{
  "title": "Zero of a Polynomial - Grade 9 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Find zero of polynomial p(x) = x - 3", "options": {"A": "3", "B": "-3", "C": "0", "D": "1"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "Find zero of polynomial p(x) = x + 2", "options": {"A": "-2", "B": "2", "C": "0", "D": "-1"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "Find zero of polynomial p(x) = 2x - 4", "options": {"A": "2", "B": "-2", "C": "4", "D": "-4"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "Find zero of polynomial p(x) = 3x + 6", "options": {"A": "-2", "B": "2", "C": "3", "D": "-3"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "Find zero of polynomial p(x) = x - 5", "options": {"A": "5", "B": "-5", "C": "0", "D": "1"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "Find zero of polynomial p(x) = x + 7", "options": {"A": "-7", "B": "7", "C": "0", "D": "-1"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "Find zero of polynomial p(x) = 4x - 8", "options": {"A": "2", "B": "-2", "C": "4", "D": "-4"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "Find zero of polynomial p(x) = 5x + 10", "options": {"A": "-2", "B": "2", "C": "5", "D": "-5"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "Find zero of polynomial p(x) = x", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "Find zero of polynomial p(x) = 2x", "options": {"A": "0", "B": "2", "C": "-2", "D": "1"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "Find zero of p(x) = x² - 4", "options": {"A": "2, -2", "B": "4, -4", "C": "0", "D": "1, -1"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "Find zero of p(x) = x² - 9", "options": {"A": "3, -3", "B": "9, -9", "C": "0", "D": "1, -1"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "Find zero of p(x) = x² - 16", "options": {"A": "4, -4", "B": "16, -16", "C": "0", "D": "2, -2"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Find zero of p(x) = x² - 25", "options": {"A": "5, -5", "B": "25, -25", "C": "0", "D": "1, -1"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "Find zero of p(x) = x² - 1", "options": {"A": "1, -1", "B": "0", "C": "2, -2", "D": "0.5, -0.5"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "Find zero of p(x) = x² + 5x + 6", "options": {"A": "-2, -3", "B": "2, 3", "C": "-1, -6", "D": "1, 6"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Find zero of p(x) = x² - 5x + 6", "options": {"A": "2, 3", "B": "-2, -3", "C": "1, 6", "D": "-1, -6"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "Find zero of p(x) = x² + x - 6", "options": {"A": "2, -3", "B": "-2, 3", "C": "1, -6", "D": "-1, 6"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "Find zero of p(x) = x² - x - 6", "options": {"A": "3, -2", "B": "-3, 2", "C": "6, -1", "D": "-6, 1"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "Find zero of p(x) = x² - 3x + 2", "options": {"A": "1, 2", "B": "-1, -2", "C": "3, 2", "D": "-3, -2"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "Find zero of p(x) = 2x² - 3x - 2", "options": {"A": "2, -1/2", "B": "-2, 1/2", "C": "1, -2", "D": "-1, 2"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Find zero of p(x) = 3x² - 5x + 2", "options": {"A": "1, 2/3", "B": "-1, -2/3", "C": "1, 3/2", "D": "-1, -3/2"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Find zero of p(x) = 4x² - 4x + 1", "options": {"A": "1/2", "B": "-1/2", "C": "1/4", "D": "-1/4"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "Find zero of p(x) = 9x² - 6x + 1", "options": {"A": "1/3", "B": "-1/3", "C": "1/9", "D": "-1/9"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Find zero of p(x) = x³ - 1", "options": {"A": "1", "B": "-1", "C": "0", "D": "2"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Find zero of p(x) = x³ + 1", "options": {"A": "-1", "B": "1", "C": "0", "D": "2"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Find zero of p(x) = x³ - 8", "options": {"A": "2", "B": "-2", "C": "0", "D": "4"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "Find zero of p(x) = x³ + 8", "options": {"A": "-2", "B": "2", "C": "0", "D": "-4"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "Find zero of p(x) = x³ - 27", "options": {"A": "3", "B": "-3", "C": "0", "D": "9"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "Find