{ "chapterId": "chapter-1-real-numbers", "chapterTitle": "Real Numbers", "concepts": [ { "id": "fundamental-theorem-arithmetic", "title": "Fundamental Theorem of Arithmetic", "icon": "🔢", "precheck": { "question": "What is the prime factorisation of 12?", "options": [ "$2^2 \\times 3$", "$2 \\times 3^2$", "$4 \\times 3$" ], "correctIndex": 0, "passMessage": "Correct! $4 \\times 3 = 12$.", "failMessage": "12 = 2 × 2 × 3." }, "learn": { "paragraphs": [ "**1. Fundamental Theorem of Arithmetic:** Every composite number can be expressed (factorised) as a product of primes uniquely.", "**2. Relationship:** For any two positive integers $a$ and $b$, $HCF(a, b) \\times LCM(a, b) = a \\times b$.", "**3. Standard Form:** $n = p_1^{a_1} \\times p_2^{a_2} \\times \\dots$" ], "formulas": [ { "rule": "HCF & LCM Product", "formula": "HCF(a, b) \\times LCM(a, b) = a \\times b", "example": "Test for 6 and 20: $HCF=2, LCM=60 \\Rightarrow 2 \\times 60 = 120 = 6 \\times 20$." } ], "boxes": [ { "type": "warning", "html": "The HCF × LCM = $a \\times b$ rule only works for two numbers!" } ] }, "practice": [ { "question": "Find prime factorisation of 156.", "options": [ "$2^2 \\times 3 \\times 13$", "$2 \\times 3^2 \\times 13$", "$2^2 \\times 3^2 \\times 13$" ], "correctIndex": 0, "solution": "$156 = 2 \\times 78 = 2^2 \\times 39 = 2^2 \\times 3 \\times 13$." }, { "question": "HCF(96, 404) is 4. Find LCM.", "options": [ "9696", "4040", "1010" ], "correctIndex": 0, "solution": "$(96 \\times 404)/4 = 96 \\times 101 = 9696$." }, { "question": "Find HCF(13, 17).", "options": [ "1", "221", "13" ], "correctIndex": 0, "solution": "Both are prime." }, { "question": "LCM of 12, 15, 21.", "options": [ "420", "180", "1260" ], "correctIndex": 0, "solution": "$2^2 \\times 3, 3 \\times 5, 3 \\times 7 \\Rightarrow 4 \\times 3 \\times 5 \\times 7 = 420$." }, { "question": "Smallest prime number.", "options": [ "2", "1", "3" ], "correctIndex": 0, "solution": "Definition." }, { "question": "Smallest composite number.", "options": [ "4", "2", "6" ], "correctIndex": 0, "solution": "Definition." }, { "question": "If $a=x^3y^2$ and $b=xy^3$, HCF is:", "options": [ "$xy^2$", "$x^3y^3$", "$x^2y$" ], "correctIndex": 0, "solution": "Lowest powers." }, { "question": "If $a=x^3y^2$ and $b=xy^3$, LCM is:", "options": [ "$x^3y^3$", "$xy^2$", "$x^2y^2$" ], "correctIndex": 0, "solution": "Highest powers." }, { "question": "Every composite number is a product of primes.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Fundamental Theorem." }, { "question": "Can $4^n$ end with digit 0?", "options": [ "No", "Yes" ], "correctIndex": 0, "solution": "No factor of 5." } ], "pyq": [ { "question": "**(2023)** Smallest 2-digit composite number.", "options": [ "10", "12", "11" ], "correctIndex": 0, "solution": "10 is composite." }, { "question": "**(2022)** Ratio of LCM and HCF of 5, 15, 20.", "options": [ "12:1", "1:12", "60:1" ], "correctIndex": 0, "solution": "LCM=60, HCF=5. Ratio = 12:1." }, { "question": "**(2021)** If $p, q$ are co-primes, find HCF.", "options": [ "1", "pq", "p+q" ], "correctIndex": 0, "solution": "Definition of co-prime." }, { "question": "**(2020)** If HCF(306, 657)=9, find LCM.", "options": [ "22338", "10000", "None" ], "correctIndex": 0, "solution": "$(306 \\times 657)/9 = 22338$." }, { "question": "**(2019)** Find exponents sum of prime factors of 196.", "options": [ "4", "2", "3" ], "correctIndex": 0, "solution": "$196 = 2^2 \\times 7^2$. Sum = $2+2=4$." }, { "question": "**(2023)** smallest prime number and smallest composite number HCF.", "options": [ "2", "1", "4" ], "correctIndex": 0, "solution": "HCF(2, 4) = 2." }, { "question": "**(2022)** Express 156 as product of primes.", "options": [ "$2^2 \\times 3 \\times 13$", "None", "1" ], "correctIndex": 0, "solution": "Standard." }, { "question": "**(2021)** If product is 1080 and HCF is 30, find LCM.", "options": [ "36", "108", "30" ], "correctIndex": 0, "solution": "$1080/30 = 36$." }, { "question": "**(2020)** Can $6^n$ end with 0?", "options": [ "No", "Yes" ], "correctIndex": 0, "solution": "Requires 2 and 