Chapter 1: Real Numbers
CBSE Official PYQs (2015-2026)
Q1. The LCM of 960 and 240 is :
Q2. If two positive integers \(a\) and \(b\) are written as \(a = x^3 \times y^2\) and \(b = x \times y^3\), where \(x, y\) are prime numbers, then HCF(a, b) is :
Q3. If two positive integers \(a\) and \(b\) are written as \(a = x^3 \times y^2\) and \(b = x \times y^3\), where \(x, y\) are prime numbers, then LCM(a, b) is :
Q4. The sum of exponents of prime factors in the prime-factorisation of 196 is :
Q5. If HCF(2520, 6600) = 40 and LCM(2520, 6600) = \(252 \times k\), then the value of \(k\) is :
Q6. Two positive numbers are in the ratio 2 : 3. If their LCM is 180, then their HCF is :
Q7. Let \(p\) be a prime number and \(k\) be a positive integer. If \(p\) divides \(k^2\), then which of the following is definitely true?
Q8. If \(p\) is a prime number, then \(\sqrt{p}\) is always :
Q9. The decimal expansion of \(\frac{23}{2^2 \times 5^3}\) terminates after how many decimal places?
Q10. The decimal expansion of the rational number \(\frac{14587}{1250}\) terminates after how many decimal places?
Q11. If \(p_1\) and \(p_2\) are two odd prime numbers such that \(p_1 > p_2\), then \(p_1^2 - p_2^2\) is always :
Q12. If the HCF of 65 and 117 is expressible in the form \(65m - 117\), then the value of \(m\) is :
Q13. The largest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is :
Q14. If two positive integers \(p\) and \(q\) are written as \(p = a \times b^2\) and \(q = a^3 \times b\), where \(a, b\) are prime numbers, then HCF(p, q) is :
Q15. If two positive integers \(p\) and \(q\) are written as \(p = a \times b^2\) and \(q = a^3 \times b\), where \(a, b\) are prime numbers, then LCM(p, q) is :
Q16. The LCM of the smallest two-digit composite number and the smallest composite number is :
Q17. The HCF of the smallest prime number and the smallest composite number is :
Q18. The product of a non-zero rational and an irrational number is always :
Q19. The exponent of 2 in the prime factorisation of 144 is :
Q20. The HCF of 135 and 225 is :
Q21. If \(a = 2^2 \times 3^3 \times 5^4\) and \(b = 2^3 \times 3^2 \times 5\), then HCF(a, b) is equal to :
Q22. The LCM of 12, 15 and 21 is :
Q23. The product of two numbers is 1600 and their HCF is 5. The LCM of the numbers is :
Q24. If \(p\) is a prime number, then HCF of \(p\) and \(p + 1\) is :
Q25. The HCF of two co-prime numbers \(a\) and \(b\) is :
Q26. If LCM(x, 18) = 36 and HCF(x, 18) = 2, then the value of \(x\) is :
Q27. The number \(\pi\) (3.14159...) is a/an :
Q28. The number \(3.\overline{27}\) is :
Q29. The LCM of three numbers \(2^3 \times 3\), \(2 \times 3 \times 5\), and \(3 \times 5 \times 7\) is :
Q30. If HCF(a, b) = 1, then the numbers \(a\) and \(b\) are called :
Q31. Prove that \(5 - \sqrt{3}\) is an irrational number.
Q32. Prove that \(3 + 2\sqrt{5}\) is an irrational number, given that \(\sqrt{5}\) is irrational.
Q33. Find the HCF and LCM of 306 and 657.
Q34. Express 5005 as a product of its prime factors.
Q35. Find HCF of 96 and 404 by prime factorisation method. Hence, find their LCM.
Q36. If HCF of 144 and 180 is expressed in the form \(13m - 16\), find the value of \(m\).
Q37. Check whether \(6^n\) can end with the digit 0 for any natural number \(n\).
Q38. Check whether \(4^n\) can end with the digit 0 for any natural number \(n\).
Q39. Find the HCF of 612 and 1314 using prime factorisation.
Q40. Explain why \(7 \times 11 \times 13 + 13\) is a composite number.
Q41. Find LCM and HCF of 120 and 144 using the Fundamental Theorem of Arithmetic.
Q42. Given that \(\sqrt{2}\) is irrational, prove that \(5 + 3\sqrt{2}\) is an irrational number.
Q43. Prove that \(\sqrt{3}\) is an irrational number.
Q44. Prove that \(\sqrt{5}\) is an irrational number.
Q45. Prove that \(\sqrt{2}\) is an irrational number.
Q46. Prove that \(\sqrt{7}\) is an irrational number.
Q47. Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times.
Q48. A sweet shopkeeper prepares 396 gulab jamuns and 342 rasgullas. He packs them into containers. Each container consists of either gulab jamuns or rasgullas but must have equal number of pieces. Find the minimum number of containers required.
Q49. In a seminar, the number of participants in Hindi, English, and Mathematics are 60, 84, and 108 respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.
Q50. Find the largest number which divides 318 and 739 leaving remainders 3 and 4 respectively.
Q51. Three bells toll together at intervals of 9, 12, and 15 minutes respectively. If they toll together now, answer the following:
(i) After how many hours will they toll together next?
(ii) How many times will they toll together in 36 hours (excluding the start)?
Q52. To enhance the reading habits of Class X students, a school manages a library. There are two sections - Section A and Section B. Section A has 32 students and Section B has 36 students. Answer the following:
(i) Find the minimum number of books required for their class library so that they can be distributed equally among students of Section A or Section B.
(ii) If the product of two positive integers is equal to the product of their HCF and LCM, find HCF(32, 36).
Q53. A circular track is around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. Answer the following:
(i) After how many minutes will they meet again at the starting point?
(ii) If another cyclist, Amit, takes 15 minutes to complete one round, after how many minutes will all three meet together at the starting point?
Q54. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. Answer the following:
(i) What is the maximum number of columns in which they can march?
(ii) What is the fundamental theorem of arithmetic statement applied here?
Q55. Prove that the square of any positive integer cannot be of the form \(5m + 2\) or \(5m + 3\) for any integer \(m\).