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CBSE Class 10 Mathematics Chapter 1 Real Numbers Worksheet with Answers

Q: What topics from NCERT Class 10 Mathematics Chapter 1 are covered in this worksheet?

This worksheet comprehensively covers Euclid's division lemma and algorithm, the Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, decimal expansions of rational numbers (terminating and non-terminating recurring), and proofs of irrationality for numbers like √2, √3, and √5, exactly as presented in the NCERT textbook.

Q: How much time should my child spend on this Real Numbers worksheet?

The worksheet is designed for a 90-minute timed session, mirroring board-exam conditions. Students should attempt all sections—MCQs, fill-in-the-blanks, matching, short answers, long answers, and the case study—in one sitting, then review their work against the answer key for another 20–30 minutes.

Q: Is the difficulty level of this worksheet appropriate for an average Class 10 student?

Yes. The difficulty ranges from moderate to challenging. Section A and B questions are accessible to average students who have read NCERT carefully, while Section E and the case study include HOTS problems that stretch higher achievers. This range ensures that every student finds both confidence-building and growth-promoting questions.

Q: How can I help my child if they get stuck on a proof question, like proving √5 is irrational?

Guide them to follow the standard contradiction method: assume √5 is rational (p/q in lowest terms), square both sides to get 5q²=p², deduce that 5 divides p, substitute p=5m, then show 5 divides q, reaching a contradiction. For instant, step-by-step help, upload the question to CBSETUTOR.ai's AI tutor via photo and receive a detailed walkthrough in seconds.

Q: Why does the worksheet include a case-study question on Real Numbers?

CBSE introduced case-study questions in 2021 to assess application skills. Real Numbers case studies often involve HCF-LCM problems in scheduling, resource distribution, or measurement contexts. Practising these builds reading comprehension and the ability to translate word problems into mathematical operations, both crucial for scoring full marks in board exams.

Q: What is the marking scheme for short-answer and long-answer questions in the CBSE board exam?

Short-answer questions (2–3 marks) award one mark for correct method and the remaining for accurate computation and final answer. Long-answer questions (5 marks) distribute marks across statement of theorem, logical steps, algebraic working, and conclusion. Partial credit is given even if the final answer is wrong, provided the method is sound.

Q: Can this worksheet be used for pre-board or mid-term revision?

Absolutely. The worksheet's structure mirrors the board-exam pattern and covers every key concept in Chapter 1. Students can use it as a diagnostic tool during Diwali or winter breaks, identify weak topics via the answer key, and then focus revision on those areas before pre-boards in January or February.

Q: How do I check if a rational number has a terminating decimal expansion?

First, reduce the fraction to its lowest terms. Then factorise the denominator. The decimal expansion is terminating if and only if the denominator is of the form 2ᵐ × 5ⁿ, where m and n are non-negative integers. If any other prime factor appears, the expansion is non-terminating recurring.

Q: What common mistakes do students make in Euclid's division algorithm questions?

Students often write the algorithm in reverse order, forget to show the final step when remainder becomes zero, or miscalculate the quotient and remainder. Always write a = bq + r clearly for each step, verify 0 ≤ r < b, and continue until r = 0. The last non-zero remainder is the HCF.

Q: How does CBSETUTOR.ai help with Real Numbers preparation beyond this worksheet?

CBSETUTOR.ai's AI tutor provides 24×7 doubt resolution via photo upload, offers unlimited practice problems with instant feedback, explains every NCERT exercise in detail, and costs just ₹999 per month for all classes 6–12. A 3-day free trial lets students experience personalized, on-demand learning before committing, making it an ideal supplement to worksheet practice.

Real Numbers is the opening chapter of CBSE Class 10 Mathematics and lays the foundation for algebra and number theory throughout secondary school. This printable worksheet is structured to give students focused, timed practice on Euclid's division lemma, the Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, and the decimal expansions of rational and irrational numbers. Each section builds progressively in difficulty, culminating in HOTS and case-study questions that mirror the 2025 board-exam pattern.