zero of p(x) = x³ + 27", "options": {"A": "-3", "B": "3", "C": "0", "D": "-9"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Find zero of p(x) = (x-1)(x-2)", "options": {"A": "1, 2", "B": "-1, -2", "C": "0", "D": "1, -2"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "Find zero of p(x) = (x+1)(x+2)", "options": {"A": "-1, -2", "B": "1, 2", "C": "0", "D": "-1, 2"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "Find zero of p(x) = (x-3)(x+4)", "options": {"A": "3, -4", "B": "-3, 4", "C": "0", "D": "3, 4"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Find zero of p(x) = (x+5)(x-6)", "options": {"A": "-5, 6", "B": "5, -6", "C": "0", "D": "-5, -6"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "Find zero of p(x) = x(x-1)(x-2)", "options": {"A": "0, 1, 2", "B": "0, -1, -2", "C": "1, 2", "D": "0"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "Find zero of p(x) = (x-1)²", "options": {"A": "1", "B": "-1", "C": "0", "D": "1, -1"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "Find zero of p(x) = (x+2)²", "options": {"A": "-2", "B": "2", "C": "0", "D": "2, -2"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Find zero of p(x) = x² + 4x + 4", "options": {"A": "-2", "B": "2", "C": "0", "D": "2, -2"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Find zero of p(x) = x² - 6x + 9", "options": {"A": "3", "B": "-3", "C": "0", "D": "3, -3"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "Find zero of p(x) = 4x² - 12x + 9", "options": {"A": "3/2", "B": "-3/2", "C": "3", "D": "-3"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "medium", "question": "If 2 is zero of p(x) = x² - 3x + k, find k", "options": {"A": "2", "B": "-2", "C": "4", "D": "-4"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "medium", "question": "If -3 is zero of p(x) = x² + 2x + k, find k", "options": {"A": "3", "B": "-3", "C": "9", "D": "-9"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "medium", "question": "If 1 is zero of p(x) = 2x² - 3x + k, find k", "options": {"A": "1", "B": "-1", "C": "2", "D": "-2"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "medium", "question": "If -2 is zero of p(x) = 3x² + 5x + k, find k", "options": {"A": "2", "B": "-2", "C": "4", "D": "-4"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "If 3 is zero of p(x) = x² - kx + 6, find k", "options": {"A": "5", "B": "-5", "C": "3", "D": "-3"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "medium", "question": "Find all zeroes of p(x) = x³ - 3x² + 2x", "options": {"A": "0, 1, 2", "B": "0, -1, -2", "C": "1, 2", "D": "0, 2"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "Find all zeroes of p(x) = x³ - 6x² + 11x - 6", "options": {"A": "1, 2, 3", "B": "-1, -2, -3", "C": "1, 2", "D": "0, 1, 2"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "medium", "question": "Find all zeroes of p(x) = x³ + 6x² + 11x + 6", "options": {"A": "-1, -2, -3", "B": "1, 2, 3", "C": "-1, -2", "D": "0, -1, -2"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "medium", "question": "Find zeroes of p(x) = x² - 2x", "options": {"A": "0, 2", "B": "0, -2", "C": "2", "D": "0"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "If α and β are zeroes of x² - 5x + 6, find α² + β²", "options": {"A": "13", "B": "25", "C": "19", "D": "7"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "hard", "question": "If α and β are zeroes of x² - 3x + 2, find α³ + β³", "options": {"A": "9", "B": "27", "C": "8", "D": "1"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "If α and β are zeroes of 2x² - 4x + 1, find 1/α + 1/β", "options": {"A": "4", "B": "2", "C": "1", "D": "1/2"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "If α and β are zeroes of 3x² - 5x + 2, find α/β + β/α", "options": {"A": "13/6", "B": "5/3", "C": "25/9", "D": "13/4"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "If zeroes of x² - px + q are in ratio 2:3, find p²:q", "options": {"A": "25:6", "B": "5:6", "C": "25:36", "D": "5:1"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "hard", "question": "If α, β are zeroes of x² - 6x + k and α² + β² = 20, find k", "options": {"A": "8", "B": "10", "C": "12", "D": "16"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "If α, β are zeroes of x² - 8x + 15 and γ = α + β, δ = αβ, find polynomial with zeroes γ, δ", "options": {"A": "x² - 23x + 120", "B": "x² - 8x + 15", "C": "x² - 15x + 56", "D": "x² - 23x + 15"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "If α, β are zeroes of x² + 4x + 3, find polynomial with zeroes 2α, 2β", "options": {"A": "x² + 8x + 12", "B": "x² + 4x + 12", "C": "x² + 8x + 3", "D": "x² + 2x + 12"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "hard", "question": "If α, β are zeroes of x² - 5x + 6, find polynomial with zeroes α², β²", "options": {"A": "x² - 13x + 36", "B": "x² - 25x + 36", "C": "x² - 13x + 12", "D": "x² - 5x + 12"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "If α, β are zeroes of x² + 2x - 1, find α⁴ + β⁴", "options": {"A": "34", "B": "14", "C": "18", "D": "2"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "hard", "question": "Find zeroes of x³ - 3x² - x + 3 if one zero is 3", "options": {"A": "3, 1, -1", "B": "3, -1, -1", "C": "3, 1, 1", "D": "3, 0, -1"}, "correct_answer": "A"},
    {"id": 62, "difficulty": "hard", "question": "Find zeroes of x³ - 4x² + x + 6 if one zero is -1", "options": {"A": "-1, 2, 3", "B": "-1, -2, -3", "C": "-1, 1, 6", "D": "-1, 2, -3"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "hard", "question": "Find zeroes of 2x³ - 3x² - 3x + 2 if one zero is 2", "options": {"A": "2, 1/2, -1", "B": "2, -1/2, 1", "C": "2, 1, -1", "D": "2, -1, -1/2"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "hard", "question": "Find zeroes of 3x³ - x² - 3x + 1 if one zero is 1", "options": {"A": "1, 1/3, -1", "B": "1, -1/3, 1", "C": "1, 1, -1", "D": "1, -1, -1/3"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "hard", "question": "If α, β, γ are zeroes of x³ - 2x² - 5x + 6, find α + β + γ", "options": {"A": "2", "B": "-2", "C": "5", "D": "-5"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "hard", "question": "If α, β, γ are zeroes of 2x³ - x² - 7x + 6, find αβ + βγ + γα", "options": {"A": "-7/2", "B": "7/2", "C": "-1/2", "D": "1/2"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "hard", "question": "If α, β, γ are zeroes of x³ - 3x² + 4, find αβγ", "options": {"A": "-4", "B": "4", "C": "3", "D": "-3"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "hard", "question": "If polynomial x³ - 4x² + x + 6 has zeroes α, β, γ and α+β=5, find γ", "options": {"A": "-1", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "hard", "question": "If two zeroes of x³ - 4x² - 3x + 18 are 3 and -2, find third zero", "options": {"A": "3", "B": "-3", "C": "2", "D": "-2"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "hard", "question": "If x-2 is factor of x³ - 3x² + 4x - 4, find all zeroes", "options": {"A": "2, 1, 2", "B": "2, 1, -2", "C": "2, -1, 2", "D": "2, 1, 1"}, "correct_answer": "A"},
    {"id": 71, "difficulty": "hard", "question": "Form quadratic polynomial with zeroes 2 and -3", "options": {"A": "x² + x - 6", "B": "x² - x - 6", "C": "x² + 5x + 6", "D": "x² - 5x + 6"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "hard", "question": "Form quadratic polynomial with zeroes 1/2 and -1/3", "options": {"A": "6x² - x - 1", "B": "6x² + x - 1", "C": "6x² - x + 1", "D": "6x² + x + 1"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "hard", "question": "Form quadratic polynomial with zeroes √2 and -√2", "options": {"A": "x² - 2", "B": "x² + 2", "C": "x² - 2x", "D": "x² + 2x"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "hard", "question": "Form quadratic polynomial with zeroes 2+√3 and 2-√3", "options": {"A": "x² - 4x + 1", "B": "x² + 4x + 1", "C": "x² - 4x - 1", "D": "x² + 4x - 1"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "hard", "question": "Form cubic polynomial with zeroes 1, 2, 3", "options": {"A": "x³ - 6x² + 11x - 6", "B": "x³ + 6x² + 11x + 6", "C": "x³ - 6x² - 11x - 6", "D": "x³ + 6x² - 11x + 6"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "hard", "question": "If α, β are zeroes of x² - 2x - 8, find α³ + β³", "options": {"A": "80", "B": "64", "C": "56", "D": "40"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "hard", "question": "If α, β are zeroes of x² + 3x - 4, find |α - β|", "options": {"A": "5", "B": "1", "C": "7", "D": "3"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "hard", "question": "If α, β are zeroes of 2x² - 4x + 5, find α² + β²", "options": {"A": "-1", "B": "1", "C": "3", "D": "5"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "hard", "question": "If one zero of 4x² - 8kx + 9 is negative of other, find k", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "hard", "question": "If α, β are zeroes of x² - 6x + a and 3α + 2β = 20, find a", "options": {"A": "-16", "B": "16", "C": "8", "D": "-8"}, "correct_answer": "A"},
