{
  "title": "Nature of Roots (Discriminant) - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "The discriminant of ax² + bx + c = 0 is:", "options": {"A": "b² - 4ac", "B": "b² + 4ac", "C": "√(b² - 4ac)", "D": "b² - ac"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "For x² - 5x + 6 = 0, discriminant D =", "options": {"A": "1", "B": "0", "C": "-1", "D": "25"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "If D > 0, roots are:", "options": {"A": "Real and equal", "B": "Real and distinct", "C": "Imaginary", "D": "None"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "If D = 0, roots are:", "options": {"A": "Real and equal", "B": "Real and distinct", "C": "Imaginary", "D": "None"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "If D < 0, roots are:", "options": {"A": "Real and equal", "B": "Real and distinct", "C": "Imaginary", "D": "None"}, "correct_answer": "C"},
    {"id": 6, "difficulty": "easy", "question": "For x² + 4x + 4 = 0, discriminant D =", "options": {"A": "0", "B": "1", "C": "-1", "D": "16"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "For x² + x + 1 = 0, discriminant D =", "options": {"A": "1", "B": "0", "C": "-3", "D": "3"}, "correct_answer": "C"},
    {"id": 8, "difficulty": "easy", "question": "Nature of roots of x² - 3x + 2 = 0 is:", "options": {"A": "Real and equal", "B": "Real and distinct", "C": "Imaginary", "D": "None"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "Nature of roots of x² + 6x + 9 = 0 is:", "options": {"A": "Real and equal", "B": "Real and distinct", "C": "Imaginary", "D": "None"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "Nature of roots of x² + x + 4 = 0 is:", "options": {"A": "Real and equal", "B": "Real and distinct", "C": "Imaginary", "D": "None"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "For 2x² - 3x + 1 = 0, D =", "options": {"A": "1", "B": "0", "C": "-1", "D": "9"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "For x² - 4x + 4 = 0, D =", "options": {"A": "0", "B": "1", "C": "4", "D": "16"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "For x² + 2x + 5 = 0, nature of roots is:", "options": {"A": "Real and equal", "B": "Real and distinct", "C": "Imaginary", "D": "None"}, "correct_answer": "C"},
    {"id": 14, "difficulty": "easy", "question": "For x² - 6x + 9 = 0, D =", "options": {"A": "0", "B": "1", "C": "9", "D": "36"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "For x² - 8x + 16 = 0, nature of roots is:", "options": {"A": "Real and equal", "B": "Real and distinct", "C": "Imaginary", "D": "None"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "For 3x² + 2x - 1 = 0, D =", "options": {"A": "16", "B": "4", "C": "0", "D": "-4"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "For x² + 8x + 16 = 0, D =", "options": {"A": "0", "B": "1", "C": "8", "D": "64"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "For x² - 10x + 25 = 0, nature of roots is:", "options": {"A": "Real and equal", "B": "Real and distinct", "C": "Imaginary", "D": "None"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "For 4x² - 12x + 9 = 0, D =", "options": {"A": "0", "B": "1", "C": "9", "D": "144"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "For x² + 3x + 2 = 0, nature of roots is:", "options": {"A": "Real and equal", "B": "Real and distinct", "C": "Imaginary", "D": "None"}, "correct_answer": "B"},
    {"id": 21, "difficulty": "medium", "question": "Find k for equal roots: x² - 4x + k = 0", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "D"},
    {"id": 22, "difficulty": "medium", "question": "Find k for equal roots: 2x² + kx + 8 = 0", "options": {"A": "±4", "B": "±6", "C": "±8", "D": "±10"}, "correct_answer": "C"},