5." }, { "question": "**(2019)** find HCF of 135 and 225.", "options": [ "45", "15", "9" ], "correctIndex": 0, "solution": "$135=3^3 \\times 5, 225=3^2 \\times 5^2 \\Rightarrow HCF = 3^2 \\times 5 = 45$." } ], "test": [ { "question": "HCF of two co-primes.", "options": [ "1", "Product", "0" ], "correctIndex": 0, "solution": "Definition." }, { "question": "Smallest odd prime.", "options": [ "3", "1", "5" ], "correctIndex": 0, "solution": "2 is even." }, { "question": "$2 \\times 3 \\times 5 \\times 7 + 7$ is:", "options": [ "Composite", "Prime", "Neither" ], "correctIndex": 0, "solution": "$7(2 \\times 3 \\times 5 + 1)$." }, { "question": "Product of HCF and LCM of 10 and 20.", "options": [ "200", "100", "400" ], "correctIndex": 0, "solution": "$10 \\times 20 = 200$." }, { "question": "Fundamental theorem applies to integers > 1.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Definition." } ] }, { "id": "hcf-lcm-applications", "title": "HCF & LCM Applications", "icon": "🏗️", "precheck": { "question": "To find 'minimum' time when items meet again, we use:", "options": [ "LCM", "HCF", "Average" ], "correctIndex": 0, "passMessage": "Exactly! Lowest Common Multiple for intervals.", "failMessage": "Minimum meeting time requires LCM." }, "learn": { "paragraphs": [ "**1. Intervals:** When bells toll or people run intervals, they meet at the **LCM** of the times.", "**2. Maximum Capacity:** To find the largest volume/length that can measure others exactly, use **HCF**.", "**3. Tilting/Rowing:** To arrange items in equal rows/tiles of maximum size, use **HCF**." ], "formulas": [ { "rule": "Application Trigger", "formula": "Min/Together \\Rightarrow LCM, Max/Equal \\Rightarrow HCF", "example": "Largest bottle size for 20L and 30L \\Rightarrow HCF(20, 30) = 10L." } ], "boxes": [ { "type": "tip", "html": "Read the question carefully for keywords like 'Maximum', 'Greatest' (HCF) or 'Minimum', 'Together again' (LCM)." } ] }, "practice": [ { "question": "Find max weight to measure 75kg and 69kg exactly.", "options": [ "3kg", "1kg", "15kg" ], "correctIndex": 0, "solution": "HCF(75, 69) = 3." }, { "question": "Sonia (18m) and Ravi (12m) meet at starting point after:", "options": [ "36m", "18m", "72m" ], "correctIndex": 0, "solution": "LCM(18, 12)." }, { "question": "Max capacity of container for 850L and 680L.", "options": [ "170L", "85L", "340L" ], "correctIndex": 0, "solution": "HCF(850, 680)." }, { "question": "Bells at 6, 8, 12 min. Toll together after:", "options": [ "24 min", "48 min", "12 min" ], "correctIndex": 0, "solution": "LCM(6, 8, 12)." }, { "question": "Three tankers 403, 434, 465 L. Max container size?", "options": [ "31 L", "13 L", "7 L" ], "correctIndex": 0, "solution": "HCF(403, 434, 465) = 31." }, { "question": "Army 616 member march behind 32 member band. Max columns?", "options": [ "8", "32", "16" ], "correctIndex": 0, "solution": "HCF(616, 32) = 8." }, { "question": "Sweet seller 420 Kaji and 130 Almond bar. Area of stack min, number max?", "options": [ "10", "20", "30" ], "correctIndex": 0, "solution": "HCF(420, 130) = 10." }, { "question": "Interval of meeting for 2, 4, 6 seconds.", "options": [ "12s", "6s", "24s" ], "correctIndex": 0, "solution": "LCM." }, { "question": "Greatest number dividing 70 and 125 leaving rem 5 and 8.", "options": [ "13", "11", "17" ], "correctIndex": 0, "solution": "HCF(65, 117) = 13." }, { "question": "Smallest number which divided by 35, 56, 91 leaves rem 7.", "options": [ "3647", "3640", "3633" ], "correctIndex": 0, "solution": "LCM(35, 56, 91) + 7 = 3647." } ], "pyq": [ { "question": "**(2023)** Three bells toll 9, 12, 15 min. Toll together after:", "options": [ "180 min", "60 min", "120 min" ], "correctIndex": 0, "solution": "LCM(9, 12, 15) = 180." }, { "question": "**(2022)** Max students in equal rows (32 in XI, 36 in X).", "options": [ "4", "8", "2" ], "correctIndex": 0, "solution": "HCF(32, 36) = 4." }, { "question": "**(2021)** Largest number divide 125, 70 leaving rem 8, 5.", "options": [ "13", "65", "117" ], "correctIndex": 0, "solution": "HCF(117, 65)." }, { "question": "**(2020)** Circular