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1. REAL NUMBERS (NCERT)

CBSE Class 10 Maths REAL NUMBERS (NCERT)

Chapter 1 : Real Numbers

 Exercise 1.1  Complete solution

 Exercise 1.2 Complete solution

    Algebraic Identities

 (i)  

 (ii)

 (iii)

 (iv)

                   

 (v)

                    

 Important notes

(ii) Fundamental Theorem of Arithmetic : Every composite number can be expressed ( factorised) as a  product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

 (iii) The prime factorisation of a natural number is unique, except for the order of its factors.

 (iv) HCF = Product of the smallest power of each common prime factor in the numbers.
 (v) LCM = Product of the greatest power of each prime factor, involved in the numbers .

 (vi) If two positive integers  and ,then .

i.e ., HCF of two numbers × LCM of two numbers = One number × Other number

(vii) The product of three numbers is not equal to the product of their HCF and LCM.

(viii) A number  is called rational if it can be written in the form ,  where p and q are integers . Example : 0 ,  – 5 , , …… etc .

(ix) A number  is called irrational if it cannot be written in the form ,   where p and q are integers  . Example :   ……… etc.

(x) Let  be a prime number. If  divides , then  divides , where  is a positive integer.

(xi)  The sum or difference of a rational and an irrational number is irrational .
(xii) The product and quotient of a non-zero rational and irrational number is irrational.

 EXERCISE 1.1

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

Solution : (i) We have,

                                         

   (ii) We have,

                               

 (iii) We have,

                                

 (iv)  We have ,  

 (v)  We have ,

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

Solution :    (i)   26 and 91 

 We have ,  and     

   HCF (26 , 91)  13   

and LCM (26 , 91)

 Now ,

 

 

     Verified .

  (ii)    510 and 92 

 We have ,  

 and       

 HCF(510 , 92)    

 and  LCM(510 , 92)  

  Now , 

  

  

      Verified .

(iii)   336 and 54

We have,

                       

 and     

 HCF(336 , 54)    

 and LCM(336 , 54)

Now , 

 

 

     Verified . 

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

Solution :    (i) 12 , 15  and 21     

  We have ,

 

   HCF (12 , 15 , 21)  3   

and LCM (12 , 15 , 21)  

(ii)   17 , 23  and 29   

  We have , 

  

  

    HCF (17 , 23 , 29)   

 and LCM (17 , 23 , 29)  

(iii)   8 , 9  and 25

 We have ,

 

  

    HCF(8 , 9 , 25)   

  and  LCM (8 , 9 , 25)  

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

Solution :  We have ,

 

                               

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

Solution :  We have , 

 

The prime factors of  does not contain  in factor , where  are positive integers .Therefore,  does not end with the digit 0 .

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

Solution:  We have ,

 is a composite number.

and 

 

 

  is a composite number .

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 ?

Solution : The required  number of minutes  is  LCM (18 , 12) . We find the LCM by prime factorization method ,

  

and  

Therefore, the LCM (18 , 12)  

Hence, Sonia and Ravi will meet again at the starting point after 36 minutes .

EXERCISE 1.2

1. Prove that  is irrational .

Solution : let, us assume to the contrary that  is rational .There exists co-prime integers and  () such that

  

 Therefore ,  is divisible by 5 . So,  is also divisible by 5 .

 Let  , for some integer c  .

 From  and  , we get

 

 

So,  is divisible by 5 . So,  is also divisible by 5 . 

Therefore,  and  have at least as a common factor . But this contradicts the fact that and  are co-prime. This contradiction has arisen because of our incorrect assumption that   is rational . So, we conclude that  is irrational .

2. Prove that  is irrational .

Solution :  let us assume , to the contrary  that   is rational .

We can find co-prime  and  ( ) such that

Since, 2 ,  and  are integers , is rational and so, is rational . But this contradicts the fact that   is irrational . So , we conclude that is irrational .

3. Prove that the following are irrationals :  (i)         (ii)       (iii)   

Solution :  (i)  Let us assume , to the contrary , that   is rational . We can find co-prime and  ( ) such that

 

Since  2 , and  are integers, is rational and so, is rational . But this contradicts the fact that  is irrational . So , is an irrational .

(ii)  Let us assume , to the contrary , that   is rational .  We can find co-prime and  ( ) such that 

 Since  7 , and  are integers ,  is rational and so, is rational . But this contradicts the fact that  is irrational .So , is an irrational .

(iii)  Let us assume , to the contrary , that   is rational . We can find co-prime and  ( ) such that

 Since and  are integers ,  is rational and so, is rational . But this contradicts the fact that  is irrational . So , is an irrational  .