Important Questions: CBSE Class 10 Mathematics Chapter 1 Real Numbers

Real Numbers appears as Chapter 1 in the NCERT Class 10 Mathematics textbook and typically contributes 6 marks to the total 80-mark theory paper. The 2024 and 2025 CBSE sample papers consistently allocated these marks as one 1-mark MCQ, one or two 2-mark questions, and one 3-mark question from this chapter. The chapter builds on Class 9 concepts of rational and irrational numbers, introducing Euclid's division algorithm, the Fundamental Theorem of Arithmetic, and systematic methods for finding HCF and LCM through prime factorisation. Questions test both procedural fluency in calculations and conceptual understanding of why certain numbers behave as they do. The chapter also introduces proof techniques, particularly proof by contradiction for establishing irrationality, which becomes important for higher mathematics. Since 2023, CBSE has occasionally included case-based questions where Real Numbers concepts are applied to real-world scenarios like cryptography basics, quality control in manufacturing involving divisibility, or financial calculations requiring LCM and HCF. Understanding decimal expansions of rational numbers and their connection to prime factors of denominators is another favourite topic for 2-mark questions. Students should allocate approximately 8-10 days to master this chapter, ensuring they can solve HCF-LCM problems within 3 minutes and irrationality proofs within 4 minutes during the board exam.

The 1-mark question from Real Numbers in the CBSE board exam invariably appears as a multiple-choice question in Section A. These MCQs test quick recall and conceptual clarity rather than lengthy calculations. Common topics include identifying rational or irrational numbers, recognising prime or composite numbers, understanding properties of HCF and LCM, determining the nature of decimal expansions, and applying the Fundamental Theorem of Arithmetic. Students often lose this easy mark through careless reading or confusion between HCF and LCM properties. Remember that for any two positive integers a and b, the relationship HCF(a,b) × LCM(a,b) = a × b always holds and is frequently tested. Questions may present a statement and ask for its validity, or provide a calculation result and ask which concept it demonstrates. The key to scoring full marks in 1-mark MCQs is reading all four options carefully before selecting, as CBSE often includes distractors that are partially correct or represent common misconceptions. Practice at least 50 MCQs from this chapter to build speed and accuracy. These questions should take no more than 30 seconds each during the actual exam, leaving you maximum time for higher-value questions.

Two-mark questions from Real Numbers require clear working and arrive at a specific numerical answer or complete a short proof. CBSE marking schemes award 1 mark for correct method and 1 mark for correct final answer, so even if your final answer has a calculation error, you can secure partial marks by showing proper steps. Typical 2-mark questions include finding HCF or LCM of two or three numbers using prime factorisation, proving a specific number is irrational, determining whether a given decimal expansion is terminating or non-terminating, expressing a number as a product of prime factors, or applying Euclid's division lemma to find HCF. When solving HCF-LCM problems, always write the prime factorisation explicitly rather than doing mental calculations, as this demonstrates your method to the examiner. For irrationality proofs, state your assumption clearly, derive a contradiction, and conclude precisely. Avoid vague statements like 'this is not possible' without explaining why it contradicts the assumption. Time management is crucial: allocate no more than 3 minutes per 2-mark question. If you are stuck beyond 4 minutes, move on and return later. Practice writing concise solutions that include all necessary steps without excessive elaboration. Remember that neatness and logical flow matter as much as correctness when partial marking is involved.

This section presents five carefully selected 2-mark questions that mirror the style and difficulty level of actual CBSE board exam papers from 2020-2024. Each question tests a distinct concept from the Real Numbers chapter, ensuring comprehensive coverage. When practising these questions, time yourself strictly at 3 minutes per question to simulate exam conditions. Write your solutions on paper rather than solving mentally, as this builds the muscle memory needed for the board exam. Pay special attention to questions involving proofs, as these require precise mathematical language that many students neglect during practice. After attempting all five questions, compare your solutions with the model answers provided here, checking not just for correctness but for clarity and completeness of explanation. If you find yourself consistently exceeding the 3-minute limit, identify which step is slowing you down: is it the prime factorisation, the logical reasoning, or the final calculation? Targeted practice on that specific skill will improve your overall speed. Remember that in the board exam, these 2-mark questions appear in Section B, and you will typically encounter two such questions from Chapter 1 Real Numbers. Scoring full marks on these questions is entirely achievable with systematic practice and attention to presentation.