path, min time to meet.", "options": [ "LCM based", "HCF based" ], "correctIndex": 0, "solution": "Standard application." }, { "question": "**(2019)** Tanker problem (620L and 840L). Max capacity?", "options": [ "20L", "40L", "10L" ], "correctIndex": 0, "solution": "HCF." }, { "question": "**(2023)** Step-count matching problem (steps 80cm, 85cm, 90cm). Min distance?", "options": [ "12240cm", "None", "1" ], "correctIndex": 0, "solution": "LCM." }, { "question": "**(2022)** Row arrangement (30 and 42 plants). Max plants per row?", "options": [ "6", "10", "30" ], "correctIndex": 0, "solution": "HCF." }, { "question": "**(2021)** Traffic lights 48, 72, 108 sec. Chance change together?", "options": [ "432 sec", "108 sec", "72 sec" ], "correctIndex": 0, "solution": "LCM." }, { "question": "**(2020)** Case study: library books (32 Eng, 36 Hin). Min shelf length?", "options": [ "288 books", "None", "1" ], "correctIndex": 0, "solution": "LCM." }, { "question": "**(2019)** Find least number of square tiles to pave room 15m17cm x 9m2cm.", "options": [ "814", "None", "1" ], "correctIndex": 0, "solution": "HCF of 1517, 902 = 41. Area / 41^2 = 814." } ], "test": [ { "question": "Keyword 'Maximum' usually implies:", "options": [ "HCF", "LCM", "Sum" ], "correctIndex": 0, "solution": "Greatest divisor." }, { "question": "Keyword 'Together again' implies:", "options": [ "LCM", "HCF", "None" ], "correctIndex": 0, "solution": "Common multiple." }, { "question": "HCF(a, b, c) is always $\\le \\dots$", "options": [ "Min(a,b,c)", "Max(a,b,c)", "a+b+c" ], "correctIndex": 0, "solution": "Property." }, { "question": "LCM(a, b, c) is always $\\ge \\dots$", "options": [ "Max(a,b,c)", "Min(a,b,c)", "0" ], "correctIndex": 0, "solution": "Property." }, { "question": "Is subtraction needed for 'remainder' word problems in HCF?", "options": [ "Yes", "No" ], "correctIndex": 0, "solution": "Subtract first, then find HCF." } ] }, { "id": "irrationality-proofs", "title": "Proofs of Irrationality", "icon": "🛡️", "precheck": { "question": "If p divides $a^2$, then p divides $a$ (whenever p is prime).", "options": [ "True", "False" ], "correctIndex": 0, "passMessage": "Correct! This theorem is the backbone of all irrationality proofs.", "failMessage": "This is a key theorem for Chapter 1." }, "learn": { "paragraphs": [ "**1. Contradiction Method:** Assume the number is rational ($\\frac{p}{q}$), then prove this assumption leads to a mathematical impossibility.", "**2. Example ($\\sqrt{2}$):**", " - Assume $\\sqrt{2} = \\frac{a}{b}$ (co-primes).", " - $2b^2 = a^2 \\Rightarrow 2$ divides $a$.", " - Let $a = 2c \\Rightarrow 2b^2 = 4c^2 \\Rightarrow b^2 = 2c^2 \\Rightarrow 2$ divides $b$.", " - Both have factor 2, contradicting 'co-prime'." ], "formulas": [ { "rule": "Theorem", "formula": "p | a^2 \\Rightarrow p | a", "example": "If 3 divides $x^2$, then 3 divides $x$ (for prime 3)." } ], "boxes": [ { "type": "warning", "html": "For the board exam, you must be able to write the full proof for $\\sqrt{2}$, $\\sqrt{3}$, or $\\sqrt{5}$." } ] }, "practice": [ { "question": "In $\\sqrt{3} = a/b$, if we square it, we get:", "options": [ "$3b^2 = a^2$", "$a^2 = b^2$", "$a = 3b$" ], "correctIndex": 0, "solution": "Algebra." }, { "question": "If 5 divides $a^2$, what can we say about $a$?", "options": [ "5 divides a", "5 is odd", "No info" ], "correctIndex": 0, "solution": "Theorem." }, { "question": "Co-prime numbers have common factor:", "options": [ "Only 1", "No factors", "2" ], "correctIndex": 0, "solution": "Definition." }, { "question": "Which of these is irrational?", "options": ["$\\sqrt{7}$", "$\\sqrt{4}$", "$\\sqrt{9}$" ], "correctIndex": 0, "solution": "Prime roots." }, { "question": "Decimal expansion of $\\sqrt{2}$ is:", "options": [ "Non-terminating non-repeating", "Terminating", "Repeating" ], "correctIndex": 0, "solution": "Definition of irrational." }, { "question": "Assumption of proof by contradiction for $\\sqrt{5}$ is:", "options": ["$\\sqrt{5}$ is rational", "$\\sqrt{5}$ is prime", "None" ], "correctIndex": 0, "solution": "Standard start." }, { "question": "If $a = 3k$, then $a^2 = \\dots$", "options": [ "$9k^2$", "$3k^2$", "$6k$" ], "correctIndex": 0, "solution": "Squaring." }, { "question": "Contradiction arises because HCF(a, b) was assumed 1.