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Key takeaways

✓Worksheet covers all four core topics: Fundamental Theorem of Arithmetic, HCF-LCM by prime factorisation, decimal expansions, and irrational numbers as per NCERT Class 10 Mathematics syllabus.
✓Section-wise practice includes 6 MCQs, 5 fill-in-the-blanks, matching/true-false, 5 short-answer (2-3 marks), and 3 long-answer (5 marks) questions plus one case-study.
✓Complete answer key with step-by-step explanations helps students self-assess and understand common board-exam error points.
✓Suggested time: 90 minutes; difficulty: Moderate to Challenging, aligned with recent CBSE board exam trends.
✓Ideal for revision during Diwali or mid-term breaks, and for identifying weak areas before pre-board exams.
✓Parents can use CBSETUTOR.ai's 24×7 AI tutor to help children solve any stuck problem by uploading a photo—₹999/month flat fee for all classes 6-12, 3-day free trial available.

Quick Chapter Recap: NCERT Class 10 Mathematics Chapter 1 Real Numbers

Real Numbers introduces students to the rigorous properties of integers and rational numbers using Euclid's division algorithm and the Fundamental Theorem of Arithmetic. Euclid's division lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. This lemma underpins methods to compute the HCF of two numbers efficiently. The Fundamental Theorem of Arithmetic asserts that every composite number can be expressed as a product of primes in a unique way, disregarding the order of factors. This theorem is the backbone for finding HCF and LCM by prime factorisation. The chapter also explores decimal expansions: a rational number has either a terminating or a repeating (non-terminating recurring) decimal expansion, while irrational numbers have non-terminating, non-repeating decimals. Understanding these distinctions is critical for proving irrationality and for simplifying surds in later algebra chapters.

Euclid's division lemma: a = bq + r, 0 ≤ r < b, used to find HCF.
Fundamental Theorem of Arithmetic: unique prime factorisation of every composite number.
HCF is the product of the smallest powers of common prime factors; LCM is the product of the greatest powers of all prime factors.
Rational numbers yield terminating or recurring decimals; irrational numbers yield non-terminating, non-repeating decimals.
Proving irrationality often uses proof by contradiction, as shown for √2, √3, and √5 in NCERT examples.

Worksheet Overview and How to Use This Resource

This worksheet is designed for a single 90-minute sitting and mirrors the structure of CBSE board examination papers for Class 10 Mathematics. It is divided into six distinct sections: Section A tests foundational recall through multiple-choice questions; Section B uses fill-in-the-blanks to reinforce terminology and key results; Section C employs matching or true-false statements to check conceptual clarity; Section D presents five short-answer questions worth 2-3 marks each, requiring justification and intermediate steps; Section E contains three long-answer or higher-order thinking skill questions worth 5 marks each; and finally, a case-study question integrates real-world contexts with multi-part sub-questions. Students should attempt all sections sequentially without referring to the answer key, then self-assess using the detailed explanations provided at the end. Parents and teachers can use the difficulty markers to decide whether remedial practice or advanced challenge sets are needed next. For instant doubt resolution, students can upload any tricky problem to CBSETUTOR.ai's AI tutor, available 24×7 at a flat subscription of ₹999 per month for Classes 6-12, with a 3-day free trial.

Suggested time: 90 minutes in one sitting; simulate exam conditions by keeping notes and textbooks aside.
Difficulty level: Moderate to Challenging—questions range from direct NCERT exercises to board-level HOTS.
Mark distribution aligns with CBSE pattern: MCQs (1 mark each), short answers (2-3 marks), long answers (5 marks).
Use the answer key only after completing all sections; compare your steps, not just final answers.
Identify recurring mistakes (e.g. sign errors in Euclid's lemma, incorrect prime factorisation) and revise those topics from NCERT Class 10 Mathematics textbook.

Section A: Multiple Choice Questions (1 mark each)

Multiple-choice questions test your ability to recall definitions, theorems, and standard results quickly. In the CBSE 2025 board exam, MCQs carry 20 marks out of 80, so mastering this format is essential. For Real Numbers, expect questions on Euclid's lemma applications, recognising terminating versus non-terminating decimals, identifying HCF-LCM relationships, and properties of irrational numbers. Read each option carefully; eliminate obviously incorrect choices first, then apply the Fundamental Theorem of Arithmetic or perform a quick mental calculation to confirm your answer. Common traps include confusing HCF with LCM, misapplying the condition for terminating decimals (the denominator in lowest terms must be of the form 2ᵐ × 5ⁿ), and incorrectly assuming that the sum of two irrationals is always irrational. Practice these six questions under timed conditions to build speed and accuracy, as each MCQ should take no more than one minute on average during the actual board examination.