    {"id": 23, "difficulty": "medium", "question": "Find k for real roots: x² + 4x + k = 0", "options": {"A": "k ≤ 4", "B": "k ≥ 4", "C": "k ≤ 2", "D": "k ≥ 2"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "Find k for imaginary roots: x² + kx + 9 = 0", "options": {"A": "-6 < k < 6", "B": "k < -6 or k > 6", "C": "k = ±6", "D": "k = ±3"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "For what k does 3x² + kx + 3 = 0 have equal roots?", "options": {"A": "±3", "B": "±6", "C": "±9", "D": "±12"}, "correct_answer": "B"},
    {"id": 26, "difficulty": "medium", "question": "Find k for real and distinct roots: kx² - 6x + 1 = 0", "options": {"A": "k < 9", "B": "k > 9", "C": "k < 9 and k ≠ 0", "D": "k > 9 and k ≠ 0"}, "correct_answer": "C"},
    {"id": 27, "difficulty": "medium", "question": "For what k does x² + 2(k-1)x + (k+5) = 0 have equal roots?", "options": {"A": "4", "B": "4, -1", "C": "-4", "D": "-4, 1"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "Find k for real roots: kx² + 4x + 1 = 0", "options": {"A": "k ≤ 4", "B": "k ≥ 4", "C": "k ≤ 4 and k ≠ 0", "D": "k ≥ 4 and k ≠ 0"}, "correct_answer": "C"},
    {"id": 29, "difficulty": "medium", "question": "For what k does 4x² - 2(k+1)x + (k+4) = 0 have equal roots?", "options": {"A": "3, 5", "B": "-3, -5", "C": "3, -5", "D": "-3, 5"}, "correct_answer": "D"},
    {"id": 30, "difficulty": "medium", "question": "Find k for imaginary roots: x² + (k+1)x + (k+4) = 0", "options": {"A": "-5 < k < 3", "B": "k < -5 or k > 3", "C": "k = -5, 3", "D": "k = 5, -3"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "For what k does (k+1)x² - 2(k-1)x + 1 = 0 have equal roots?", "options": {"A": "0, 3", "B": "0, -3", "C": "1, 3", "D": "1, -3"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "Find k for real roots: x² + 2kx + (k² - k + 2) = 0", "options": {"A": "k ≤ 2", "B": "k ≥ 2", "C": "k ≤ 1", "D": "k ≥ 1"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "For what k does x² + k(2x-1) + 2 = 0 have equal roots?", "options": {"A": "2, 4", "B": "-2, -4", "C": "2, -4", "D": "-2, 4"}, "correct_answer": "D"},
    {"id": 34, "difficulty": "medium", "question": "Find k for real and distinct roots: (k-1)x² - 2(k-1)x + 1 = 0", "options": {"A": "k < 2", "B": "k > 2", "C": "k < 2 and k ≠ 1", "D": "k > 2 and k ≠ 1"}, "correct_answer": "C"},
    {"id": 35, "difficulty": "medium", "question": "For what k does (k-2)x² - 2(2k-3)x + (5k-6) = 0 have equal roots?", "options": {"A": "1, 3", "B": "-1, -3", "C": "1, -3", "D": "-1, 3"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "Find k for imaginary roots: kx² + 3x + 2 = 0", "options": {"A": "k > 9/8", "B": "k < 9/8", "C": "k > 9/8 and k ≠ 0", "D": "k < 9/8 and k ≠ 0"}, "correct_answer": "C"},
    {"id": 37, "difficulty": "medium", "question": "For what k does (k+2)x² + 4x + (k+3) = 0 have equal roots?", "options": {"A": "1, -4", "B": "-1, 4", "C": "1, 4", "D": "-1, -4"}, "correct_answer": "B"},
    {"id": 38, "difficulty": "medium", "question": "Find k for real roots: x² - 2kx + (k² - k - 2) = 0", "options": {"A": "k ≥ -2", "B": "k ≤ -2", "C": "k ≥ 2", "D": "k ≤ 2"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "For what k does 9x² + 6kx + 4 = 0 have equal roots?", "options": {"A": "±1", "B": "±2", "C": "±3", "D": "±4"}, "correct_answer": "B"},