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Logic step." }, { "question": "$\\sqrt{p}$ is irrational if p is prime.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Theorem." }, { "question": "Is $\\pi$ irrational?", "options": [ "Yes", "No" ], "correctIndex": 0, "solution": "Fact." } ], "pyq": [ { "question": "**(2024)** Prove $\\sqrt{2}$ is irrational.", "options": [ "Done in board tutorial", "None" ], "correctIndex": 0, "solution": "Follow standard steps." }, { "question": "**(2023)** Prove $\\sqrt{5}$ is irrational.", "options": [ "Standard Proof", "None" ], "correctIndex": 0, "solution": "Contradiction method." }, { "question": "**(2022)** Which theory is used in proving irrational roots?", "options": [ "Contradiction", "Induction", "Direct" ], "correctIndex": 0, "solution": "Method name." }, { "question": "**(2021)** If p is prime, $\\sqrt{p}$ is irrational.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "General form." }, { "question": "**(2020)** Proof for $\\sqrt{3}$ involves squaring denominator.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Standard algebra." }, { "question": "**(2023)** 'p divides a' implies p is prime in Chapter 1?", "options": [ "Yes", "No" ], "correctIndex": 0, "solution": "Assumption." }, { "question": "**(2022)** Is 22/7 a value of $\\pi$ or approximation?", "options": [ "Approximation", "Exact" ], "correctIndex": 0, "solution": "22/7 is rational." }, { "question": "**(2021)** Every rational number can be a fraction.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Definition." }, { "question": "**(2020)** Decimal form of irrational numbers.", "options": [ "Unending, non-patterned", "Repeating" ], "correctIndex": 0, "solution": "Terminology." }, { "question": "**(NCERT)** prove $\\sqrt{2} + \\sqrt{3}$ is irrational.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Complex proof." } ], "test": [ { "question": "Assume $\\sqrt{x} = p/q$. Next step?", "options": [ "Square both sides", "Divide", "Multiply" ], "correctIndex": 0, "solution": "Removes root." }, { "question": "Can an irrational number be zero?", "options": [ "No", "Yes" ], "correctIndex": 0, "solution": "0 is rational." }, { "question": "Value of $\\sqrt{12}$ is irrational.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "$2\\sqrt{3}$." }, { "question": "Why is proof needed for board?", "options": [ "Conceptual check", "None" ], "correctIndex": 0, "solution": "Core curriculum." }, { "question": "If p divides $a^2$, does p divide $a^4$?", "options": [ "Yes", "No" ], "correctIndex": 0, "solution": "By progression." } ] }, { "id": "combinations-irrationals", "title": "Irrational Combinations", "icon": "🧬", "precheck": { "question": "The sum of a rational and an irrational number is:", "options": [ "Always irrational", "Always rational", "Zero" ], "correctIndex": 0, "passMessage": "Correct! It cannot be rationalized.", "failMessage": "Rational + Irrational = Irrational." }, "learn": { "paragraphs": [ "**1. Basic Rules:**", " - Rational + Irrational = Irrational.", " - Rational × Irrational = Irrational (non-zero rational).", "**2. Proof Strategy:**", " - Let $3 + \\sqrt{2} = r$ (where r is rational).", " - $\\sqrt{2} = r - 3$.", " - Since $r - 3$ is rational, $\\sqrt{2}$ must be rational. Contradiction!" ], "formulas": [ { "rule": "General Outcome", "formula": "r + i = i, r \\times i = i", "example": "7√5 is irrational." } ], "boxes": [ { "type": "success", "html": "These proofs are much shorter than $\\sqrt{2}$ proofs!" } ] }, "practice": [ { "question": "$3 + 2\\sqrt{5}$ is irrational.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Combination rule." }, { "question": "$7\\sqrt{5}$ is irrational.