Three-mark questions from Real Numbers constitute the longest questions you will face from this chapter in the CBSE Class 10 board exam. These questions demand multi-step reasoning, clear logical flow, and often combine two concepts. Typical 3-mark questions include proving irrationality of expressions like 5 - √3 or 1/√2, finding HCF and LCM of three numbers using prime factorisation, application problems requiring both HCF and LCM concepts, questions based on Euclid's division lemma with explanation, or proving divisibility properties using the Fundamental Theorem of Arithmetic. The CBSE marking scheme for 3-mark questions typically allocates 1 mark for correct method or initial steps, 1 mark for intermediate working or key insight, and 1 mark for final answer or conclusion. This means partial marks are very much available even if you cannot complete the entire solution. When attempting 3-mark questions, always write clear sub-headings or step numbers to help the examiner follow your logic and award partial marks easily. For proof questions, ensure you state what you are proving, what you are assuming, where the contradiction arises, and what you conclude. Application problems require you to first identify whether the situation needs HCF or LCM or both, then solve systematically. Allocate 5-6 minutes per 3-mark question during the exam. If you are running short on time, write the method and key steps even if you cannot complete calculations, as this secures at least 1-2 marks.

Here are five high-quality 3-mark questions that test deeper understanding of Real Numbers concepts. These questions represent the most challenging items you will encounter from this chapter in the board exam, often appearing as Question 18 or 19 in Section C. When practising these questions, focus not just on reaching the correct answer but on presenting your solution in a structured, examiner-friendly format. Use clear mathematical notation, define any variables you introduce, and ensure every step follows logically from the previous one. Many students lose marks on 3-mark questions not because their mathematics is wrong but because their presentation is unclear or they skip crucial justification steps. For instance, when proving irrationality, you must explicitly state why a certain conclusion contradicts rationality rather than assuming the examiner will fill in the gaps. Similarly, when using the Fundamental Theorem of Arithmetic, mention the theorem by name to signal to the examiner that you understand which mathematical principle you are applying. After attempting these questions, review the model answers carefully, paying attention to the exact phrasing and structure used. Try to internalise this pattern so that during the exam, your hand automatically produces well-structured solutions. Remember that presentation clarity can be the difference between scoring 2 out of 3 and securing full marks on these questions.

While Chapter 1 Real Numbers traditionally does not contribute full 5-mark questions independently, from 2023 onwards CBSE has introduced case-based or integrated questions worth 4-5 marks that may draw upon Real Numbers concepts alongside other chapters. These questions present a real-world scenario or a passage followed by 3-4 sub-questions. For instance, a case study might describe a manufacturing process requiring parts of certain lengths to be cut from standard rods, and ask students to find the maximum length that can be cut equally from multiple rod lengths using HCF, calculate total number of pieces using LCM, and prove some property using number theory. Another common format involves cryptography basics where prime factorisation and properties of numbers are applied. When attempting case-based questions, read the entire passage carefully before attempting any sub-question, as information needed for later parts is often embedded early in the scenario. Answer all sub-questions even if you are unsure, as these are typically structured with increasing difficulty, and the first 1-2 parts are usually straightforward. The final sub-question often requires synthesis of concepts or a proof, carrying 2-3 marks. Allocate 7-8 minutes total for a complete case-based question with 4 parts. Even though Real Numbers alone rarely generates a full case study, its concepts frequently appear as one component within a larger problem, so mastery of HCF-LCM and irrationality proofs remains essential for scoring well across the entire paper.

This final question bank provides five additional exam-style questions spanning all mark categories and difficulty levels. These questions are designed to fill gaps in the previous question sets and ensure comprehensive coverage of every NCERT topic within Real Numbers. Question 14 tests understanding of when a decimal expansion terminates, a concept that appears regularly in board exams. Question 15 examines Euclid's division lemma in a pure number theory context, requiring students to demonstrate divisibility properties. Question 16 is a 2-mark application problem requiring LCM, common in questions about events repeating simultaneously. Question 17 challenges students with a slightly complex irrationality proof involving a fraction. Question 18 combines HCF and word problem application, testing whether students can translate a practical scenario into mathematical operations. When practising these, simulate exam conditions strictly: no textbook, no calculator beyond what you are allowed in the exam, strict time limits, and no erasing or restarting if you make an error halfway through. Instead, strike through incorrect work and continue, as you would in the actual exam. This builds the resilience and time management skills essential for board exam success. After completing all 18 questions in this resource, attempt previous years' CBSE question papers for Chapter 1, and you will find your speed and accuracy have improved significantly. Remember that Real Numbers questions reward precision and clear communication as much as mathematical knowledge.

