Class 10 Maths Chapter 1 Real Numbers PYQ

Q1. If (−1) + (−1) = 0, then n is: (a) any positive integer (b) any negative integer (c) any odd number (d) any even numberCBSE 2025 | 1M

Ans: (c) any odd number [CBSE 2025 | 1 Mark] n

Q2. Which of the following cannot be the unit digit of 8 , where n is a natural number? (a) 4 (b) 2 (c) 0 (d) 6CBSE 2025 | 1M

Ans: (c) 0 [CBSE 2025 | 1 Mark]

Q3. If x is the LCM of 4, 6, 8 and y is the LCM of 3, 5, 7 and p is the LCM of x and y, then which is true? (a) p = 35x (b) p = 4y (c) p = 8x (d) p = 16yCBSE 2025 | 1M

Ans: (a) p = 35x UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com Amitesh Nagar, Indore (M.P.) [CBSE 2025 | 1 Mark]

Q4. If HCF(98, 28) = m and LCM(98, 28) = n, then the value of n − 7m is: (a) 0 (b) 28 (c) 98 (d) 198CBSE 2025 | 1M

Ans: (c) 98 [CBSE 2025 | 1 Mark]

Q5. The greatest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is: (a) 13 (b) 65 (c) 875 (d) 1750CBSE 2025 | 1M

Ans: (a) 13 [CBSE 2025 | 1 Mark]

Q6. A rational number between √3 and √5 is: (a) 1.4142387954012... (b) 1.7320508... (c) π (d) 1.857142CBSE 2025 | 1M

Ans: (d) 1.857142 [CBSE 2024 | 1 Mark]

Q7. The smallest irrational number by which √20 should be multiplied so as to get a rational number, is: (a) √20 (b) √2 (c) 5 (d) √5CBSE 2024 | 1M

Ans: (d) √5 [CBSE 2024 | 1 Mark]

Q8. The LCM of two prime numbers p and q (p > q) is 221. Then the value of 3p − q is: (a) 4 (b) 28 (c) 38 (d) 48CBSE 2024 | 1M

Ans: (c) 38 [CBSE 2024 | 1 Mark]

Q9. A pair of irrational numbers whose product is a rational number is: (a) (√16, √4) (b) (√5, √2) (c) (√3, √27) (d) (√36, √2)CBSE 2024 | 1M

Ans: (c) (√3, √27) UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com Amitesh Nagar, Indore (M.P.) [CBSE 2024 | 1 Mark]

Q10. Given HCF(2520, 6600) = 40, LCM(2520, 6600) = 252 × k, then k is: (a) 1650 (b) 1600 (c) 165 (d) 1625CBSE 2024 | 1M

Ans: (a) 1650 [CBSE 2024 | 1 Mark]

Q11. If p = 18a²b and q = 20a³b² (a, b are prime numbers), then LCM(p, q) is: (a) 2a²b² (b) 180a²b² (c) 12a²b² (d) 180a³b²CBSE 2024 | 1M

Ans: (d) 180a³b² [CBSE 2024 | 1 Mark]

Q12. LCM(850, 500) is: (a) 850 × 50 (b) 17 × 500 (c) 17 × 5² × 2² (d) 17 × 5³ × 2CBSE 2024 | 1M

Ans: (b) 17 × 500 = 8500 [CBSE 2023 | 1 Mark]

Q13. The ratio of HCF to LCM of the least composite number and the least prime number is: (a) 1 : 2 (b) 2 : 1 (c) 1 : 1 (d) 1 : 3CBSE 2023 | 1M

Ans: (a) 1 : 2 [CBSE 2022 | 1 Mark]

Q14. Two positive numbers have their HCF as 12 and their product as 6336. The number of pairs possible is: (a) 2 (b) 3 (c) 4 (d) 1CBSE 2022 | 1M

Ans: (a) 2 [CBSE 2022 | 1 Mark]