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Product rule." }, { "question": "Is $\\sqrt{2} + \\sqrt{2}$ irrational?", "options": [ "Yes (2√2)", "No" ], "correctIndex": 0, "solution": "Sum of irrationals." }, { "question": "Is $\\sqrt{2} \\times \\sqrt{2}$ irrational?", "options": [ "No (2)", "Yes" ], "correctIndex": 0, "solution": "Product of irrationals can be rational." }, { "question": "Is $2 + \\pi$ irrational?", "options": [ "Yes", "No" ], "correctIndex": 0, "solution": "$\\pi$ is irrational." }, { "question": "Difference of rational and irrational is:", "options": [ "Irrational", "Rational" ], "correctIndex": 0, "solution": "Inverse of sum." }, { "question": "Reciprocal of an irrational number is:", "options": [ "Irrational", "Rational" ], "correctIndex": 0, "solution": "$1/\\sqrt{2}$." }, { "question": "In $3+2\\sqrt{5} = a/b$, isolate $\\sqrt{5}$:", "options": ["$\\sqrt{5} = (a-3b)/(2b)$", "$\\sqrt{5} = a/b - 3$" ], "correctIndex": 0, "solution": "Algebra." }, { "question": "Non-zero rational times irrational is never rational.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Standard theorem." }, { "question": "Is $(\\sqrt{2}-2)^2$ rational?", "options": [ "No (6-4√2)", "Yes" ], "correctIndex": 0, "solution": "Contains cross product." } ], "pyq": [ { "question": "**(2023)** Prove $3+2\\sqrt{5}$ is irrational given $\\sqrt{5}$ is irrational.", "options": [ "Standard Contradiction", "None" ], "correctIndex": 0, "solution": "Short method." }, { "question": "**(2022)** If $\\sqrt{3}$ is irr, prove $5-2\\sqrt{3}$ is irr.", "options": [ "Subtractive proof", "None" ], "correctIndex": 0, "solution": "Isolate $\\sqrt{3} = (5-r)/2$." }, { "question": "**(2021)** $2\\sqrt{3}/3$ is irrational.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Simplified $\\frac{2}{\\sqrt{3}}$." }, { "question": "**(2020)** Prove $1/\\sqrt{2}$ is irrational.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Rationalize to $\\sqrt{2}/2$." }, { "question": "**(2019)** sum of rational and irrational.", "options": [ "Always irrational", "None" ], "correctIndex": 0, "solution": "One marker question." }, { "question": "**(2023)** Case study: $h + \\sqrt{v}$ where v is speed.", "options": [ "Irrational if v non-square", "None" ], "correctIndex": 0, "solution": "Application." }, { "question": "**(2022)** Why is 0.1201200... irrational?", "options": [ "Non-repeating", "None" ], "correctIndex": 0, "solution": "Pattern check." }, { "question": "**(2021)** which is rational: (i) $2-\\sqrt{3}$ (ii) $\\sqrt{2}+\\sqrt{2}$ (iii) $(\\sqrt{2}+\\sqrt{3})(\\sqrt{2}-\\sqrt{3})$", "options": [ "(iii)", "All", "None" ], "correctIndex": 0, "solution": "$2-3 = -1$." }, { "question": "**(2020)** $(\\sqrt{3}- \\sqrt{2})(\\sqrt{3}+ \\sqrt{2}) = \\dots$", "options": [ "1", "5", "None" ], "correctIndex": 0, "solution": "Difference of squares." }, { "question": "**(NCERT)** Is $2 + \\sqrt{0.09}$ rational?", "options": [ "Yes (2.3)", "No" ], "correctIndex": 0, "solution": "$\\sqrt{0.09} = 0.3$." } ], "test": [ { "question": "Is $0$ rational?", "options": [ "Yes", "No" ], "correctIndex": 0, "solution": "$0/1$." }, { "question": "$(\\text{Rational}) / (\\text{Irrational})$ is:", "options": [ "Irrational", "Rational" ], "correctIndex": 0, "solution": "Usually." }, { "question": "Is $\\sqrt{2} \\times \\sqrt{8}$ rational?", "options": [ "Yes (4)", "No" ], "correctIndex": 0, "solution": "$\\sqrt{16}=4$." }, { "question": "If $x$ is rational, $x + y$ irr \\Rightarrow $y$ is?", "options": [ "Irrational", "Rational" ], "correctIndex": 0, "solution": "Subtraction logic." }, { "question": "Can two irrationals sum to rational?", "options": [ "Yes", "No" ], "correctIndex": 0, "solution": "$\\sqrt{2} + (-\\sqrt{2}) = 0$." } ] } ], "pyqComplete": { "title": "CBSE Class 10 Real Numbers PYQ – Complete (2017–2026)", "description": "Sectioned PYQs (Sec A–E) including Case-Based and 5-mark items. Euclid's Division Algorithm deleted from syllabus.", "questions": [ { "id": "Q1", "year": 2026, "marks": 1, "section": "A", "text": "Assertion (A): HCF(36m², 18m) = 18m, where m is a prime number. Reason (R): HCF of two numbers is always less than or equal to the smaller number.", "options": null, "answer": "(A) Both A and R true; R is the correct explanation", "topic": "Fundamental Theorem [A-R]" }, { "id": "Q2", "year": 2025, "marks": 1, "section": "A", "text": "√0.4 is a/an:", "options": ["natural number","integer","rational","irrational"], "answer": "(D) irrational number", "topic": "Irrationality [MCQ]" }, { "id": "Q3", "year": 2025, "marks": 1, "section": "A", "text": "Which of the following CANNOT be the unit digit of 8^n (n is a natural number)?", "options": ["4","2","0","6"], "answer": "(C) 0", "topic": "Fundamental Theorem [MCQ]" }, { "id": "Q4", "year": 2025, "marks": 1, "section": "A", "text": "Assertion (A): For primes x, y with x < y: HCF(x,y) = x and LCM(x,y) = y. Reason (R): HCF(x,y) is always ≤ LCM(x,y) for any two natural numbers.", "options": null, "answer": "(D) A is false, but R is true", "topic": "Fundamental Theorem [A-R]" }, { "id": "Q5", "year": 2025, "marks": 1, "section": "A", "text": "Assertion (A): For any two natural numbers, HCF × LCM = product of the numbers. Reason (R): LCM(x,y) is always a multiple of HCF(x,y).", "options": null, "answer": "(B) Both true; R is NOT the correct explanation", "topic": "Fundamental Theorem [A-R]" }, { "id": "Q6", "year": 2024, "marks": 1, "section": "A", "text": "LCM of two numbers is 14 times their HCF. Sum of LCM and HCF is 600. If one number is 280, find the other.", "options": null, "answer": "VSA (compute other number = 120)", "topic": "Fundamental Theorem [VSA]" }, { "id": "Q7", "year": 2024, "marks": 1, "section": "A", "text": "If p = a²b³ and q = a³b (a, b are prime numbers), then HCF(p, q) is:", "options": ["ab","ab²","a²b","a²b²"], "answer": "(B) ab²", "topic": "Fundamental Theorem [MCQ]" }, { "id": "Q8", "year": 2023, "marks": 1, "section": "A", "text": "The LCM of the smallest prime number and the smallest composite number is __.", "options": null, "answer": "LCM(2,4)=4", "topic": "Fundamental Theorem [VSA]" }, { "id": "Q9", "year": 2023, "marks": 1, "section": "A", "text": "If HCF(a, b) = 12 and a × b = 1800, then LCM(a, b) = __.", "options": null, "answer": "1800/12 = 150", "topic": "Fundamental Theorem [VSA]" }, { "id": "Q10", "year": 2022, "marks": 1, "section": "A", "text": "How many decimal places will the expansion of 23 / (2² × 5) have?", "options": null, "answer": "2 decimal places", "topic": "Decimal Expansion [VSA]" }, { "id": "Q11", "year": 2021, "marks": 1, "section": "A", "text": "LCM of two numbers is 182 and HCF is 13. If one number is 26, find the other.", "options": null, "answer": "Other = (13*182)/26 = 91", "topic": "Fundamental Theorem [VSA]" }, { "id": "Q12", "year": 2020, "marks": 1, "section": "A", "text": "The decimal expansion of 17/8 will terminate after how many decimal places?", "options": null, "answer": "3 places (since 8 = 2^3)", "topic": "Decimal Expansion [VSA]" }, { "id": "Q13", "year": 2019, "marks": 1, "section": "A", "text": "Express 98 as a product of its prime factors.", "options": null, "answer": "98 = 2 × 7^2", "topic": "Fundamental Theorem [VSA]" }, { "id": "Q14", "year": 2018, "marks": 1, "section": "A", "text": "If HCF(26, 169) = 13, then LCM(26, 169) = __.", "options": null, "answer": "(26×169)/13 = 338", "topic": "Fundamental Theorem [VSA]" }, { "id": "Q15", "year": 2017, "marks": 1, "section": "A", "text": "Is 7 × 11 × 13 × 15 + 15 a composite number? State True/False with justification.", "options": null, "answer": "True (factor 15)", "topic": "Fundamental Theorem [VSA]" }, { "id": "Q16", "year": 2017, "marks": 1, "section": "A", "text": "Does (2√45 + 3√20) / (2√5) simplify to a rational or irrational number?", "options": null, "answer": "Rational (simplifies)", "topic": "Decimal Expansion [VSA]" }, { "id": "Q17", "year": 2024, "marks": 2, "section": "B", "text": "Prove that √5 is an irrational number.", "options": null, "answer": "Proof by contradiction using prime factor argument", "topic": "Irrationality Proof [Proof]" }, { "id": "Q18", "year": 2023, "marks": 2, "section": "B", "text": "Prove that √2 is irrational.", "options": null, "answer": "Standard contradiction proof", "topic": "Irrationality Proof [Proof]" }, { "id": "Q19", "year": 2022, "marks": 2, "section": "B", "text": "Find the