Important Question of Class 10 CBSE ch1 : Real Number

1. Check whether 6n can end with the digit 0 for any natural number n.

2. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
(i) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 
(i) 43.123456789
(ii) 0.120120012000120000. . .

3.  Given that HCF (306, 657) = 9, find LCM (306, 657).

4.  Express each number as a product of its prime factors:
(i) 140
(ii) 156
(iii) 3825
(iv) 5005
(v) 7429

5. Which of the following rational numbers have the terminating decimal representation?
(i) 3/5
(ii) 7/20
(iii) 2/13
(iv) 27/40
(v) 133/125

6. Give an example to show that the product of a rational number and an irrational number may be a rational number.

7. Mention whether the following numbers are rational or irrational:
(i) (√2 + 2)
(ii) (2 – √2) x (2 + √2)
(iii) (√2 + √3)2
(iv) 6/3√2

8. HCF and LCM of two numbers is 9 and 459 respectively. If one of the numbers is 27, find the other number.

9. Find the LCM of 96 and 360 by using fundamental theorem of arithmetic.

10. Find the HCF (865, 255) using Euclid’s division lemma.

11. Find the largest number which divides 70 and 125 leaving remainder 5 and 8 respectively.

12.  Complete the following factor tree and find the composite number x.

13. Prove that 3 + 2√5 is irrational.

14.  The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p/q what can you say about the prime factors of q?

15. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

16. Show that 3√7 is an irrational number. 

17. Can two numbers have 15 as their HCF and 175 as their LCM? Give reasons.

18. There are 104 students in class X and 96 students in class IX in a school. In a house examination, the students are to be evenly seated in parallel rows such that no two adjacent rows are of the same class.

(a) Find the maximum number of parallel rows of each class for the seating arrange¬ment.
(b) Also, find the number of students of class IX and also of class X in a row.
(c) What is the objective of the school administration behind such an arrangement?

19. Show that any positive odd integer is of the form 41 + 1 or 4q + 3 where q is a positive integer.

20.  If two positive integers x and y are expressible in terms of primes as x = p2q3 and y = p3q, what can you say about their LCM and HCF. Is LCM a multiple of HCF? Explain