{
  "title": "Irrational Numbers Review - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Which of these is irrational?", "options": {"A": "√4", "B": "√9", "C": "√2", "D": "√16"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "π is:", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Natural"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "Which is rational?", "options": {"A": "√3", "B": "√5", "C": "√8", "D": "√25"}, "correct_answer": "D"},
    {"id": 4, "difficulty": "easy", "question": "0.333... is:", "options": {"A": "Irrational", "B": "Rational", "C": "Integer", "D": "None"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "√7 is:", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "None"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "Which is irrational?", "options": {"A": "0.5", "B": "3/4", "C": "√10", "D": "2"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "1.4142135... (√2) is:", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Natural"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "22/7 is:", "options": {"A": "Irrational", "B": "Rational", "C": "Equal to π", "D": "Integer"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "√(9/16) is:", "options": {"A": "Irrational", "B": "Rational", "C": "Integer", "D": "None"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "√12 is:", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Natural"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "Which is rational?", "options": {"A": "√13", "B": "√17", "C": "√49", "D": "√50"}, "correct_answer": "C"},
    {"id": 12, "difficulty": "easy", "question": "0.1010010001... is:", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Natural"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "√(1/4) is:", "options": {"A": "Irrational", "B": "Rational", "C": "Integer", "D": "None"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "3√2 is:", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Natural"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "Which is irrational?", "options": {"A": "0", "B": "1", "C": "√18", "D": "4"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "√27 is:", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Natural"}, "correct_answer": "B"},
    {"id": 17, "difficulty": "easy", "question": "0.75 is:", "options": {"A": "Irrational", "B": "Rational", "C": "Integer", "D": "None"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "√0.04 is:", "options": {"A": "Irrational", "B": "Rational", "C": "Integer", "D": "Natural"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "2+√3 is:", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Natural"}, "correct_answer": "B"},
    {"id": 20, "difficulty": "easy", "question": "√100 is:", "options": {"A": "Irrational", "B": "Rational", "C": "Integer", "D": "Both B and C"}, "correct_answer": "D"},
    
    {"id": 21, "difficulty": "medium", "question": "Prove √2 is irrational.", "options": {"A": "Assume √2=a/b in lowest terms", "B": "Assume √2 rational", "C": "Assume √2 integer", "D": "Assume √2 prime"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Prove √3 is irrational.", "options": {"A": "Assume √3=a/b in lowest terms", "B": "Assume √3 rational", "C": "Assume √3 integer", "D": "Assume √3 prime"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "Prove √5 is irrational.", "options": {"A": "Assume √5=a/b in lowest terms", "B": "Assume √5 rational", "C": "Assume √5 integer", "D": "Assume √5 prime"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "Prove 2√3 is irrational.", "options": {"A": "Assume rational, then √3 rational", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Prove 3+√2 is irrational.", "options": {"A": "Assume rational, then √2 rational", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Prove 5-√3 is irrational.", "options": {"A": "Assume rational, then √3 rational", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Prove √2+√3 is irrational.", "options": {"A": "Assume rational, square both sides", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "Prove 1/√2 is irrational.", "options": {"A": "Assume rational, then √2 rational", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "Prove √6 is irrational.", "options": {"A": "Assume √6=a/b in lowest terms", "B": "Assume √6 rational", "C": "Assume √6 integer", "D": "Assume √6 prime"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "Prove √7 is irrational.", "options": {"A": "Assume √7=a/b in lowest terms", "B": "Assume √7 rational", "C": "Assume √7 integer", "D": "Assume √7 prime"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Is √(4/9) rational or irrational?", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Natural"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "Is √(25/36) rational?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Cannot say"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "Is √(2/3) rational?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Cannot say"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "Is 0.010010001... rational?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Cannot say"}, "correct_answer": "B"},
    {"id": 35, "difficulty": "medium", "question": "Is 1.234567891011... rational?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Cannot say"}, "correct_answer": "B"},
    {"id": 36, "difficulty": "medium", "question": "Is 0.123123123... rational?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Cannot say"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "Is 2.718281828... (e) rational?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Cannot say"}, "correct_answer": "B"},