HCF and LCM of 404 and 96 using the Fundamental Theorem of Arithmetic.", "options": null, "answer": "HCF=4, LCM=(404×96)/4 = 9696", "topic": "Fundamental Theorem [SA]" }, { "id": "Q20", "year": 2021, "marks": 2, "section": "B", "text": "Show that (5 − √3) is irrational, given that √3 is irrational.", "options": null, "answer": "If 5−√3 were rational then √3 would be rational; contradiction", "topic": "Irrationality Proof [Proof]" }, { "id": "Q21", "year": 2020, "marks": 2, "section": "B", "text": "Find HCF of 96 and 404 by prime factorisation. Hence, find their LCM.", "options": null, "answer": "HCF=4, LCM=9696", "topic": "Fundamental Theorem [SA]" }, { "id": "Q22", "year": 2019, "marks": 2, "section": "B", "text": "Given HCF(306, 657) = 9, find LCM(306, 657).", "options": null, "answer": "LCM=(306×657)/9 = 22338", "topic": "Fundamental Theorem [SA]" }, { "id": "Q23", "year": 2019, "marks": 2, "section": "B", "text": "Find HCF and LCM of 306 and 54. Verify HCF × LCM = Product of both numbers.", "options": null, "answer": "HCF = 6, LCM = (306×54)/6 = 2754", "topic": "Fundamental Theorem [SA]" }, { "id": "Q24", "year": 2018, "marks": 2, "section": "B", "text": "Prove that √2 is irrational.", "options": null, "answer": "Standard proof", "topic": "Irrationality Proof [Proof]" }, { "id": "Q25", "year": 2017, "marks": 2, "section": "B", "text": "Without long division, state whether 17/3125 is terminating or non-terminating repeating. Give reasons.", "options": null, "answer": "Terminating (denominator = 5^5)", "topic": "Decimal Expansion [SA]" }, { "id": "Q26", "year": 2024, "marks": 3, "section": "C", "text": "Prove that (3 + 2√5) is irrational, given √5 is irrational.", "options": null, "answer": "Assume rational then isolate √5; contradiction", "topic": "Irrationality Proof [Proof]" }, { "id": "Q27", "year": 2023, "marks": 3, "section": "C", "text": "Prove that (2 + 3√5) is irrational.", "options": null, "answer": "Same strategy as above", "topic": "Irrationality Proof [Proof]" }, { "id": "Q28", "year": 2022, "marks": 3, "section": "C", "text": "Find the LCM and HCF of 12, 15 and 21 by prime factorisation method.", "options": null, "answer": "HCF=3, LCM=420", "topic": "Fundamental Theorem [SA]" }, { "id": "Q29", "year": 2020, "marks": 3, "section": "C", "text": "Prove √5 is irrational. Hence show (3 + √5) is also irrational.", "options": null, "answer": "Two-step contradiction proof", "topic": "Irrationality Proof [Proof]" }, { "id": "Q30", "year": 2018, "marks": 3, "section": "C", "text": "Explain why (7×11×13 + 13) and (7×6×5×4×3×2×1 + 5) are composite numbers.", "options": null, "answer": "Both have a common factor hence composite", "topic": "Fundamental Theorem [SA]" }, { "id": "Q31", "year": 2017, "marks": 3, "section": "C", "text": "Show that √2 is irrational.", "options": null, "answer": "Standard contradiction proof", "topic": "Irrationality Proof [Proof]" }, { "id": "Q32", "year": 2023, "marks": 5, "section": "D", "text": "(a) Three alarm clocks ring at intervals of 4, 12 and 20 minutes. If they ring together at 12:00 noon, at what time will they next ring together? (b) Find the HCF of 612 and 1314 using the prime factorisation method.", "options": null, "answer": "(a) LCM(4,12,20)=60 → 1:00 pm; (b) HCF(612,1314)=6", "topic": "Fundamental Theorem [LA]" }, { "id": "Q33", "year": 2022, "marks": 5, "section": "D", "text": "(a) Find the largest number that divides 125, 162 and 259 leaving remainders 5, 6 and 7 respectively. (b) An army contingent of 678 soldiers marches behind a band of 36 members. Both groups march in the same number of columns. Find the maximum number of columns.", "options": null, "answer": "(a) Convert to 120,156,252 then HCF = 12; (b) HCF(678,36)=6", "topic": "Fundamental Theorem [LA]" }, { "id": "Q34", "year": 2021, "marks": 5, "section": "D", "text": "(a) Steps 40 cm, 42 cm and 45 cm. Minimum distance so that each covers complete steps? (b) 576 boys and 448 girls divided into equal sections of either boys or girls only. Find total sections.", "options": null, "answer": "(a) LCM(40,42,45)=1260 