Class 10 Maths Real Numbers Worksheet — Free Printable PDF

Section A: Multiple Choice Questions

Q1. The decimal expansion of 17/8 is:

a) terminating

b) non-terminating repeating

c) non-terminating non-repeating

d) none of these

Q2. The HCF of 96 and 404 is:

a) 2

b) 4

c) 6

d) 8

Q3. Which of the following is an irrational number?

a) sqrt(16)

b) sqrt(25)

c) sqrt(36)

d) sqrt(7)

Q4. The prime factorisation of 144 is:

a) 2 raised to 3 multiplied by 3 raised to 2

b) 2 raised to 4 multiplied by 3 raised to 2

c) 2 raised to 2 multiplied by 3 raised to 4

d) 2 raised to 3 multiplied by 3 raised to 3

Q5. If HCF of two numbers is 4 and their LCM is 36, and one number is 12, the other number is:

a) 9

b) 12

c) 16

d) 18

Section B: Fill in the Blanks

Q6. The product of two irrational numbers is ________.

Q7. The HCF of two prime numbers is always ________.

Q8. Every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of factors. This is called the ________.

Q9. The decimal expansion of 13/625 is ________.

Q10. For any two positive integers a and b, HCF multiplied by LCM equals ________.

Section C: Short Answer Questions

Q11. Show that 5 minus sqrt(3) is irrational.

Assume 5 minus sqrt(3) is rational. Then 5 minus sqrt(3) equals p/q where p and q are integers and q is not equal to 0. This gives sqrt(3) equals 5 minus p/q, which is rational. But sqrt(3) is irrational, which is a contradiction. Therefore 5 minus sqrt(3) is irrational.

Q12. Find the HCF and LCM of 12, 15 and 21 using prime factorisation.

12 equals 2 raised to 2 multiplied by 3

15 equals 3 multiplied by 5

21 equals 3 multiplied by 7

HCF equals 3

LCM equals 2 raised to 2 multiplied by 3 multiplied by 5 multiplied by 7 equals 420

Q13. Prove that sqrt(2) is irrational.

Assume sqrt(2) is rational. Then sqrt(2) equals p/q where HCF of p and q is 1. Squaring both sides gives 2 equals p squared divided by q squared, so p squared equals 2q squared. This means 2 divides p squared, so 2 divides p. Let p equals 2m. Then 4m squared equals 2q squared, giving q squared equals 2m squared. So 2 divides q as well. This contradicts HCF of p and q being 1. Therefore sqrt(2) is irrational.

Q14. Check whether 6 raised to n can end with the digit 0 for any natural number n.

For a number to end with 0, it must have 2 and 5 as prime factors. The prime factorisation of 6 raised to n is 2 raised to n multiplied by 3 raised to n. Since 5 is not a factor of 6 raised to n, it cannot end with the digit 0 for any natural number n.

Q15. Find the LCM of 96 and 360 using prime factorisation.

96 equals 2 raised to 5 multiplied by 3

360 equals 2 raised to 3 multiplied by 3 raised to 2 multiplied by 5

LCM equals 2 raised to 5 multiplied by 3 raised to 2 multiplied by 5 equals 32 multiplied by 9 multiplied by 5 equals 1440