Q15. The number 385 can be expressed as the product of prime factors as: (a) 5 × 11 × 13 (b) 5 × 7 × 11 (c) 5 × 7 × 13 (d) 5 × 11 × 17CBSE 2022 | 1M

Ans: (b) 5 × 7 × 11 UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com Amitesh Nagar, Indore (M.P.) [CBSE 2020 | 1 Mark]

Q16. The HCF and the LCM of 12, 21 and 15 respectively, are: (a) 3, 140 (b) 12, 420 (c) 3, 420 (d) 420, 3CBSE 2020 | 1M

Ans: (c) 3, 420 Assertion-Reason Questions (1 Mark) [CBSE 2025 | 1 Mark]

Q17. Assertion (A): For any two prime numbers p and q, their HCF is 1 and LCM is p + q. Reason (R): For any two natural numbers, HCF × LCM = product of numbers. (a) Both A and R true, R is correct explanation of A (b) Both A and R true, R is not correct explanation of A (c) A is true, R is false (d) A is false, R is trueCBSE 2025 | 1M

Ans: (d) A is false, R is true 2 Mark Questions (SA-I) [CBSE 2025 | 2 Marks]

Q18. Find the smallest number that is divisible by both 644 and 462.CBSE 2025 | 2M

Ans: LCM(644, 462) = 21252 [CBSE 2025 | 2 Marks]

Q19. Two numbers are in the ratio 4 : 5 and their HCF is 11. Find the LCM of these numbers.CBSE 2025 | 2M

Ans: Numbers = 44 and 55. LCM = 220. [CBSE 2024 | 2 Marks]

Q20. Show that 11 × 19 × 23 + 3 × 11 is not a prime number.CBSE 2024 | 2M

Ans: = 11(19 × 23 + 3) = 11 × 440. Has factors other than 1 and itself, hence not prime. [CBSE 2024 | 2 Marks]

Q21. Show that 5 × 11 × 17 + 3 × 11 is a composite number.CBSE 2024 | 2M

Ans: = 11(85 + 3) = 11 × 88 = 2³ × 11². Has more than two factors, hence composite. [CBSE 2020 | 2 Marks]

Q22. The LCM of two numbers is 182 and their HCF is 13. If one number is 26, find the other.CBSE 2020 | 2M

Ans: Other number = (13 × 182) / 26 = 91 UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com Amitesh Nagar, Indore (M.P.) [CBSE 2021 | 2 Marks]

Q23. Explain why 2 × 3 × 5 + 5 and 5 × 7 × 11 + 7 × 5 are composite numbers.CBSE 2021 | 2M

Ans: 2 × 3 × 5 + 5 = 5 × 7 = 35 (composite). 5 × 7 × 11 + 7 × 5 = 5 × 7 × 12 = 420 (composite). [CBSE 2023 | 2 Marks]

Q24. Find the least number which when divided by 12, 16, and 24 leaves remainder 7 in each case.CBSE 2023 | 2M

Ans: LCM(12, 16, 24) = 48. Required number = 48 + 7 = 55. 3 Mark Questions (SA-II) [CBSE 2023 | 3 Marks]

Q25. Prove that √3 is an irrational number.CBSE 2023 | 3M

Ans: Assume √3 = a/b (co-prime). Then a² = 3b² ⇒ 3|a. Let a = 3c, then b² = 3c² ⇒ 3|b. Contradicts co-prime. Hence √3 is irrational. [CBSE 2024 | 3 Marks]

Q26. Prove that 6 − 4√5 is an irrational number, given that √5 is irrational.CBSE 2024 | 3M

Ans: Assume rational. Then √5 = (6b − a)/(4b) = rational. Contradiction. Hence irrational. [CBSE 2024 | 3 Marks]

Q27. Prove that 5 − 2√3 is an irrational number, given that √3 is irrational.CBSE 2024 | 3M

Ans: Assume rational. Then √3 = (5b − a)/(2b) = rational. Contradiction. Hence irrational. [CBSE 2025 | 3 Marks]

