{
  "title": "Remainder Theorem - Grade 9 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Find remainder when x²+3x+2 is divided by x+1", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "Find remainder when x²+5x+6 is divided by x+2", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "Find remainder when x²-3x+2 is divided by x-1", "options": {"A": "0", "B": "1", "C": "2", "D": "-1"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "Find remainder when x²-5x+6 is divided by x-2", "options": {"A": "0", "B": "1", "C": "2", "D": "-2"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "Find remainder when x²-7x+12 is divided by x-3", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "Find remainder when x²+x-6 is divided by x+3", "options": {"A": "0", "B": "1", "C": "-6", "D": "6"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "Find remainder when x²+2x-15 is divided by x-3", "options": {"A": "0", "B": "1", "C": "-15", "D": "15"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "Find remainder when x²-4 is divided by x+2", "options": {"A": "0", "B": "1", "C": "-4", "D": "4"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "Find remainder when x²-9 is divided by x-3", "options": {"A": "0", "B": "1", "C": "-9", "D": "9"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "Find remainder when x²-16 is divided by x+4", "options": {"A": "0", "B": "1", "C": "-16", "D": "16"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "Find remainder when 2x+3 is divided by x+1", "options": {"A": "1", "B": "2", "C": "3", "D": "0"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "Find remainder when 3x-4 is divided by x-2", "options": {"A": "2", "B": "3", "C": "-4", "D": "0"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "Find remainder when 4x+5 is divided by x+2", "options": {"A": "-3", "B": "4", "C": "5", "D": "0"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Find remainder when 5x-6 is divided by x-3", "options": {"A": "9", "B": "5", "C": "-6", "D": "0"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "Find remainder when 6x+7 is divided by x+3", "options": {"A": "-11", "B": "6", "C": "7", "D": "0"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "Find remainder when x³-1 is divided by x-1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Find remainder when x³+1 is divided by x+1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "Find remainder when x³-8 is divided by x-2", "options": {"A": "0", "B": "8", "C": "-8", "D": "2"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "Find remainder when x³+8 is divided by x+2", "options": {"A": "0", "B": "8", "C": "-8", "D": "2"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "Find remainder when x³-27 is divided by x-3", "options": {"A": "0", "B": "27", "C": "-27", "D": "3"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "Find remainder when 2x²+3x+1 is divided by x+1", "options": {"A": "0", "B": "1", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Find remainder when 3x²-2x-1 is divided by x-1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Find remainder when 4x²+4x+1 is divided by 2x+1", "options": {"A": "0", "B": "1", "C": "4", "D": "2"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "Find remainder when 9x²-6x+1 is divided by 3x-1", "options": {"A": "0", "B": "1", "C": "9", "D": "3"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Find remainder when 4x²-12x+9 is divided by 2x-3", "options": {"A": "0", "B": "9", "C": "4", "D": "3"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Find remainder when 2x³-3x²+4x-5 is divided by x-1", "options": {"A": "-2", "B": "2", "C": "-5", "D": "5"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Find remainder when 3x³+2x²-x+7 is divided by x+1", "options": {"A": "5", "B": "-5", "C": "7", "D": "-7"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "Find remainder when x³-3x²+3x-1 is divided by x-1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "Find remainder when x³+6x²+11x+6 is divided by