Q1. The HCF of 52 and 130 is: (a) 13 (b) 26 (c) 52 (d) 130
Q2. The decimal expansion of 17/8 is: (a) terminating (b) non-terminating recurring (c) non-terminating non-recurring (d) none
Q3. If n is any natural number, then 6n cannot end with the digit: (a) 0 (b) 2 (c) 4 (d) 6
Q4. The product of HCF and LCM of 12 and 18 is: (a) 72 (b) 108 (c) 216 (d) 324
Q5. Which of the following is irrational? (a) √4 (b) √9 (c) √16 (d) √7
Q6. Euclid's division lemma states that for positive integers a and b, there exist unique q and r such that a = bq + r and: (a) 0 < r < b (b) 0 ≤ r < b (c) 0 ≤ r ≤ b (d) 0 < r ≤ b

Section B: Fill in the Blanks (1 mark each)

Fill-in-the-blank questions reinforce precise mathematical terminology and standard forms. CBSE examiners often test whether students can state the Fundamental Theorem of Arithmetic correctly, recall the condition for terminating decimals, or complete a step in Euclid's algorithm. Write your answers in the blank spaces provided; if printed, use a pen. Avoid vague or incomplete phrases—for instance, writing 'unique factorisation' instead of 'unique prime factorisation' may lose you the mark. These five questions cover the breadth of Chapter 1, from basic definitions to algorithmic steps. After completing this section, cross-check your answers with the NCERT Class 10 Mathematics textbook glossary and theorem statements. If you consistently struggle with terminology, create flashcards for each theorem, lemma, and definition, and review them daily. Remember, precision in language translates directly to marks in board examinations, especially in sections that award one mark per blank without partial credit for nearly correct answers.

Q1. The Fundamental Theorem of Arithmetic states that every composite number can be factorised as a product of __________ in a unique way.
Q2. A rational number p/q (in lowest terms) has a terminating decimal expansion if and only if the prime factorisation of q is of the form __________.
Q3. The HCF of two numbers is 12 and their LCM is 180. If one number is 36, the other number is __________.
Q4. If a and b are coprime, then HCF(a, b) = __________.
Q5. The decimal expansion of an irrational number is __________ and __________.

Section C: Match the Following and True/False (1 mark each)

This section assesses your ability to connect statements, recognize equivalent forms, and verify the truth value of propositions. Matching questions require you to pair a theorem or property in Column A with its correct consequence or example in Column B. True-or-false statements test common misconceptions, such as whether the product of two irrationals is always irrational (false: √2 × √2 = 2, which is rational) or whether HCF × LCM always equals the product of two numbers (true only for pairs of integers). Read each statement twice before marking; board examiners often include subtle negations or qualifiers like 'always', 'sometimes', or 'never'. When matching, eliminate obvious pairs first, then use logical deduction for the remainder. If the worksheet is printed, draw neat lines or write the matched pairs clearly. For true-false, write T or F in the space provided and, if instructed, provide a one-line justification. Practising these formats builds the careful reading habits essential for scoring full marks in objective sections of the CBSE Class 10 Mathematics board paper.

Match the Following: (A) Euclid's division lemma → (i) Unique prime factorisation; (B) Fundamental Theorem of Arithmetic → (ii) a = bq + r, 0 ≤ r < b; (C) Rational number → (iii) Terminating or non-terminating recurring decimal; (D) Irrational number → (iv) Non-terminating non-recurring decimal.
True or False: (1) The sum of two irrational numbers is always irrational. (2) HCF(a, b) × LCM(a, b) = a × b for any two positive integers a and b. (3) The number 3.141141114… is rational. (4) If p is a prime and p divides a², then p divides a.