    {"id": 40, "difficulty": "medium", "question": "Find k for real and distinct roots: 3x² + 2kx + 3 = 0", "options": {"A": "-3 < k < 3", "B": "k < -3 or k > 3", "C": "k = ±3", "D": "k ≠ ±3"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "medium", "question": "For what k does x² + 2(k-4)x + 2k = 0 have equal roots?", "options": {"A": "4, 8", "B": "-4, 8", "C": "4, -8", "D": "-4, -8"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "medium", "question": "Find k for imaginary roots: 2x² + kx + 8 = 0", "options": {"A": "-8 < k < 8", "B": "k < -8 or k > 8", "C": "k = ±8", "D": "k ≠ ±8"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "medium", "question": "For what k does kx² - 2(k-2)x + (k+2) = 0 have equal roots?", "options": {"A": "2", "B": "2, 3", "C": "3", "D": "1, 2"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "medium", "question": "Find k for real roots: 4x² + kx + 9 = 0", "options": {"A": "k ≤ -12 or k ≥ 12", "B": "-12 ≤ k ≤ 12", "C": "k = ±12", "D": "k ≠ ±12"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "For what k does (k-3)x² - 4x + k = 0 have equal roots?", "options": {"A": "4", "B": "-1, 4", "C": "1, -4", "D": "-1, -4"}, "correct_answer": "B"},
    {"id": 46, "difficulty": "medium", "question": "Find k for real and distinct roots: x² - kx + 9 = 0", "options": {"A": "-6 < k < 6", "B": "k < -6 or k > 6", "C": "k = ±6", "D": "k ≠ ±6"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "medium", "question": "For what k does 4x² + kx + 1 = 0 have equal roots?", "options": {"A": "±1", "B": "±2", "C": "±3", "D": "±4"}, "correct_answer": "D"},
    {"id": 48, "difficulty": "medium", "question": "Find k for imaginary roots: x² - 4x + k = 0", "options": {"A": "k > 4", "B": "k < 4", "C": "k = 4", "D": "k ≠ 4"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "medium", "question": "For what k does x² + k(2x+3) + 4 = 0 have equal roots?", "options": {"A": "1, 4", "B": "-1, -4", "C": "1, -4", "D": "-1, 4"}, "correct_answer": "D"},
    {"id": 50, "difficulty": "medium", "question": "Find k for real roots: kx² - 5x + k = 0", "options": {"A": "k ≤ -5/2 or k ≥ 5/2", "B": "-5/2 ≤ k ≤ 5/2", "C": "k = ±5/2", "D": "k ≠ ±5/2"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "For what k does (k+1)x² + 2(k+3)x + (k+8) = 0 have equal roots?", "options": {"A": "1, -3", "B": "-1, 3", "C": "1, 3", "D": "-1, -3"}, "correct_answer": "B"},
    {"id": 52, "difficulty": "medium", "question": "Find k for real and distinct roots: 2x² + kx + 2 = 0", "options": {"A": "-4 < k < 4", "B": "k < -4 or k > 4", "C": "k = ±4", "D": "k ≠ ±4"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "medium", "question": "For what k does 9x² - 3kx + 1 = 0 have equal roots?", "options": {"A": "±1", "B": "±2", "C": "±3", "D": "±4"}, "correct_answer": "B"},
    {"id": 54, "difficulty": "medium", "question": "Find k for imaginary roots: 3x² - 2kx + 3 = 0", "options": {"A": "-3 < k < 3", "B": "k < -3 or k > 3", "C": "k = ±3", "D": "k ≠ ±3"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "For what k does x² - 2x(1+3k) + 7(3+2k) = 0 have equal roots?", "options": {"A": "2, -10/9", "B": "-2, 10/9", "C": "2, 10/9", "D": "-2, -10/9"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "Find k for real roots: x² + 2(k-1)x + k+5 = 0", "options": {"A": "k ≤ -1 or k ≥ 4", "B": "-1 ≤ k ≤ 4", "C": "k = -1, 4", "D": "k ≠ -1, 4"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "For what k does (k-2)x² + 2(2k-3)x + 5k-6 = 0 have equal roots?", "options": {"A": "1, 2", "B": "1, 3", "C": "2, 3", "D": "1, 4"}, "correct_answer": "B"},
    {"id": 58, "difficulty": "medium", "question": "Find k for real and distinct roots: kx² + 6x + 1 = 0", "options": {"A": "k < 9", "B": "k > 9", "C": "k < 9 and k ≠ 0", "D": "k > 9 and k ≠ 0"}, "correct_answer": "C"},
    {"id": 59, "difficulty": "medium", "question": "For what k does 4x² + 4kx + (k² - k + 2) = 0 have equal roots?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "medium", "question": "Find k for imaginary roots: x² + (3-k)x + k = 0", "options": {"A": "1 < k < 9", "B": "k < 1 or k > 9", "C": "k = 1, 9", "D": "k ≠ 1, 9"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "medium", "question": "For what k does x² + k(2x-3) + 4 = 0 have equal roots?", "options": {"A": "1, 4", "B": "-1, -4", "C": "1, -4", "D": "-1, 4"}, "correct_answer": "D"},