x+1", "options": {"A": "0", "B": "6", "C": "-6", "D": "1"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "Find remainder when x³-6x²+11x-6 is divided by x-1", "options": {"A": "0", "B": "-6", "C": "6", "D": "1"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "If x³+3x²+3x+1 is divided by x+1, remainder is?", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "If 8x³+12x²+6x+1 is divided by 2x+1, remainder is?", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "If 27x³-27x²+9x-1 is divided by 3x-1, remainder is?", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "If x⁴-1 is divided by x-1, remainder is?", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "If x⁴-16 is divided by x-2, remainder is?", "options": {"A": "0", "B": "16", "C": "-16", "D": "2"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "If polynomial p(x)=x²-5x+6 is divided by x-2, remainder is?", "options": {"A": "0", "B": "2", "C": "6", "D": "-6"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "If polynomial p(x)=x²-3x-10 is divided by x+2, remainder is?", "options": {"A": "0", "B": "2", "C": "-10", "D": "10"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "If polynomial p(x)=2x²+x-3 is divided by x-1, remainder is?", "options": {"A": "0", "B": "1", "C": "-3", "D": "3"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "If polynomial p(x)=3x²-5x+2 is divided by x-1, remainder is?", "options": {"A": "0", "B": "2", "C": "-2", "D": "1"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "If polynomial p(x)=4x²-4x+1 is divided by 2x-1, remainder is?", "options": {"A": "0", "B": "1", "C": "4", "D": "2"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "medium", "question": "Find k if x³+3x²+3x+k has x+1 as factor", "options": {"A": "1", "B": "-1", "C": "3", "D": "-3"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "medium", "question": "Find k if x³-3x²+3x-k has x-1 as factor", "options": {"A": "1", "B": "-1", "C": "3", "D": "-3"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "medium", "question": "Find k if 2x³+x²-3x+k has x-1 as factor", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "medium", "question": "Find k if 3x³-2x²+x-k has x+1 as factor", "options": {"A": "-6", "B": "6", "C": "3", "D": "-3"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "Find k if x³+kx²+3x-4 has x+2 as factor", "options": {"A": "3", "B": "-3", "C": "4", "D": "-4"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "medium", "question": "Find remainder when x⁴+x³+x²+x+1 is divided by x-1", "options": {"A": "5", "B": "1", "C": "0", "D": "4"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "Find remainder when x⁴-x³+x²-x+1 is divided by x+1", "options": {"A": "5", "B": "1", "C": "0", "D": "4"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "medium", "question": "Find remainder when x⁴+2x³+3x²+2x+1 is divided by x+1", "options": {"A": "1", "B": "0", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "medium", "question": "Find remainder when x⁴-2x³+3x²-2x+1 is divided by x-1", "options": {"A": "1", "B": "0", "C": "2", "D": "3"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "medium", "question": "Find remainder when 2x⁴-3x³+4x²-5x+6 is divided by x-1", "options": {"A": "4", "B": "6", "C": "0", "D": "2"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "Find remainder when x⁵-1 is divided by x-1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "Find remainder when x⁵+1 is divided by x+1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "Find remainder when x⁵-32 is divided by x-2", "options": {"A": "0", "B": "32", "C": "-32", "D": "2"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "Find remainder when x⁵+32 is divided by x+2", "options": {"A": "0", "B": "32", "C": "-32", "D": "2"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "Find remainder when x⁶-1 is divided by x-1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "Find remainder when polynomial p(x)=x³-ax²+2x-a is divided by x-a", "options": {"A": "a", "B": "2a", "C": "a²", "D": "a³"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "Find remainder