Q28. Prove that 3 + 2√5 is irrational, given that √5 is irrational.CBSE 2025 | 3M

Ans: Assume rational = a/b. Then √5 = (a − 3b)/(2b) = rational. Contradiction. Hence irrational. [CBSE 2025 | 3 Marks]

Q29. Prove that 2 − √3/5 is an irrational number, given that √3 is irrational.CBSE 2025 | 3M

Ans: Assume rational = a/b. Then √3 = 5(2b − a)/b = rational. Contradiction. Hence irrational. Case Study / 4-5 Mark Questions [CBSE 2024 | 4 Marks]

Q30. Case Study: Ms. Mukta announced the number 2 and asked students to multiply it by a prime number and pass it on. The last student got 173250. (A) What is the least prime number used by students? [1] (B) How many students are in the class? [1] (C) Which prime number has been used maximum times?CBSE 2024 | 5M

Ans: 173250 = 2 × 3² × 5³ × 7 × 11. (A) Least prime by students = 3. (B) 7 students. (C) 5 used 3 times (maximum). UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com Amitesh Nagar, Indore (M.P.) [CBSE 2025 | 4 Marks]

Q31. Three sets of Physics, Chemistry and Mathematics books have to be stacked such that all books are stored subject-wise and height of each stack is same. Physics = 144, Chemistry = 180, Maths = 192. Find number of stacks of each subject.CBSE 2025 | 4M

Ans: HCF(144, 180, 192) = 12. Physics = 12 stacks, Chemistry = 15 stacks, Maths = 16 stacks. [CBSE 2024 | 3 Marks]

Q32. In a teachers’ workshop, teachers of French = 48, Hindi = 80, English = 144. Find minimum rooms required if same number of teachers per room, all of same subject.CBSE 2024 | 3M

Ans: HCF(48, 80, 144) = 16. Total rooms = 3 + 5 + 9 = 17. [CBSE 2025 | 5 Marks]

Q33. Let p, q, r be three distinct prime numbers. Check whether p·q·r + q is composite. Give examples: (i) p·q·r + 1 is composite (ii) p·q·r + 1 is prime.CBSE 2025 | 5M

Ans: pqr + q = q(pr + 1) — composite. (i) p=3, q=5, r=7: 106 (composite). (ii) p=2, q=3, r=5: 31 (prime). UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com Amitesh Nagar, Indore (M.P.) CHAPTER SUMMARY: PYQ Analysis (As per CBSE 2025-26 Syllabus | Euclid's Division Lemma Excluded) Topic Years Asked Frequency Marks HCF & LCM by Prime Factorisation 2020, 2022, 2023, 2024, 2025 10+ 1–3 Fundamental Theorem of Arithmetic 2022, 2024, 2025 5+ 1–2 Composite/Prime Number Proof 2021, 2024, 2025 4+ 2 Irrational Number Proof 2023, 2024, 2025 6+ 3 HCF × LCM = Product Property 2020, 2024, 2025 4+ 1–2 Word Problems (HCF/LCM) 2024, 2025 3+ 3–5 Assertion-Reason (HCF/LCM) 2024, 2025 2+ 1 Case Study (Prime Factorisation) 2024, 2025 2+ 4 Key Observations: • HCF and LCM by prime factorisation is the most frequently asked topic — appears every year. • Irrationality proofs (√2, √3, √5 and expressions like 3+2√5) are asked as 3-mark questions consistently. • Case study questions on prime factorisation are a newer pattern (2024–2025). • Assertion-Reason on HCF/LCM properties is a recent trend. • Euclid's Division Lemma is DELETED from 2025-26 syllabus — no questions from this topic. • Expected marks from this chapter: 5–6 marks. "Practice makes perfect. Solve PYQs to master your Board Exam!" Best Wishes for Your Board Exam! Visit: www.uniquestudyonline.com Download Our App: Search "Unique Study Point" on Google Play Store UNIQUE STUDY POINT | Amitesh Nagar, Indore (M.P.) | www.uniquestudyonline.com