    {"id": 81, "difficulty": "hard", "question": "Find zeroes of x⁴ - 5x² + 4", "options": {"A": "1, -1, 2, -2", "B": "1, 2", "C": "1, -1", "D": "2, -2"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "Find zeroes of x⁴ - 13x² + 36", "options": {"A": "2, -2, 3, -3", "B": "2, 3", "C": "2, -2", "D": "3, -3"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "Find zeroes of x⁴ - 10x² + 9", "options": {"A": "1, -1, 3, -3", "B": "1, 3", "C": "1, -1", "D": "3, -3"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Find zeroes of x⁴ - 17x² + 16", "options": {"A": "1, -1, 4, -4", "B": "1, 4", "C": "1, -1", "D": "4, -4"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "Find zeroes of x⁴ - 5x² - 36", "options": {"A": "3, -3", "B": "2, -2", "C": "3, -3, 2i, -2i", "D": "3, -3, 2, -2"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "If α, β are zeroes of ax² + bx + c, find α⁻¹ + β⁻¹", "options": {"A": "-b/c", "B": "b/c", "C": "-c/b", "D": "c/b"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "If α, β are zeroes of ax² + bx + c, find α²β + αβ²", "options": {"A": "-bc/a²", "B": "bc/a²", "C": "-b/a", "D": "c/a"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "If α, β are zeroes of x² - px + q, find polynomial with zeroes α/β, β/α", "options": {"A": "qx² - (p²-2q)x + q", "B": "x² - (p²/q)x + 1", "C": "qx² - p²x + q", "D": "x² - (p/q)x + 1"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "If α, β are zeroes of x² + 2x - 1, find polynomial with zeroes α², β²", "options": {"A": "x² - 6x + 1", "B": "x² + 6x + 1", "C": "x² - 2x + 1", "D": "x² + 2x + 1"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "If α, β, γ are zeroes of x³ - 2x² + 3x - 4, find Σα²", "options": {"A": "-2", "B": "2", "C": "4", "D": "-4"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "If zeroes of x³ - 12x² + 39x - 28 are in AP, find them", "options": {"A": "1, 4, 7", "B": "2, 4, 6", "C": "1, 3, 9", "D": "3, 4, 5"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "hard", "question": "If zeroes of x³ - 6x² + 3x + 10 are in AP, find them", "options": {"A": "-1, 2, 5", "B": "1, 2, 3", "C": "2, 3, 4", "D": "-2, 0, 2"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "If α, β are zeroes of x² - 2x + 3, find polynomial with zeroes α-1, β-1", "options": {"A": "x² + 3", "B": "x² - 4x + 6", "C": "x² + 4x + 6", "D": "x² - 3"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "If α, β are zeroes of x² - x - 1, find polynomial with zeroes 2α+1, 2β+1", "options": {"A": "x² - 4x - 1", "B": "x² - 4x + 1", "C": "x² + 4x - 1", "D": "x² + 4x + 1"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "If product of zeroes of (k-1)x² + kx + 1 is -2, find k", "options": {"A": "-1", "B": "1", "C": "2", "D": "-2"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "If sum of zeroes of kx² + 2x + 3k is equal to product, find k", "options": {"A": "-2/3", "B": "2/3", "C": "-3/2", "D": "3/2"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "If α, β are zeroes of x² - 4x + 1, find α⁶ + β⁶", "options": {"A": "2702", "B": "1348", "C": "676", "D": "338"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "If α, β, γ are zeroes of x³ - x² - 4x + 4, find Σ1/α", "options": {"A": "1", "B": "-1", "C": "1/4", "D": "-1/4"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "Find cubic polynomial with zeroes 1/2, 1, 2", "options": {"A": "2x³ - 7x² + 7x - 2", "B": "2x³ + 7x² + 7x + 2", "C": "x³ - 7x² + 7x - 2", "D": "x³ + 7x² + 7x + 2"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "If α, β are zeroes of ax² + bx + c, find polynomial with zeroes α², β²", "options": {"A": "a²x² - (b²-2ac)x + c²", "B": "a²x² + (b²-2ac)x + c²", "C": "a²x² - b²x + c²", "D": "a²x² + b²x + c²"}, "correct_answer": "A"}
  ]
}