CBSE Class 10 Mathematics Worksheet

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CBSE Class 10 Maths Worksheet – Real Numbers (Chapter 1) Important Questions

CBSE Class 10 Mathematics Worksheet

Chapter 1: Real Numbers – Important Practice Questions

Time: 60 minutes
Marks: 25
Academic Year: 2024-25

General Instructions:

  1. All questions are compulsory
  2. Questions 1-5 carry 1 mark each
  3. Questions 6-8 carry 2 marks each
  4. Questions 9-10 carry 3 marks each
  5. Question 11 carries 5 marks
1. State the Fundamental Theorem of Arithmetic. [1]
Solution:

The Fundamental Theorem of Arithmetic states that every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

2. Find the HCF of 96 and 404 by prime factorization method. [1]
Solution:

96 = 25 × 3

404 = 22 × 101

HCF = Product of smallest power of common prime factors = 22 = 4

3. Without actual division, state whether the rational number 17/8 has terminating or non-terminating decimal expansion. [1]
Solution:

17/8 = 17/23

Since denominator is of form 2n, it has terminating decimal expansion.

4. Express 156 as a product of its prime factors. [1]
Solution:

156 = 2 × 2 × 3 × 13 = 22 × 3 × 13

5. Write the condition for the decimal expansion of a rational number to be terminating. [1]
Solution:

A rational number p/q has terminating decimal expansion if q can be expressed in the form 2n × 5m where n, m are non-negative integers.

6. Use Euclid’s division algorithm to find the HCF of 135 and 225. [2]
Solution:

225 = 135 × 1 + 90

135 = 90 × 1 + 45

90 = 45 × 2 + 0

HCF = 45

7. Check whether 6n can end with the digit 0 for any natural number n. [2]
Solution:

6n = (2 × 3)n = 2n × 3n

For a number to end with 0, its prime factors must include both 2 and 5.

Since there’s no factor of 5, 6n cannot end with digit 0.

8. Explain why 7 × 11 × 13 + 13 is a composite number. [2]
Solution:

7 × 11 × 13 + 13 = 13(7 × 11 + 1) = 13 × 78 = 13 × 2 × 3 × 13

Since it has more than two distinct prime factors (2, 3, and 13), it is a composite number.

9. Prove that √5 is irrational. [3]
Solution:

Assume √5 is rational. Then √5 = a/b where a, b are coprime integers.

Squaring: 5 = a²/b² ⇒ 5b² = a² ⇒ a² is divisible by 5 ⇒ a is divisible by 5.

Let a = 5k. Then 5b² = (5k)² ⇒ 5b² = 25k² ⇒ b² = 5k² ⇒ b is divisible by 5.

This contradicts that a, b are coprime. Hence, √5 is irrational.

10. Find the LCM and HCF of 12, 15 and 21 by prime factorization method. [3]
Solution:

12 = 22 × 3

15 = 3 × 5

21 = 3 × 7

HCF = 3 (only common prime factor)

LCM = 22 × 3 × 5 × 7 = 420

11. (a) Use Euclid’s division lemma to show that the square of any positive integer is either of form 3m or 3m + 1 for some integer m. [5]
(b) There is a circular path around a sports field. Sonia takes 18 minutes to drive one round while Ravi takes 12 minutes. If they both start at the same point and time, after how many minutes will they meet again at the starting point?
Solution:

(a) Let a be any positive integer and b = 3. By Euclid’s division lemma:
a = 3q + r where r = 0, 1, or 2
Case 1: r = 0 ⇒ a = 3q ⇒ a² = 9q² = 3(3q²) = 3m
Case 2: r = 1 ⇒ a = 3q + 1 ⇒ a² = 9q² + 6q + 1 = 3(3q² + 2q) + 1 = 3m + 1
Case 3: r = 2 ⇒ a = 3q + 2 ⇒ a² = 9q² + 12q + 4 = 3(3q² + 4q + 1) + 1 = 3m + 1
Hence proved.

(b) Time until they meet = LCM(18, 12)
18 = 2 × 32
12 = 22 × 3
LCM = 22 × 32 = 36 minutes

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