    {"id": 62, "difficulty": "medium", "question": "Find k for real roots: 2x² - 4x + k = 0", "options": {"A": "k ≤ 2", "B": "k ≥ 2", "C": "k ≤ 1", "D": "k ≥ 1"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "medium", "question": "For what k does (k+1)x² - 2(3k+1)x + (8k+1) = 0 have equal roots?", "options": {"A": "0, 3", "B": "0, -3", "C": "1, 3", "D": "1, -3"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "medium", "question": "Find k for real and distinct roots: x² - 2kx + (k² - k - 2) = 0", "options": {"A": "k > -2", "B": "k < -2", "C": "k > 2", "D": "k < 2"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "For what k does 9x² + 3kx + 4 = 0 have equal roots?", "options": {"A": "±2", "B": "±3", "C": "±4", "D": "±6"}, "correct_answer": "C"},
    {"id": 66, "difficulty": "medium", "question": "Find k for imaginary roots: kx² + 4x + k = 0", "options": {"A": "-2 < k < 2", "B": "k < -2 or k > 2", "C": "k = ±2", "D": "k ≠ ±2"}, "correct_answer": "B"},
    {"id": 67, "difficulty": "medium", "question": "For what k does x² + 2(k-3)x + k+3 = 0 have equal roots?", "options": {"A": "2, 6", "B": "-2, -6", "C": "2, -6", "D": "-2, 6"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "medium", "question": "Find k for real roots: 3x² + kx + 12 = 0", "options": {"A": "k ≤ -12 or k ≥ 12", "B": "-12 ≤ k ≤ 12", "C": "k = ±12", "D": "k ≠ ±12"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "medium", "question": "For what k does (k-1)x² + kx + 1 = 0 have equal roots?", "options": {"A": "2±2√2", "B": "2±√2", "C": "1±√2", "D": "1±2√2"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "medium", "question": "Find k for real and distinct roots: 4x² + kx + 1 = 0", "options": {"A": "-4 < k < 4", "B": "k < -4 or k > 4", "C": "k = ±4", "D": "k ≠ ±4"}, "correct_answer": "B"},
    {"id": 71, "difficulty": "medium", "question": "For what k does x² - 2kx + k² - k - 2 = 0 have equal roots?", "options": {"A": "-1", "B": "-2", "C": "-3", "D": "-4"}, "correct_answer": "B"},
    {"id": 72, "difficulty": "medium", "question": "Find k for imaginary roots: 2x² - 5x + k = 0", "options": {"A": "k > 25/8", "B": "k < 25/8", "C": "k = 25/8", "D": "k ≠ 25/8"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "For what k does kx² + 2(k-4)x + 2 = 0 have equal roots?", "options": {"A": "2, 8", "B": "-2, -8", "C": "2, -8", "D": "-2, 8"}, "correct_answer": "D"},
    {"id": 74, "difficulty": "medium", "question": "Find k for real roots: x² + kx + 16 = 0", "options": {"A": "k ≤ -8 or k ≥ 8", "B": "-8 ≤ k ≤ 8", "C": "k = ±8", "D": "k ≠ ±8"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "For what k does (k-2)x² + 4x + (k+3) = 0 have equal roots?", "options": {"A": "3±√13", "B": "3±√11", "C": "2±√13", "D": "2±√11"}, "correct_answer": "B"},
    {"id": 76, "difficulty": "medium", "question": "Find k for real and distinct roots: x² + (k+1)x + k = 0", "options": {"A": "k ≠ 1", "B": "k < 1", "C": "k > 1", "D": "k = 1"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "For what k does 4x² - 4kx + (k² - k + 2) = 0 have equal roots?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 78, "difficulty": "medium", "question": "Find k for imaginary roots: x² + 4x + k = 0", "options": {"A": "k > 4", "B": "k < 4", "C": "k = 4", "D": "k ≠ 4"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "medium", "question": "For what k does x² + k(3x-1) + 2 = 0 have equal roots?", "options": {"A": "1, 8", "B": "-1, -8", "C": "1, -8", "D": "-1, 8"}, "correct_answer": "D"},
    {"id": 80, "difficulty": "medium", "question": "Find k for real roots: kx² - 6x + 1 = 0", "options": {"A": "k ≤ 9", "B": "k ≥ 9", "C": "k ≤ 9 and k ≠ 0", "D": "k ≥ 9 and k ≠ 0"}, "correct_answer": "C"},