when polynomial p(x)=x³+ax²+bx+c is divided by x+1", "options": {"A": "a-b+c-1", "B": "a+b+c+1", "C": "a-b+c+1", "D": "a+b+c-1"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "Find remainder when polynomial p(x)=ax³+bx²+cx+d is divided by x-1", "options": {"A": "a+b+c+d", "B": "a-b+c-d", "C": "a+b-c-d", "D": "a-b-c+d"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "Find remainder when polynomial p(x)=ax³-bx²+cx-d is divided by x+1", "options": {"A": "-a-b-c-d", "B": "a+b+c+d", "C": "-a+b-c+d", "D": "a-b+c-d"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "Find remainder when polynomial p(x)=x⁴+ax³+bx²+cx+d is divided by x-1", "options": {"A": "1+a+b+c+d", "B": "1-a+b-c+d", "C": "1+a-b+c-d", "D": "1-a-b-c-d"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "hard", "question": "Find remainder when x¹⁰⁰+1 is divided by x+1", "options": {"A": "2", "B": "0", "C": "1", "D": "-1"}, "correct_answer": "A"},
    {"id": 62, "difficulty": "hard", "question": "Find remainder when x¹⁰⁰-1 is divided by x-1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "hard", "question": "Find remainder when x⁹⁹+1 is divided by x+1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "hard", "question": "Find remainder when x⁹⁹-1 is divided by x-1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "hard", "question": "Find remainder when x¹⁰¹+1 is divided by x+1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "hard", "question": "Find remainder when x¹⁰¹-1 is divided by x-1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "hard", "question": "Find remainder when x²⁰²³+1 is divided by x+1", "options": {"A": "2", "B": "0", "C": "1", "D": "-1"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "hard", "question": "Find remainder when x²⁰²⁴-1 is divided by x-1", "options": {"A": "0", "B": "1", "C": "-1", "D": "2"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "hard", "question": "Find remainder when (x+1)¹⁰⁰ is divided by x", "options": {"A": "1", "B": "0", "C": "100", "D": "x"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "hard", "question": "Find remainder when (x-1)¹⁰⁰ is divided by x", "options": {"A": "1", "B": "0", "C": "100", "D": "x"}, "correct_answer": "A"},
    {"id": 71, "difficulty": "hard", "question": "Find remainder when (x+2)¹⁰⁰ is divided by x", "options": {"A": "2¹⁰⁰", "B": "0", "C": "100", "D": "x"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "hard", "question": "Find remainder when (x-2)¹⁰⁰ is divided by x", "options": {"A": "2¹⁰⁰", "B": "0", "C": "100", "D": "x"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "hard", "question": "If x³+ax²+bx+6 has x-2 as factor and leaves remainder 3 when divided by x-3, find a and b", "options": {"A": "a=-3, b=-1", "B": "a=3, b=1", "C": "a=-1, b=-3", "D": "a=1, b=3"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "hard", "question": "If x³+ax²+bx+10 has x+2 as factor and leaves remainder -2 when divided by x-1, find a and b", "options": {"A": "a=-7, b=14", "B": "a=7, b=-14", "C": "a=-14, b=7", "D": "a=14, b=-7"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "hard", "question": "If polynomial x⁴-6x³+16x²-25x+10 is divided by x²-2x+k, remainder is x+a, find k and a", "options": {"A": "k=5, a=-5", "B": "k=5, a=5", "C": "k=-5, a=5", "D": "k=-5, a=-5"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "hard", "question": "If both x-2 and x-1/2 are factors of px²+5x+r, show that p=r. Find p and r if sum is 5", "options": {"A": "p=2, r=2", "B": "p=3, r=3", "C": "p=4, r=4", "D": "p=5, r=5"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "hard", "question": "What must be added to x⁴+2x³-2x²+x-1 so that result is exactly divisible by x²+2x-3?", "options": {"A": "-x+4", "B": "x-4", "C": "-x-4", "D": "x+4"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "hard", "question": "What must be subtracted from x⁴+2x³-13x²-12x+21 so that result is exactly divisible by x²-4x+3?", "options": {"A": "2x-3", "B": "2x+3", "C": "3x-2", "D": "3x+2"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "hard", "question": "If x²-1 is factor of ax⁴+bx³+cx²+dx+e, show that a+c+e=b+d=0. Find a,b,c,d,e if all are 1", "options": {"A": "a=1,c=1,e=1,b=-1,d=-1", "B": "a=1,c=-1,e=1,b=1,d=1", "C": "a=-1,c=1,e=-1,b=1,d=1", "D": "a=1,c=1,e=-1,b=-1,d=1"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "hard", "question": "Find remainder when x⁵-3x²+2x-1 is divided by x²-1", "options": {"A": "2x-2", "B": "2x+2", "C": "-2x+2", "D": "-2x-2"}, "correct_answer": "A"},
