Real Numbers

1. Insert 4 rational numbers between 1 and without using formula

 Sol: we have to insert 4 rational numbers between 1 and  and 1

4 +1 = 5

1 can be written as 

Lcm of (4, 5) = 20

2. The prime factorisation of a natural number (n) is 2³ × 3² × 5² × 7. How many consecutive zeros will it have at the end of it? Justify your answer.

Sol:

 n = 2³ × 3² × 5² × 7

    = 2 × 2² × 3² × 5² × 7

    = 2 × 3² × 7 (5² × 2²)

    = 2 × 3² × 7 × (10)² = 12600

∴ n has 2 consecutive zeros at the end

3. Find the value of

Sol:

4. Write any two irrational numbers between 3 and 4

Sol: 

 5. Find the value of

 6.  Find the HCF and LCM of 90, 144 by using the prime factorisation method

Sol:

7.  Is  rational or irrational?  justify your answer

Sol:

8. Expand

Sol:

9. Find the HCF of 24 and 33 by using division method

Sol:

33 > 24

 33 = 24 × 1 + 9

 24 = 9 × 2 + 6

  9 = 6 × 1 + 3

  6 = 3 × 2 + 0

∴ HCF of 24 and 33 = 3

10. Find the value of

Sol:

11. Ramu says, “if = 0, the value of x is 0”. Do you agree with him? Give reason.

Sol:

1. Write any three numbers of two digits. Find the LCM and HCF for the above numbers by the prime factorisation method.

Sol: let 10, 12, 16 be three two-digit numbers

         Prime factarisation of 10 = 2 × 5 = 2¹× 5¹

         Prime factarisation of 12 = 2 × 2× 3 = 2² × 3

         Prime factarisation of 16 = 2 × 2× 2× 2 = 2⁴

HCF of 10, 12 and 16 = 2¹ = 2

LCM of 10, 12 and 16 = 2⁴ × 3× 5 = 8 × 3× 5 = 120

2. Give an example for each of the following

(i) The product of two irrational numbers is a rational number

(ii) The product of two irrational numbers is an irrational number

Sol:

3. State with reasons which of the following are rational numbers and which are irrational numbers

4. Expand

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