Section D: Short Answer Questions (2–3 marks each)

Short-answer questions in CBSE Class 10 Mathematics typically carry 2 or 3 marks and require you to show all intermediate steps, state the theorem or formula used, and arrive at a clear final answer. For Real Numbers, expect tasks such as using Euclid's division algorithm to find the HCF of two numbers, proving that a given number is irrational, expressing a number in prime-factorised form and then computing HCF or LCM, or determining whether a given rational fraction has a terminating or non-terminating decimal expansion. Marks are awarded for method as well as accuracy, so even if your arithmetic slips, a correct method can earn you partial credit. Always begin by writing down the relevant theorem or lemma; for example, 'By Euclid's division lemma, a = bq + r, 0 ≤ r < b' signals to the examiner that you understand the underlying principle. Show every division or factorisation step, box or underline your final answer, and double-check units or simplifications before moving on. These five questions are drawn from past board papers and NCERT exercises, representing the core skills tested every year in Chapter 1.

Q1. Use Euclid's division algorithm to find the HCF of 455 and 42. (2 marks)
Q2. Show that 5 − √3 is irrational, given that √3 is irrational. (2 marks)
Q3. Find the LCM and HCF of 12, 15 and 21 by prime factorisation method. (3 marks)
Q4. Explain why 7 × 11 × 13 + 13 is a composite number. (2 marks)
Q5. Without actual division, determine whether the rational number 64/455 will have a terminating or non-terminating decimal expansion. (2 marks)

Section E: Long Answer and HOTS Questions (5 marks each)

Long-answer questions carry 5 marks and demand multi-step reasoning, proof construction, or the synthesis of several concepts. In the CBSE board exam, these questions separate average students from high scorers. For Real Numbers, classic 5-mark problems include proving that certain square roots (√2, √3, √5) are irrational using contradiction, deriving HCF and LCM for three numbers and verifying the product relationship, or solving word problems that combine divisibility, remainders, and Euclid's lemma. Higher-order thinking skill (HOTS) questions may ask you to justify why no natural number can have its square ending in 2, 3, 7, or 8, or to construct an example demonstrating a particular property. Structure your answer in clear paragraphs: state what is given, what is to be proved, outline your approach, execute each step with justification, and conclude with a boxed final statement. Examiners award marks for logical flow, so even if you cannot reach the final answer, a well-reasoned partial solution earns significant credit. Allocate at least 10 minutes per question in this section and review your work for algebraic slips before submitting.

Q1. Prove that √5 is irrational. (5 marks)
Q2. Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of a container that can measure the petrol in each tanker an exact number of times. Use Euclid's division algorithm. (5 marks)
Q3. Show that the square of any positive integer is of the form 3m or 3m + 1 for some integer m. Hence prove that the square of any positive integer cannot be of the form 3m + 2. (5 marks)

Case-Study Question: Real-World Application of HCF and LCM (4 marks)

Case-study questions were introduced in the CBSE 2021 board exams and now appear in every Class 10 Mathematics paper. They present a real-life scenario—such as arranging students in rows and columns, scheduling buses, or distributing resources—followed by three or four sub-questions that apply chapter concepts. For Real Numbers, a typical case study involves finding the smallest or largest common measure, using HCF or LCM to solve optimization problems, or interpreting remainders in Euclid's algorithm. Each sub-question is worth 1 mark and tests a distinct skill: one may be a direct calculation, another a conceptual true-false, and a third a short justification. Read the passage carefully, underline key numerical data, and identify which quantity (HCF, LCM, or Euclid quotient/remainder) the question seeks. Write answers in the space provided; if no space is marked, write them in your answer booklet with clear sub-question labels (i, ii, iii, iv). Case studies reward careful reading and the ability to translate words into mathematical operations, so practice at least five such passages from sample papers before your board exam.

