{
  "title": "Remainder Theorem - Grade 10 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Find remainder when x³ - 3x² + 2x - 1 is divided by (x-1). 🧮", "options": {"A": "-1", "B": "0", "C": "1", "D": "2"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "What is the remainder when x² + 5x + 6 is divided by (x+2)? ✨", "options": {"A": "0", "B": "2", "C": "4", "D": "6"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "If f(x)=2x-4 is divided by (x-2), the remainder is? ➗", "options": {"A": "0", "B": "2", "C": "-2", "D": "4"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "Remainder when x⁴ - 16 is divided by (x-2) is? 📏", "options": {"A": "0", "B": "8", "C": "16", "D": "-16"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "Find remainder: (x³ + 8) ÷ (x+2) 🎯", "options": {"A": "0", "B": "8", "C": "-8", "D": "16"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "What is remainder when x² - 9 is divided by (x+3)? 🔢", "options": {"A": "0", "B": "3", "C": "-6", "D": "9"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "Remainder of x² - 5x + 6 divided by (x-3) is? ✔️", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "If polynomial P(x) divided by (x-a) gives remainder R, then R = ? 🎲", "options": {"A": "P(a)", "B": "P(-a)", "C": "P(0)", "D": "P(1)"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "Remainder when 2x² + 3x - 5 is divided by (x-1) is? 📝", "options": {"A": "0", "B": "1", "C": "-1", "D": "5"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "Find remainder: (x³ - 27) ÷ (x-3) 🔄", "options": {"A": "0", "B": "27", "C": "-27", "D": "9"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "Remainder of x² + 7x + 12 divided by (x+4) is? ➕", "options": {"A": "0", "B": "4", "C": "12", "D": "-12"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "If f(x)=x²-4x+4 is divided by (x-2), remainder = ? ⚖️", "options": {"A": "0", "B": "2", "C": "4", "D": "-4"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "Remainder when x⁵ - 1 is divided by (x-1) is? 🔧", "options": {"A": "0", "B": "1", "C": "-1", "D": "5"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Find remainder: (3x² - 2x + 1) ÷ (x-1) 📊", "options": {"A": "2", "B": "1", "C": "0", "D": "3"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "Remainder of x³ + 3x² + 3x + 1 divided by (x+1) is? ✨", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "If (x-4) divides f(x) exactly, then remainder is? 🏷️", "options": {"A": "0", "B": "4", "C": "f(4)", "D": "f(-4)"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Remainder when x² - 10x + 25 is divided by (x-5) is? 🎲", "options": {"A": "0", "B": "5", "C": "10", "D": "25"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "Find remainder: (x⁴ - 1) ÷ (x-1) 🔁", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "If f(2)=5 for polynomial f(x), remainder when divided by (x-2) is? 🤔", "options": {"A": "5", "B": "2", "C": "0", "D": "Cannot say"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "Remainder of 4x² - 9 divided by (2x-3) is? 📐", "options": {"A": "0", "B": "3", "C": "9", "D": "-9"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "Find remainder when x³ - 2x² + x - 3 is divided by (x-2). 🧠", "options": {"A": "-1", "B": "0", "C": "1", "D": "3"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "If (x-1) leaves remainder 3 when dividing f(x)=x³+ax²+bx+1, find a+b. ⚡", "options": {"A": "-1", "B": "0", "C": "1", "D": "2"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Remainder when 2x³ - 3x² + 4x - 5 is divided by (x+1) is? 🔄", "options": {"A": "-14", "B": "-10", "C": "10", "D": "14"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "If polynomial x⁴ - 3x³ + 2x² - x + 7 divided by (x-2) gives remainder R, find R. 📈", "options": {"A": "9", "B": "10", "C": "11", "D": "12"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Find remainder when x³ + 2x² - 5x - 6 is divided by (x-2). 🧮", "options": {"A": "0", "B": "4", "C": "-4", "D": "6"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "When f(x)=x³+px²+qx+6 is divided by (x-2), remainder is 12. When divided by (x+1), remainder is 0. Find p. 🔑", "options": {"A": "-1", "B": "0", "C": "1", "D": "2"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Remainder of x⁴ - 5x³ + 3x² + 4x - 2 divided by (x-3) is? 📊", "options": {"A": "-5", "B": "-3", "C": "0", "D": "5"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "If (x+3) divides x³ + kx² - 4x - 12 exactly, find k. 🎯", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "Find remainder when 3x³ - 4x² + 7x - 5 is divided by (x-1). ➗", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "When polynomial P(x) divided by (x-1) gives remainder 2, and by (x-2) gives remainder 1. Find remainder when divided by (x-1)(x-2). 🧩", "options": {"A": "-x+3", "B": "x+1", "C": "x-3", "D": "-x+1"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Remainder when x³ - 3x² - x + 3 is divided by (x+1) is? 🔍", "options": {"A": "0", "B": "2", "C": "-2", "D": "4"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "If f(x)=2x³+ax²-5x+2 leaves remainder 10 when divided by (x-2), find a. ⚙️", "options": {"A": "-1", "B": "0", "C": "1", "D": "2"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "Remainder of x⁴ + x³ - 2x² + x + 1 divided by (x-1) is? ✨", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Find k if remainder of x³ + 3x² - kx + 4 divided by (x-2) is 0. 🔢", "options": {"A": "12", "B": "10", "C": "8", "D": "6"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "When x³ + ax² + bx + 6 divided by (x-2) leaves remainder 12, and by (x+1) leaves remainder 6. Find a. 📐", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "B"},