    {"id": 81, "difficulty": "hard", "question": "Find remainder when x⁶-1 is divided by x²-1", "options": {"A": "0", "B": "1", "C": "x-1", "D": "x+1"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "Find remainder when x⁶+1 is divided by x²+1", "options": {"A": "0", "B": "1", "C": "2", "D": "x"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "Find remainder when x⁸-1 is divided by x²-1", "options": {"A": "0", "B": "1", "C": "x-1", "D": "x+1"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Find remainder when x⁸+1 is divided by x²+1", "options": {"A": "0", "B": "1", "C": "2", "D": "x"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "If polynomial x⁴-6x³+16x²-25x+10 is divided by x²-2x+k gives remainder x+a, and when divided by x-1 gives remainder 1, find k and a", "options": {"A": "k=5, a=-5", "B": "k=5, a=5", "C": "k=3, a=-3", "D": "k=3, a=3"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "Find remainder when (x+1)ⁿ is divided by x", "options": {"A": "1", "B": "0", "C": "n", "D": "x"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "Find remainder when (x-1)ⁿ is divided by x", "options": {"A": "(-1)ⁿ", "B": "1", "C": "0", "D": "n"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "Find remainder when (x+2)ⁿ is divided by x", "options": {"A": "2ⁿ", "B": "0", "C": "n", "D": "x"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "Find remainder when (x-2)ⁿ is divided by x", "options": {"A": "(-2)ⁿ", "B": "2ⁿ", "C": "0", "D": "n"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "If x³+ax²-bx+10 is divisible by x²-3x+2, find a and b", "options": {"A": "a=-8, b=17", "B": "a=8, b=-17", "C": "a=-17, b=8", "D": "a=17, b=-8"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "Find remainder when x²⁰²³+x²⁰²²+x²⁰²¹+...+x²+x+1 is divided by x-1", "options": {"A": "2024", "B": "2023", "C": "1", "D": "0"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "hard", "question": "Find remainder when x²⁰²³-x²⁰²²+x²⁰²¹-...+x²-x+1 is divided by x+1", "options": {"A": "2024", "B": "2023", "C": "1", "D": "0"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "If polynomial x⁴+px³+qx²+rx+5 is divisible by x²-3x+2, find p, q, r", "options": {"A": "p=-5, q=11, r=-15", "B": "p=5, q=11, r=15", "C": "p=-5, q=-11, r=15", "D": "p=5, q=-11, r=-15"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "Find remainder when x¹⁰⁰-2x⁹⁹+3x⁹⁸-4x⁹⁷+...+100x-101 is divided by x+1", "options": {"A": "-51", "B": "51", "C": "101", "D": "-101"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "Find remainder when x⁹⁹+2x⁹⁸+3x⁹⁷+...+99x+100 is divided by x-1", "options": {"A": "5050", "B": "4950", "C": "5000", "D": "5100"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "If x²+x+1 is factor of ax³+bx²+cx+d, then which relation holds?", "options": {"A": "a+c=b+d", "B": "a+b=c+d", "C": "a+d=b+c", "D": "a=b=c=d"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "Find remainder when (x+1)(x+2)(x+3)(x+4) is divided by x+5", "options": {"A": "24", "B": "120", "C": "0", "D": "1"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "Find remainder when (x-1)(x-2)(x-3)(x-4) is divided by x-5", "options": {"A": "24", "B": "120", "C": "0", "D": "1"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "If polynomial x⁴+px³+qx²+rx+s is divisible by x²+1, then which relation holds?", "options": {"A": "p=r and q=s+1", "B": "p=r and q=s-1", "C": "p=-r and q=s+1", "D": "p=-r and q=s-1"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "Find remainder when x¹⁰⁰⁰-1 is divided by x⁴-1", "options": {"A": "0", "B": "1", "C": "x²-1", "D": "x²+1"}, "correct_answer": "A"}
  ]
}