Vardaan Learning Institute

Chapter Practice Sheet: Real Numbers

Class: 10 (CBSE) Subject: Mathematics Max. Marks: 50
SECTION A: OBJECTIVE TYPE QUESTIONS (1 Mark Each)
1. The HCF of 135 and 225 is:
  1. 35
  2. 45
  3. 55
  4. 65
[1]
2. Which of the following is an irrational number?
  1. $\sqrt{4}$
  2. $3.141414...$
  3. $\pi$
  4. $0.375$
[1]
3. The product of a non-zero rational and an irrational number is:
  1. Always irrational
  2. Always rational
  3. Rational or irrational
  4. One
[1]
4. The LCM of the smallest prime number and the smallest composite number is:
  1. 2
  2. 4
  3. 6
  4. 8
[1]
5. If HCF(a, b) = 12 and $a \times b = 1800$, then LCM(a, b) is:
  1. 3600
  2. 900
  3. 150
  4. 1500
[1]
6. The decimal expansion of the rational number $\frac{14587}{1250}$ will terminate after:
  1. One decimal place
  2. Two decimal places
  3. Three decimal places
  4. Four decimal places
[1]
7. If LCM(x, 18) = 36 and HCF(x, 18) = 2, then x is:
  1. 2
  2. 3
  3. 4
  4. 1
[1]
8. The ratio of LCM and HCF of the least composite and the least prime numbers is:
  1. 1:2
  2. 2:1
  3. 1:1
  4. 1:3
[1]
9. $\sqrt{p}$ is an irrational number if p is:
  1. A square of a prime
  2. A prime number
  3. An even number
  4. An odd number
[1]
10. Assertion (A): The number $5^n$ cannot end with the digit 0, where n is a natural number.
Reason (R): Prime factorisation of 5 has only two factors, 1 and 5.
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
[1]
SECTION B: SHORT ANSWER TYPE QUESTIONS (2 Marks Each)
11. Using prime factorization, find the HCF and LCM of 72 and 120.
[2]
12. Check whether $6^n$ can end with the digit 0 for any natural number $n$.
[2]
13. Explain why $7 \times 11 \times 13 + 13$ is a composite number.
[2]
14. Prove that $3 + \sqrt{2}$ is irrational, given that $\sqrt{2}$ is irrational.
[2]
SECTION C: SHORT ANSWER TYPE II QUESTIONS (3 Marks Each)
15. Prove that $\sqrt{5}$ is an irrational number.
[3]
16. Find the largest number that divides 245 and 1029 leaving remainder 5 in each case.
[3]
17. 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. What is the maximum number of columns in which they can march?
[3]
18. Three bells ring at intervals of 4, 7 and 14 minutes. All three rang at 6 AM. When will they ring together again?
[3]
19. Find HCF and LCM of 404 and 96 and verify that HCF $\times$ LCM = Product of the two numbers.
[3]
20. Prove that $5 - \sqrt{3}$ is an irrational number.
[3]
SECTION D: LONG ANSWER TYPE QUESTIONS (5 Marks Each)
21. Prove that $\sqrt{2} + \sqrt{5}$ is irrational.
[5]
22. A sweets seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of that can be placed in each stack for this purpose?
[5]
SECTION E: CASE STUDY (4 Marks)
23. Case Study: Seminar Hall
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.

(i) What is the maximum number of participants that can be seated in each room? (1 Mark)
(ii) What is the LCM of 60, 84 and 108? (1 Mark)
(iii) Find the total number of rooms required. (2 Marks)
[4]