Passage: A school is planning a sports day. The organizers have 56 cricket bats and 70 footballs. They want to divide these items into the maximum number of identical kits, with each kit containing the same number of bats and footballs (and no item left over).
(i) How many kits can be made? (1 mark)
(ii) How many cricket bats will each kit contain? (1 mark)
(iii) How many footballs will each kit contain? (1 mark)
(iv) If the organizers want to pack these kits into cartons holding 7 kits each, how many cartons are needed? (1 mark)

Complete Answer Key with Step-by-Step Explanations

A robust answer key is more than a list of final answers; it provides the reasoning, intermediate steps, and common pitfalls for each question. Below, every MCQ choice is justified, every fill-in-the-blank is completed with the exact NCERT terminology, matching and true-false items include brief explanations, short answers show full working with theorem references, long answers present complete proofs or derivations, and the case-study sub-questions are solved in sequence. Use this key to self-assess: award yourself marks strictly according to the marking scheme, deduct half-marks for missing steps or incorrect justifications, and note recurring errors in a revision log. If you score below seventy percent on this worksheet, revisit the relevant NCERT Class 10 Mathematics sections, watch concept videos, and reattempt similar questions. For personalized help, CBSETUTOR.ai offers a 24×7 AI tutor that solves doubts by photo upload, explains every step in Indian English, and costs just ₹999 per month for all classes from 6 to 12, with a 3-day free trial to start. Regular practice with immediate feedback is the fastest route to board-exam confidence and top scores in Real Numbers.

Tips for Scoring Full Marks in Real Numbers

Maximizing your score in CBSE Class 10 Mathematics Chapter 1 requires more than rote learning; it demands strategic practice and attention to detail. First, memorize the exact statements of Euclid's division lemma and the Fundamental Theorem of Arithmetic as they appear in NCERT—examiners expect verbatim theorem statements in proofs. Second, practice prime factorisation daily until it becomes second nature; errors here cascade into wrong HCF and LCM. Third, for irrationality proofs, follow the standard contradiction template: assume rational p/q in lowest terms, derive that a prime divides both p and q, conclude contradiction. Fourth, in Euclid's algorithm, show every division step with remainder explicitly; partial working earns partial marks. Fifth, double-check the condition for terminating decimals: reduce the fraction fully, then verify the denominator is 2ᵐ×5ⁿ. Sixth, manage time by spending one minute per MCQ, three minutes per short answer, and ten minutes per long answer. Seventh, review past years' board papers to identify question patterns—Real Numbers questions recur with minor numerical changes. Eighth, use CBSETUTOR.ai's AI tutor to clarify doubts instantly; upload a photo of any problem, receive a step-by-step solution in seconds, and practice similar variants. Finally, simulate exam conditions weekly: print this worksheet, set a 90-minute timer, and self-grade strictly using the answer key. Consistent, timed practice is the surest path to a perfect score in Real Numbers.

Write theorem statements verbatim from NCERT to earn statement marks in proofs.
Master prime factorisation with numbers up to 1000; use factor trees for speed.
For Euclid's algorithm, show all divisions: a=bq+r lines until remainder is zero.
In terminating-decimal questions, always reduce the fraction to lowest terms first.
Allocate time: 6 min for Section A, 5 min for Section B, 5 min for Section C, 15 min for Section D, 30 min for Section E, 10 min for case study, leaving 19 min for review.
Practise at least 10 case-study passages from sample papers to build reading speed.
Use CBSETUTOR.ai for instant doubt solving—₹999/month, 3-day free trial, all classes 6-12.

How CBSETUTOR.ai Supports Your Real Numbers Preparation

Preparing for CBSE Class 10 board exams can be stressful, especially when doubts arise late at night or during weekend revision sessions. CBSETUTOR.ai offers a 24×7 AI-powered tutor that fits in your pocket, ready to solve any Real Numbers problem the moment you need help. Simply snap a photo of a worksheet question, an NCERT exercise, or a confusing proof, and the AI tutor delivers a step-by-step solution in clear Indian English within seconds. Unlike generic video lectures, CBSETUTOR.ai tailors explanations to your specific question, highlighting exactly where you went wrong and offering tips to avoid similar mistakes. The platform covers all of Class 10 Mathematics—from Real Numbers and Polynomials to Trigonometry and Statistics—at one flat fee of ₹999 per month, with no hidden charges or per-question costs. Parents appreciate the transparent pricing and the peace of mind that their child has expert help around the clock, eliminating dependency on expensive local coaching centres. Students love the instant feedback loop: ask, learn, practice, repeat. A 3-day free trial lets you test the service risk-free; most families continue because the tutor becomes an indispensable study companion. Whether you are stuck on proving √5 irrational, confused about Euclid's lemma, or need a quick check of your HCF-LCM working, CBSETUTOR.ai is there—anytime, anywhere, for every class from 6 to 12.