cm; (b) HCF(576,448)=64 → sections = 576/64 + 448/64 = 9 + 7 = 16", "topic": "Fundamental Theorem [LA]" }, { "id": "Q35", "year": 2024, "marks": 4, "section": "E", "text": "PASSAGE: Meera collected 72 red, 48 blue and 108 green buttons. She wants to pack them into packets such that each packet has the same number of buttons of the same colour. (i) Find the HCF of 72, 48 and 108. (ii) Find the LCM of 72, 48 and 108. (iii) What is the maximum number of buttons in each packet? Also find the total number of packets formed.", "options": null, "answer": "(i) HCF=12, (ii) LCM=432, (iii) max per packet=12; total packets = 72/12 + 48/12 + 108/12 = 6 + 4 + 9 = 19", "topic": "Fundamental Theorem [Case Study]" }, { "id": "Q36", "year": 2023, "marks": 4, "section": "E", "text": "PASSAGE: Students in Class 6,7,8 are 96, 240 and 336 respectively. Principal wants the maximum number of students in each row with each row consisting of students of the same class only. (i) HCF of 96 and 240. (ii) HCF of 96, 240 and 336. (iii) Maximum number of rows formed in the march.", "options": null, "answer": "(i) HCF(96,240)=48, (ii) HCF(96,240,336)=48, (iii) rows = 96/48 + 240/48 + 336/48 = 2 + 5 + 7 = 14", "topic": "Fundamental Theorem [Case Study]" }, { "id": "Q37", "year": 2022, "marks": 4, "section": "E", "text": "PASSAGE: Steps measure 80 cm, 85 cm and 90 cm. (i) Find the HCF of step lengths 80 cm and 85 cm. (ii) What is the minimum distance Anjali should walk so that she completes her steps exactly? (iii) What is the minimum distance all three should walk so that each completes their steps exactly? Express your answer in metres.", "options": null, "answer": "(i) HCF(80,85)=5 cm, (ii) Anjali (80 cm) minimum distance = 80 cm, (iii) LCM(80,85,90)=? = 61200 cm = 612 m", "topic": "Fundamental Theorem [Case Study]" }, { "id": "Q38", "year": 2021, "marks": 4, "section": "E", "text": "PASSAGE: A librarian has 204 English books, 306 Hindi books and 136 Sanskrit books. She wants to arrange the books in shelves such that each shelf contains the same number of books and all books on a shelf are of the same language. (i) Find the prime factorisation of 204. (ii) Find the HCF of 204, 306 and 136. (iii) What is the maximum number of books on each shelf? Find also the minimum number of shelves required.", "options": null, "answer": "(i) 204 = 2^2 × 3 × 17, (ii) HCF(204,306,136)=2, (iii) max per shelf=2 → shelves = 204/2 + 306/2 + 136/2 = 102 + 153 + 68 = 323", "topic": "Fundamental Theorem [Case Study]" } ] }, "chapterTest": { "title": "Chapter 1 Mastery Exam", "description": "30 mins · 10 questions", "passPercent": 80, "questions": [ { "question": "HCF(9, 27).", "options": [ "9", "3", "27" ], "correctIndex": 0, "solution": "9 divides 27." }, { "question": "HCF(a, b) = 4, product = 400. LCM?", "options": [ "100", "400", "4" ], "correctIndex": 0, "solution": "$400/4 = 100$." }, { "question": "Smallest composite number.", "options": [ "4", "2", "6" ], "correctIndex": 0, "solution": "4." }, { "question": "$\\sqrt{3}$ is:", "options": [ "Irrational", "Rational" ], "correctIndex": 0, "solution": "Prime root." }, { "question": "HCF of two primes.", "options": [ "1", "None" ], "correctIndex": 0, "solution": "Property." }, { "question": "LCM(6, 15).", "options": [ "30", "15", "60" ], "correctIndex": 0, "solution": "$2 \\times 3, 3 \\times 5 \\Rightarrow 30$." }, { "question": "Interval meeting 10m and 15m.", "options": [ "30m", "5m" ], "correctIndex": 0, "solution": "LCM." }, { "question": "Max container size for 10L and 15L.", "options": [ "5L", "30L" ], "correctIndex": 0, "solution": "HCF." }, { "question": "$\\sqrt{2} + 5$ is irrational.", "options": [ "True", "False" ], "correctIndex": 0, "solution": "Rule." }, { "question": "Real master status?", "options": [ "Achieved", "Not yet" ], "correctIndex": 0, "solution": "Done." } ] }, "completion": { "title": "Real Master!", "message": "You have mastered the foundation of arithmetic and irrationality. Numbers have no secrets for you now!", "nextChapter": { "label": "Polynomials", "url": "/class-10-maths/chapter-2-polynomials.html" } } }