Understanding the patterns and preferences of CBSE question setters can significantly boost your preparation efficiency for Chapter 1 Real Numbers. Analysis of the last ten years of board papers reveals several consistent trends. First, approximately 70% of questions involve prime factorisation in some form, whether directly asking for HCF-LCM or embedding it within proofs and applications. CBSE prefers numbers that factorise into 2, 3, 5, 7, 11 and occasionally 13, avoiding very large primes that make mental calculation difficult. Second, irrationality proofs almost always involve √2, √3, √5 or simple expressions combining these with integers. You will rarely if ever see √7 or higher roots. Third, application problems are deliberately worded to test reading comprehension as much as mathematics: the words 'largest', 'smallest', 'maximum', 'minimum', 'exactly' are clues indicating HCF or LCM. Fourth, CBSE increasingly favours questions that require students to verify their results, testing both forward calculation and reverse checking. Fifth, the Fundamental Theorem of Arithmetic is tested indirectly through questions about uniqueness of prime factorisation or divisibility properties rather than asking students to state the theorem. Sixth, decimal expansion questions always require simplification of the fraction first before examining the denominator, testing alertness and not just formula application. Finally, since 2021, CBSE has moved away from purely theoretical questions towards those with a thin application layer, even if artificial, so expect even proof questions to be situated in a context. Practising with an awareness of these patterns allows you to predict question types and prepare targeted responses.

Even well-prepared students lose marks on Real Numbers questions through a handful of recurring errors. First and most frequent: confusing HCF with LCM. Remember HCF is the highest common factor (smaller number) used when you need to divide or break things evenly into maximum-sized groups, while LCM is the least common multiple (larger number) used when you need things to align or repeat together. Second major error: incorrect prime factorisation, especially missing a factor or writing 2×2×3 = 12 but forgetting to write it as 2²×3. Always express factorisation in exponential form clearly. Third: incomplete proofs of irrationality. Students often write 'Let √2 = a/b' and jump straight to 'this is irrational' without showing the contradiction. You must derive that both a and b are even, which contradicts coprimality. Fourth: misunderstanding decimal expansion rules. After simplifying a fraction p/q to lowest terms, the decimal terminates if and only if q = 2^m × 5^n. Students often check the original denominator before simplification, leading to wrong conclusions. Fifth: Euclid's algorithm errors where students stop one step early or make arithmetic mistakes in division. Show each step explicitly: dividend = divisor × quotient + remainder. Sixth: in application problems, answering with the wrong number. If the question asks for total items, do not stop after finding HCF or LCM but complete the calculation to the final required quantity. Seventh: poor time management, spending 6-7 minutes on a 2-mark question. Practice with a timer to build speed. Eighth: not reading whether the question says 'find' or 'prove' and giving the wrong response format. These errors are entirely preventable with conscious attention during practice and paper-solving.

While practising these important questions systematically builds exam readiness, many students find themselves stuck on specific steps or unable to identify exactly where their proof goes wrong. This is where CBSETUTOR.ai becomes invaluable as your 24×7 personal mathematics tutor. Unlike static solution PDFs, CBSETUTOR.ai allows you to photograph any Real Numbers problem you are struggling with, whether from your NCERT textbook, reference book, or previous year papers, and upload it directly. Within seconds, the AI tutor provides a step-by-step solution tailored to CBSE marking schemes, explaining not just what to do but why each step follows. For Chapter 1 specifically, students benefit immensely from the platform's ability to break down irrationality proofs into granular logical steps, making the proof structure crystal clear. The AI tutor also catches common errors in your working: if you confuse HCF and LCM, it explains the conceptual difference immediately. If your prime factorisation is incomplete, it identifies the missing factor and explains how to systematically factorise any number. The platform covers Classes 6 through 12, making it valuable beyond just Class 10, as your younger sibling can use the same subscription for Class 8 mathematics or you can use it next year for Class 11. At ₹999 per month for unlimited questions across all subjects and classes, with a 3-day free trial to test the platform risk-free, CBSETUTOR.ai offers exceptional value for families serious about academic excellence. The photo-upload feature is particularly powerful during homework or self-study sessions when parents may not remember Class 10 proofs and coaching classes are closed. Real Numbers becomes significantly less intimidating when you have an expert available anytime you are studying, whether that is early morning revision or late-night doubt clearing before the exam.