    {"id": 36, "difficulty": "medium", "question": "Remainder when 4x³ - 3x² + 2x - 1 is divided by (x+½) is? ➖", "options": {"A": "-3", "B": "-2", "C": "0", "D": "2"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "If (x-1) and (x+2) are factors of x³ + ax² + bx - 4, find remainder when divided by (x-3). 🔗", "options": {"A": "20", "B": "18", "C": "16", "D": "14"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Remainder of 2x⁴ - 3x³ + 5x² - 7x + 1 divided by (x-1) is? 📍", "options": {"A": "-2", "B": "0", "C": "2", "D": "4"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Find remainder when x³ - 6x² + 11x - 6 is divided by (x-4). 🎲", "options": {"A": "6", "B": "8", "C": "10", "D": "12"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "If polynomial gives remainder 3 and 5 when divided by (x-1) and (x-2). Find remainder when divided by (x-1)(x-2). 🌟", "options": {"A": "2x+1", "B": "2x-1", "C": "x+2", "D": "x-2"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "hard", "question": "Find remainder when x⁴ - 3x³ + 2x² - 5x + 7 is divided by (x² - 4). 🧮", "options": {"A": "4x+15", "B": "4x-15", "C": "15x+4", "D": "15x-4"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "If f(x) divided by (x-1)(x-2) leaves remainder (3x-2), find remainder when divided by (x-1). 🔄", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "Polynomial f(x) divided by (x²-1) gives remainder (2x+3). Find remainder when divided by (x-1). 📊", "options": {"A": "5", "B": "4", "C": "3", "D": "2"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "When x³ + ax² + bx + c is divided by (x-1), (x-2), (x-3) remainders are 1, 2, 3 respectively. Find a. ⚙️", "options": {"A": "-6", "B": "-3", "C": "0", "D": "3"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "hard", "question": "Find remainder when x⁴ + x³ - 2x² + x + 1 is divided by (x² - 1). 🔍", "options": {"A": "x+2", "B": "2x+1", "C": "x-2", "D": "2x-1"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "hard", "question": "Polynomial leaves remainder 2 when divided by (x-1) and remainder 1 when divided by (x-2). If divided by (x-1)(x-2), remainder is ax+b. Find a+b. ➕", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "hard", "question": "If f(x)=x⁴ - 3x³ + 2x² - x + 5 divided by (x² + x + 1) gives remainder px+q. Find p+q. 🎯", "options": {"A": "6", "B": "7", "C": "8", "D": "9"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "hard", "question": "When f(x) divided by (x-α)(x-β) leaves remainder ax+b, then f(α)=? and f(β)=? 📐", "options": {"A": "aα+b, aβ+b", "B": "α, β", "C": "0, 0", "D": "b, b"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "Find remainder when 2x⁴ - 3x³ + 4x² - 5x + 6 is divided by (x² - 3x + 2). 🧩", "options": {"A": "3x+2", "B": "2x+3", "C": "3x-2", "D": "2x-3"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "If polynomial f(x) divided by (x-1)² leaves remainder (x+1) and divided by (x-2) leaves remainder 4. Find remainder when divided by (x-1)²(x-2). 🌈", "options": {"A": "x² - 2x + 3", "B": "x² + 2x - 3", "C": "x² - 3x + 2", "D": "x² + 3x - 2"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "Find remainder of x¹⁰⁰ + x⁹⁹ + ... + x + 1 divided by (x-1). 🔢", "options": {"A": "101", "B": "100", "C": "99", "D": "1"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "hard", "question": "When f(x) divided by (x² + 1) leaves remainder (2x-3). Find remainder when divided by (x-1). ➗", "options": {"A": "-1", "B": "0", "C": "1", "D": "2"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "Polynomial f(x) gives remainder 2x+3 when divided by (x²-4). Find f(2) and f(-2). ⚡", "options": {"A": "7, -1", "B": "7, 1", "C": "-7, -1", "D": "-7, 1"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "If x³ + ax² + bx + c leaves same remainder when divided by (x-1) and (x+1). Find relation between a and c. 🔗", "options": {"A": "a+c=0", "B": "a-c=0", "C": "a+b=0", "D": "b+c=0"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "Find remainder when x⁵ - 4x³ + x² - 3x + 2 is divided by (x² - x - 2). 📦", "options": {"A": "5x-2", "B": "5x+2", "C": "2x+5", "D": "2x-5"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "hard", "question": "When f(x) divided by (x-1)(x-2)(x-3) leaves remainder 2x² - 3x + 4. Find remainder when divided by (x-2). 🎲", "options": {"A": "6", "B": "5", "C": "4", "D": "3"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "If f(x)=x⁴ + px³ + qx² + rx + s divided by (x²+1) leaves remainder (x+2). Find p+q+r+s. ➕", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "Find remainder when x⁴⁴ + x³³ + x²² + x¹¹ + 1 is divided by (x⁴ + x³ + x² + x + 1). 🌀", "options": {"A": "0", "B": "1", "C": "x", "D": "x+1"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "hard", "question": "Polynomial f(x) leaves remainder 3 when divided by (x-1) and remainder 5 when divided by (x-2). Find f(x) if it's a quadratic polynomial. ✨", "options": {"A": "2x+1", "B": "2x-1", "C": "x+2", "D": "x-2"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "When f(x) divided by (x-a)(x-b) leaves remainder px+q. If f(a)=A and f(b)=B, find p and q in terms of a,b,A,B. 🧮", "options": {"A": "p=(A-B)/(a-b), q=(aB-bA)/(a-b)", "B": "p=(A+B)/(a+b), q=(aB+bA)/(a+b)", "C": "p=(A-B)/(b-a), q=(bA-aB)/(b-a)", "D": "p=(A+B)/(a-b), q=(aB+bA)/(a-b)"}, "correct_answer": "A"}
  ]
}