    {"id": 81, "difficulty": "hard", "question": "For what values of k does (k+1)x² - 2(3k+1)x + (8k+1) = 0 have real and equal roots?", "options": {"A": "0, 1", "B": "0, 3", "C": "1, 3", "D": "2, 3"}, "correct_answer": "B"},
    {"id": 82, "difficulty": "hard", "question": "Find k such that x² + 2x(k-1) + (k+5) = 0 has no real roots", "options": {"A": "k < -1", "B": "-1 < k < 4", "C": "k > 4", "D": "k < -1 or k > 4"}, "correct_answer": "B"},
    {"id": 83, "difficulty": "hard", "question": "For what k does (k-3)x² - 4x + k = 0 have roots which are real and distinct?", "options": {"A": "k < -1 or k > 4", "B": "-1 < k < 4", "C": "k ≤ -1 or k ≥ 4", "D": "-1 ≤ k ≤ 4"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Find k such that (k+2)x² + 4x + (k+3) = 0 has imaginary roots", "options": {"A": "k < -4 or k > 1", "B": "-4 < k < 1", "C": "k ≤ -4 or k ≥ 1", "D": "-4 ≤ k ≤ 1"}, "correct_answer": "B"},
    {"id": 85, "difficulty": "hard", "question": "For what k does x² + 2(k-4)x + 2k = 0 have roots which are equal in magnitude but opposite in sign?", "options": {"A": "0", "B": "2", "C": "4", "D": "8"}, "correct_answer": "C"},
    {"id": 86, "difficulty": "hard", "question": "Find k such that 9x² + 3kx + 4 = 0 has one root double the other", "options": {"A": "±6", "B": "±8", "C": "±10", "D": "±12"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "For what k does (k-1)x² + kx + 1 = 0 have roots which are reciprocal of each other?", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "Find k such that x² - 2kx + k² - k - 2 = 0 has roots whose difference is 4", "options": {"A": "±3", "B": "±4", "C": "±5", "D": "±6"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "For what k does kx² + 2(k-4)x + 2 = 0 have roots which are in ratio 2:3?", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "B"},
    {"id": 90, "difficulty": "hard", "question": "Find k such that (k-2)x² + 4x + (k+3) = 0 has one root positive and one negative", "options": {"A": "k < -3", "B": "-3 < k < 2", "C": "k > 2", "D": "k < -3 or k > 2"}, "correct_answer": "B"},
    {"id": 91, "difficulty": "hard", "question": "For what k does x² + k(2x-3) + 4 = 0 have roots whose sum is equal to their product?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 92, "difficulty": "hard", "question": "Find k such that 4x² - 4kx + (k² - k + 2) = 0 has roots which are consecutive integers", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "For what k does (k+1)x² - 2(3k+1)x + (8k+1) = 0 have roots which are in arithmetic progression?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 94, "difficulty": "hard", "question": "Find k such that x² + (3-k)x + k = 0 has roots whose difference is 1", "options": {"A": "2, 5", "B": "3, 6", "C": "4, 7", "D": "5, 8"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "For what k does kx² - 6x + 1 = 0 have roots which are in geometric progression?", "options": {"A": "8", "B": "9", "C": "10", "D": "11"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "Find k such that x² - 2x(1+3k) + 7(3+2k) = 0 has roots which are in ratio 3:4", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 97, "difficulty": "hard", "question": "For what k does (k-2)x² - 2(2k-3)x + (5k-6) = 0 have roots which are equal in magnitude?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "C"},
    {"id": 98, "difficulty": "hard", "question": "Find k such that x² + 2(k-3)x + k+3 = 0 has roots whose squares sum to 34", "options": {"A": "4, -2", "B": "5, -3", "C": "6, -4", "D": "7, -5"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "For what k does x² + k(3x-1) + 2 = 0 have roots which are reciprocals of each other?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "Find k such that 9x² + 6kx + 4 = 0 has roots which differ by 2/3", "options": {"A": "±1", "B": "±2", "C": "±3", "D": "±4"}, "correct_answer": "B"}
  ]
}