    {"id": 38, "difficulty": "medium", "question": "Is 1.618033... (φ) rational?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Cannot say"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "Is 0.10110111011110... rational?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Cannot say"}, "correct_answer": "B"},
    {"id": 40, "difficulty": "medium", "question": "Is √0.09 rational?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Cannot say"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "medium", "question": "Decimal expansion of irrational number is:", "options": {"A": "Terminating", "B": "Non-terminating repeating", "C": "Non-terminating non-repeating", "D": "Integer"}, "correct_answer": "C"},
    {"id": 42, "difficulty": "medium", "question": "Which statement is true?", "options": {"A": "All irrationals are real", "B": "All reals are irrational", "C": "No rational is real", "D": "All integers are irrational"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "medium", "question": "Between two rational numbers, there are:", "options": {"A": "Finite irrationals", "B": "Infinite irrationals", "C": "No irrationals", "D": "Exactly one irrational"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "medium", "question": "Sum of rational and irrational is:", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Natural"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "medium", "question": "Product of non-zero rational and irrational is:", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Natural"}, "correct_answer": "B"},
    {"id": 46, "difficulty": "medium", "question": "Sum of two irrationals can be:", "options": {"A": "Always irrational", "B": "Always rational", "C": "Either rational or irrational", "D": "Always integer"}, "correct_answer": "C"},
    {"id": 47, "difficulty": "medium", "question": "Product of two irrationals can be:", "options": {"A": "Always irrational", "B": "Always rational", "C": "Either rational or irrational", "D": "Always integer"}, "correct_answer": "C"},
    {"id": 48, "difficulty": "medium", "question": "Difference of two irrationals can be:", "options": {"A": "Always irrational", "B": "Always rational", "C": "Either rational or irrational", "D": "Always zero"}, "correct_answer": "C"},
    {"id": 49, "difficulty": "medium", "question": "Quotient of two irrationals can be:", "options": {"A": "Always irrational", "B": "Always rational", "C": "Either rational or irrational", "D": "Always one"}, "correct_answer": "C"},
    {"id": 50, "difficulty": "medium", "question": "Which is always irrational?", "options": {"A": "√2+√3", "B": "√4+√9", "C": "√16+√25", "D": "√1+√0"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "Find irrational between 1 and 2.", "options": {"A": "√2", "B": "1.5", "C": "1.75", "D": "1.8"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "Find irrational between 0 and 1.", "options": {"A": "1/√2", "B": "0.5", "C": "0.75", "D": "0.25"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "Find irrational between 2 and 3.", "options": {"A": "√5", "B": "2.5", "C": "2.75", "D": "2.8"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "Find irrational between 3/5 and 4/5.", "options": {"A": "√(1/2)", "B": "0.7", "C": "0.72", "D": "0.75"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "medium", "question": "Find irrational between √2 and √3.", "options": {"A": "√2.5", "B": "1.6", "C": "1.7", "D": "1.8"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "Express 0.001 as p/q.", "options": {"A": "1/1000", "B": "1/100", "C": "1/10", "D": "1/10000"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "Express 0.375 as p/q.", "options": {"A": "3/8", "B": "5/16", "C": "7/20", "D": "9/25"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "Express 0.625 as p/q.", "options": {"A": "5/8", "B": "7/12", "C": "9/16", "D": "11/18"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "Express 0.875 as p/q.", "options": {"A": "7/8", "B": "5/6", "C": "9/10", "D": "11/12"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "Express 0.0625 as p/q.", "options": {"A": "1/16", "B": "1/8", "C": "1/32", "D": "1/64"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "medium", "question": "Express 0.444... as p/q.", "options": {"A": "4/9", "B": "4/10", "C": "4/11", "D": "4/12"}, "correct_answer": "A"},
    {"id": 62, "difficulty": "medium", "question": "Express 0.2727... as p/q.", "options": {"A": "3/11", "B": "27/99", "C": "2727/9999", "D": "27/100"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "medium", "question": "Express 0.1666... as p/q.", "options": {"A": "1/6", "B": "16/99", "C": "166/999", "D": "16/100"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "medium", "question": "Express 0.8333... as p/q.", "options": {"A": "5/6", "B": "83/99", "C": "833/999", "D": "83/100"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "Express 0.142857142857... as p/q.", "options": {"A": "1/7", "B": "142857/999999", "C": "2/14", "D": "1/7 (simplest)"}, "correct_answer": "D"},
    {"id": 66, "difficulty": "medium", "question": "Classify: √2×√8", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 67, "difficulty": "medium", "question": "Classify: √3×√12", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 68, "difficulty": "medium", "question": "Classify: √5×√20", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 69, "difficulty": "medium", "question": "Classify: √7×√28", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 70, "difficulty": "medium", "question": "Classify: √2+√8", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Cannot determine"}, "correct_answer": "B"},
