NCERT Solution for Class 10 Mathematics Chapter 1 Real Numbers


Chapter Name

NCERT Solution for Class 10 Maths Chapter 1 Real Numbers

Topics Covered

  • Short Revision Notes for the Chapter
  • NCERT Exercise Solution

Related Study

  • NCERT Solution for Class 10 Maths
  • NCERT Revision Notes for Class 10 Maths
  • Important Questions for Class 10 Maths
  • MCQ for Class 10 Maths
  • NCERT Exemplar Questions For Class 10 Maths

Short Revision for Real Numbers

  1. Numbers are divided into two parts: real numbers and imaginary numbers. 
  2. Real numbers can be divided into two parts : rationals and irrationals. 
  3. Any rational number can be put in the form P/Q, where p and Q are integers with Q ≠ 0. But irrational number cannot be put in this form. 
  4. Every integer is a rational number. 
  5. Zero (0) is a rational number. 
  6. π is irrational as 22/7 is not an exact value of π. 
  7. Square root of every prime number is irrational. 
  8. √2, √3, √101 are examples of irrationals. 
  9. If a prime p divides a2 then p divides a, where a is a positive integer. 
  10. Each of prime numbers has only two factors : one factor is 1 and other one is number itself. 
  11. Two numbers are called coprime if they have no common factor other than 1. e.g., 2 and 5. 
  12. Rational numbers have either a terminating decimal expansion or a non - terminating repeating (recurring) decimal expansion. 
  13. The denominator of a rational number having terminating decimal expansion has only powers of 2, or powers of 5, or both. 
  14. 0.428571 , -21.037  and 23.3408 are examples of rationls. 
  15. The sum or difference of a rational and an irrational is irrational. 
  16. The product or quotient of a non - zero rational and an irrational is irrational. 
  17. If a and b be two positive integers, then  a × b = HCF(a, b) × LCM (a, b). 
  18. Every composite number has more than two factors. 
  19. Every composite number can be expressed as a product of primes. 
  20. 0120012000120....... is an irrational number. 
  21. If two positive numbers a and b are given, then there exist two unique whole numbers q and r satisfying the relation a = b × q + r; 0 ≤ r < b. 

NCERT Exercises Solution

Exercise 1.1

1. Use Euclid’s division algorithm to find the HCF of:
i. 135 and 225
ii. 196 and 38220
iii. 867 and 255


2. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.


3. 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?


4. 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.


5. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.


Exercise 1.2 

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


2. Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
(i) 26 and 91
(ii) 510 and 92
(iii) 336 and 54



3. Find the LCM and HCF of the following integers by applying the prime factorisation method.
(i) 12, 15 and 21
(ii) 17, 23 and 29
(iii) 8, 9 and 25


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


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


6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.


7. There is a circular path 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. After how many minutes will they meet again at the starting point?

Exercise 1.3 

1. Prove that √is irrational.


2. Prove that 3 + 2√5 + is irrational.


3. Prove that the following are irrationals:
(i) 1/√2
(ii) 7√5
(iii) 6 + 2


Exercise 1.4 

1. 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
(v) 29/343
(vi) 23/(2352)
(vii) 129/(225775)
(viii) 6/15
(ix) 35/50
(x) 77/210


2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.



(iii), (v), (vii),(viii), (x) has Non terminating decimal expansion.


3. 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?
(i) 43.123456789
(ii) 0.120120012000120000. . . 
(iii) 43.123456789

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