D": "54"
      },
      "correct_answer": "A"
    },
    {
      "id": 42,
      "difficulty": "hard",
      "question": "Simplify: (x + y + z)³ - (x - y - z)³",
      "options": {
        "A": "2y³ + 2z³ + 6x²(y + z) + 6x(y² + z²)",
        "B": "2(y + z)(3x² + y² + z² + 3x(y + z))",
        "C": "2(y³ + z³) + 6x²(y + z)",
        "D": "2(y + z)(x² + y² + z² + xy + xz)"
      },
      "correct_answer": "B"
    },
    {
      "id": 43,
      "difficulty": "hard",
      "question": "If x = √3 + √2 and y = √3 - √2, find x⁴ + y⁴",
      "options": {
        "A": "98",
        "B": "100",
        "C": "102",
        "D": "104"
      },
      "correct_answer": "A"
    },
    {
      "id": 44,
      "difficulty": "hard",
      "question": "Simplify: (a + b + c)³ - a³ - b³ - c³",
      "options": {
        "A": "3(a + b)(b + c)(c + a)",
        "B": "3(a + b + c)(ab + bc + ca)",
        "C": "3abc",
        "D": "3(a²b + b²c + c²a)"
      },
      "correct_answer": "B"
    },
    {
      "id": 45,
      "difficulty": "hard",
      "question": "If x + y + z = 0, prove that (x² + y² + z²)² = 2(x⁴ + y⁴ + z⁴). Which identity helps prove this?",
      "options": {
        "A": "(x + y + z)² identity",
        "B": "x³ + y³ + z³ identity",
        "C": "(x² + y² + z²)² expansion",
        "D": "x⁴ + y⁴ + z⁴ identity"
      },
      "correct_answer": "C"
    },
    {
      "id": 46,
      "difficulty": "hard",
      "question": "If a² + b² + c² = ab + bc + ca, prove that a = b = c. This uses which algebraic manipulation?",
      "options": {
        "A": "Multiplying by 2 and rearranging",
        "B": "Using difference of squares",
        "C": "Using cube identity",
        "D": "Direct substitution"
      },
      "correct_answer": "A"
    },
    {
      "id": 47,
      "difficulty": "hard",
      "question": "Simplify: (x + y)⁵ - (x - y)⁵",
      "options": {
        "A": "2y(5x⁴ + 10x²y² + y⁴)",
        "B": "2x(5y⁴ + 10x²y² + x⁴)",
        "C": "10x⁴y + 5x²y³ + y⁵",
        "D": "10xy⁴ + 5x³y² + x⁵"
      },
      "correct_answer": "A"
    },
    {
      "id": 48,
      "difficulty": "hard",
      "question": "If x + 1/x = √5, find x³ + 1/x³",
      "options": {
        "A": "2√5",
        "B": "3√5",
        "C": "4√5",
        "D": "5√5"
      },
      "correct_answer": "A"
    },
    {
      "id": 49,
      "difficulty": "hard",
      "question": "Simplify: (a + b + c)⁴ - (a - b - c)⁴",
      "options": {
        "A": "8a(b + c)(a² + b² + c² + ab + ac + bc)",
        "B": "8bc(a + b + c)(a² + b² + c²)",
        "C": "4a(b + c)(2a² + b² + c²)",
        "D": "4(b + c)(a³ + b³ + c³)"
      },
      "correct_answer": "A"
    },
    {
      "id": 50,
      "difficulty": "hard",
      "question": "If a³ + b³ + c³ = 3abc and a + b + c ≠ 0, then which relation holds?",
      "options": {
        "A": "a = b = c",
        "B": "a + b + c = 0",
        "C": "a² + b² + c² = ab + bc + ca",
        "D": "Both A and C"
      },
      "correct_answer": "D"
    },
    {
      "id": 51,
      "difficulty": "hard",
      "question": "Simplify: (x² + y² + z²)² - (x² - y² - z²)²",
      "options": {
        "A": "4x²(y² + z²)",
        "B": "4x²y² + 4x²z²",
        "C": "2x²(y² + z²)",
        "D": "x⁴ - y⁴ - z⁴"
      },
      "correct_answer": "A"
    },
    {
      "id": 52,
      "difficulty": "hard",
      "question": "If x + y + z = 9 and xy + yz + zx = 23, find x³ + y³ + z³ - 3xyz",
      "options": {
        "A": "108",
        "B": "144",
        "C": "216",
        "D": "324"
      },
      "correct_answer": "A"
    },
    {
      "id": 53,
      "difficulty": "hard",
      "question": "Simplify: (a + b)⁶ - (a - b)⁶",
      "options": {
        "A": "4ab(5a⁴ + 10a²b² + 5b⁴)",
        "B": "2b(15a⁴ + 15a²b² + b⁴)",
        "C": "4ab(3a⁴ + 10a²b² + 3b⁴)",
        "D": "2b(3a⁴ + 10a²b² + 3b⁴)"
      },
      "correct_answer": "C"
    },
    {
      "id": 54,
      "difficulty": "hard",
      "question": "If a + b + c = 0, find (b + c)²/bc + (c + a)²/ca + (a + b)²/ab",
      "options": {
        "A": "3",
        "B": "6",
        "C": "9",
        "D": "12"
      },
      "correct_answer": "C"
    },
    {
      "id": 55,
      "difficulty": "hard",
      "question": "Simplify: (x + y + z)² - (x - y - z)² + (y + z - x)² - (z + x - y)²",
      "options": {
        "A": "8xy + 8xz",
        "B": "4xy + 4xz",
        "C": "8xyz",
        "D": "4(xy + xz + yz)"
      },
      "correct_answer": "A"
    },
    {
      "id": 56,
      "difficulty": "hard",
      "question": "If a² + b² + c² = 2(a - b - c) - 3, find a + b + c",
      "options": {
        "A": "1",
        "B": "0",
        "C": "-1",
        "D": "2"
      },
      "correct_answer": "B"
    },
    {
      "id": 57,
      "difficulty": "hard",
      "question": "Simplify: (a + b + c)³ + (a - b - c)³",
      "options": {
        "A": "2a³ + 6a(b² + c²)",
        "B": "2a(a² + 3b² + 3c²)",
        "C": "2a³ + 2b³ + 2c³ + 6abc",
        "D": "2a(a² + 3b² + 3c² + 6bc)"
      },
      "correct_answer": "B"
    },
    {
      "id": 58,
      "difficulty": "hard",
      "question": "If x = 2 + √3, find x³ + 1/x³",
      "options": {
        "A": "52",
        "B": "56",
        "C": "60",
        "D": "64"
      },
      "correct_answer": "A"
    },
    {
      "id": 59,
      "difficulty": "hard",
      "question": "Simplify: (p + q + r)⁴ - (p - q - r)⁴",
      "options": {
        "A": "8p(q + r)(p² + q² + r² + pq + pr + qr)",
        "B": "8pqr(p + q + r)",
        "C": "4p(q + r)(2p² + q² + r²)",
        "D": "4(p + q + r)(p³ + q³ + r³)"
      },
      "correct_answer": "A"
    },
    {
      "id": 60,
      "difficulty": "hard",
      "question": "If a + b + c = 0, find a²/bc + b²/ca + c²/ab + 3",
      "options": {
        "A": "0",
        "B": "3",
        "C": "6",
        "D": "9"
      },
      "correct_answer": "C"
    }
  ]
}