    {"id": 71, "difficulty": "medium", "question": "Classify: √27/√3", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 72, "difficulty": "medium", "question": "Classify: √50/√2", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 73, "difficulty": "medium", "question": "Classify: √75/√3", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 74, "difficulty": "medium", "question": "Classify: √98/√2", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 75, "difficulty": "medium", "question": "Classify: (√3+√2)(√3-√2)", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 76, "difficulty": "medium", "question": "Classify: (√5+√3)(√5-√3)", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 77, "difficulty": "medium", "question": "Classify: (√7+√2)(√7-√2)", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 78, "difficulty": "medium", "question": "Classify: (√6+1)(√6-1)", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 79, "difficulty": "medium", "question": "Classify: (√11+√7)(√11-√7)", "options": {"A": "Rational", "B": "Irrational", "C": "Integer", "D": "Both A and C"}, "correct_answer": "D"},
    {"id": 80, "difficulty": "medium", "question": "Insert 2 irrationals between √2 and √3.", "options": {"A": "√2.2, √2.5", "B": "√2.1, √2.8", "C": "√2.25, √2.75", "D": "√2.3, √2.7"}, "correct_answer": "C"},
    
    {"id": 81, "difficulty": "hard", "question": "Prove √p is irrational where p is prime.", "options": {"A": "Assume √p=a/b in lowest terms", "B": "Assume √p rational", "C": "Assume √p integer", "D": "Assume √p natural"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "Prove 3-2√2 is irrational.", "options": {"A": "Assume rational, then √2 rational", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "Prove √2+√5 is irrational.", "options": {"A": "Assume rational, square both sides", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Prove √3-√2 is irrational.", "options": {"A": "Assume rational, then contradiction", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "Prove 1/(2+√3) is irrational.", "options": {"A": "Rationalize and assume rational", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "Prove √(2/3) is irrational.", "options": {"A": "Assume rational, then √6 rational", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "Prove (√2+1)/(√2-1) is irrational.", "options": {"A": "Simplify to 3+2√2, then irrational", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "Prove (√3+√2)/(√3-√2) is irrational.", "options": {"A": "Simplify to 5+2√6, then irrational", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "Prove √2+√3+√5 is irrational.", "options": {"A": "Assume rational, then contradiction", "B": "Assume integer", "C": "Assume prime", "D": "Assume natural"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "Prove √n is irrational if n is not perfect square.", "options": {"A": "Assume √n=a/b in lowest terms", "B": "Assume √n rational", "C": "Assume √n integer", "D": "Assume √n natural"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "Find 3 different irrationals between 1/7 and 2/7.", "options": {"A": "√(1/50), √(1/30), √(1/20)", "B": "0.15, 0.2, 0.25", "C": "1/6, 1/5, 1/4", "D": "0.142, 0.143, 0.144"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "hard", "question": "Prove sum of rational and irrational is irrational.", "options": {"A": "Assume sum rational, contradiction", "B": "Direct proof", "C": "Counterexample", "D": "By definition"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "Prove product of non-zero rational and irrational is irrational.", "options": {"A": "Assume product rational, contradiction", "B": "Direct proof", "C": "Counterexample", "D": "By definition"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "Give example where sum of two irrationals is rational.", "options": {"A": "√2 and -√2", "B": "√3 and √5", "C": "π and e", "D": "√2 and √3"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "Give example where product of two irrationals is rational.", "options": {"A": "√2 and √8", "B": "√3 and √5", "C": "π and e", "D": "√2 and √3"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "Show that set of irrationals is not closed under addition.", "options": {"A": "√2+(-√2)=0 rational", "B": "√3+√5 irrational", "C": "π+e irrational", "D": "All sums irrational"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "Show that set of irrationals is not closed under multiplication.", "options": {"A": "√2×√8=4 rational", "B": "√3×√5 irrational", "C": "π×e irrational", "D": "All products irrational"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "Prove: If x is irrational, then 1/x is irrational (x≠0).", "options": {"A": "Assume 1/x rational, contradiction", "B": "Direct proof", "C": "Counterexample", "D": "By definition"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "Prove: If x² is irrational, then x is irrational.", "options": {"A": "Contrapositive: if x rational, x² rational", "B": "Direct proof", "C": "Counterexample", "D": "By definition"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "Prove: There exist irrationals a,b such that aᵇ is rational.", "options": {"A": "Consider √2^√2", "B": "Consider π^e", "C": "Consider e^π", "D": "All of above"}, "correct_answer": "A"}
  ]
}