24×7 photo-upload doubt solving: snap, send, and receive step-by-step solutions in seconds.
Flat ₹999/month for Classes 6–12, covering all subjects including NCERT Class 10 Mathematics.
No per-question fees, no hidden charges—one price for unlimited doubt resolution.
Explanations in Indian English, aligned with CBSE terminology and NCERT textbook language.
3-day free trial available; most students continue due to instant feedback and clarity.
Ideal supplement to this worksheet—upload any question from Sections A–E for guided solutions.

Frequently asked questions

What topics from NCERT Class 10 Mathematics Chapter 1 are covered in this worksheet?+

This worksheet comprehensively covers Euclid's division lemma and algorithm, the Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, decimal expansions of rational numbers (terminating and non-terminating recurring), and proofs of irrationality for numbers like √2, √3, and √5, exactly as presented in the NCERT textbook.

How much time should my child spend on this Real Numbers worksheet?+

The worksheet is designed for a 90-minute timed session, mirroring board-exam conditions. Students should attempt all sections—MCQs, fill-in-the-blanks, matching, short answers, long answers, and the case study—in one sitting, then review their work against the answer key for another 20–30 minutes.

Is the difficulty level of this worksheet appropriate for an average Class 10 student?+

Yes. The difficulty ranges from moderate to challenging. Section A and B questions are accessible to average students who have read NCERT carefully, while Section E and the case study include HOTS problems that stretch higher achievers. This range ensures that every student finds both confidence-building and growth-promoting questions.

How can I help my child if they get stuck on a proof question, like proving √5 is irrational?+

Guide them to follow the standard contradiction method: assume √5 is rational (p/q in lowest terms), square both sides to get 5q²=p², deduce that 5 divides p, substitute p=5m, then show 5 divides q, reaching a contradiction. For instant, step-by-step help, upload the question to CBSETUTOR.ai's AI tutor via photo and receive a detailed walkthrough in seconds.

Why does the worksheet include a case-study question on Real Numbers?+

CBSE introduced case-study questions in 2021 to assess application skills. Real Numbers case studies often involve HCF-LCM problems in scheduling, resource distribution, or measurement contexts. Practising these builds reading comprehension and the ability to translate word problems into mathematical operations, both crucial for scoring full marks in board exams.

What is the marking scheme for short-answer and long-answer questions in the CBSE board exam?+

Short-answer questions (2–3 marks) award one mark for correct method and the remaining for accurate computation and final answer. Long-answer questions (5 marks) distribute marks across statement of theorem, logical steps, algebraic working, and conclusion. Partial credit is given even if the final answer is wrong, provided the method is sound.

Can this worksheet be used for pre-board or mid-term revision?+

Absolutely. The worksheet's structure mirrors the board-exam pattern and covers every key concept in Chapter 1. Students can use it as a diagnostic tool during Diwali or winter breaks, identify weak topics via the answer key, and then focus revision on those areas before pre-boards in January or February.

How do I check if a rational number has a terminating decimal expansion?+

First, reduce the fraction to its lowest terms. Then factorise the denominator. The decimal expansion is terminating if and only if the denominator is of the form 2ᵐ × 5ⁿ, where m and n are non-negative integers. If any other prime factor appears, the expansion is non-terminating recurring.

What common mistakes do students make in Euclid's division algorithm questions?+

Students often write the algorithm in reverse order, forget to show the final step when remainder becomes zero, or miscalculate the quotient and remainder. Always write a = bq + r clearly for each step, verify 0 ≤ r < b, and continue until r = 0. The last non-zero remainder is the HCF.

How does CBSETUTOR.ai help with Real Numbers preparation beyond this worksheet?+

CBSETUTOR.ai's AI tutor provides 24×7 doubt resolution via photo upload, offers unlimited practice problems with instant feedback, explains every NCERT exercise in detail, and costs just ₹999 per month for all classes 6–12. A 3-day free trial lets students experience personalized, on-demand learning before committing, making it an ideal supplement to worksheet practice.

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