Real Numbers Class 10 Extra Questions

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Real Numbers Class 10 Extra Questions - Free PDF

Author at PW

July 24, 2025

Real Numbers Class 10 Extra Questions: The chapter Real Numbers is very important for CBSE Class 10 students. Every year, several questions are asked from this chapter in the board exams. That’s why it’s important to understand the concepts properly and practise enough questions. Topics like Euclid’s Division Lemma, HCF, LCM, irrational numbers, and decimal expansions are all covered here. To help practise in a more good way, students can consider mind map for class 10 cbse maths real numbers during revision sessions.

Many students feel this chapter is easy, but the questions can sometimes be tricky. So, solving real numbers important questions Class 10 helps you become more confident. In this article, you’ll find extra questions of real numbers Class 10, along with key formulas, tips to solve them, and all the important points from the chapter. These class 10 real numbers important questions will not only help you score better in exams but also improve your overall understanding of the topic. A CBSE Class 10 revision book will also help you cover the full syllabus properly.

Important Concepts Covered in Real Numbers: Class 10

Before solving the real numbers important questions Class 10, students must first understand the main concepts of this chapter. This chapter is not just a new topic, but a continuation of what was learnt in Class 9. Here are the key topics covered in this chapter:

Euclid’s Division Lemma – This is used to find the HCF of two numbers. It is based on the formula:

a = bq + r, where 0 ≤ r < b

Fundamental Theorem of Arithmetic – This tells us that every number greater than 1 can be written as a product of prime numbers, and this method is unique.
Proofs of Irrational Numbers – You learn how to prove that numbers like √2, √3, and √5 are irrational, meaning they cannot be written as p/q.
Decimal Expansions of Rational Numbers – This helps in identifying if a rational number has a terminating or non-terminating decimal.

For structured practice, many students buy study materials like PW CBSE Class 10 Mind Maps Book, which often include worked-out examples and shortcut strategies.

Why Practising Extra Questions is Important for Real Numbers?

Many students think this chapter is simple, but questions can come in different forms in the exam. That is why it is important to solve extra questions of real numbers Class 10.

It helps you understand concepts better by applying them in different types of questions.
It makes you faster and more accurate, especially when solving difficult or tricky problems.
It covers all kinds of questions that can appear in the exam, like multiple choice, long answer, and case-based.
It also prepares you for competitive exams like NTSE, where similar logic-based questions are asked.

Check Out: CBSE Class 10th Question Banks

Real Numbers Class 10 Extra Questions

Are you preparing for your Class 10 Maths exam and looking for more practise on Chapter 1- Real Numbers? PW is here with a helpful PDF filled with extra questions of real numbers Class 10. This PDF includes all types of questions, whether it's for 1 mark, 2 marks, or 5 marks, so you can be ready for any type of exam question.

Since this chapter is important and many real numbers questions in Class 10 are seen in the board papers every year, regular practice becomes a must. These class 10th real numbers extra questions are designed to make your revision easy and strong. From Euclid’s Division Lemma to HCF and decimal expansions, every topic is covered in a clear and useful way. Just download the PDF and start solving the questions.

Q.1: 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.

Solution:

Let x be any positive integer and y = 3. By Euclid’s division algorithm; x =3q + r (for some integer q ≥ 0 and r = 0, 1, 2 as r ≥ 0 and r < 3) Therefore, x = 3q, 3q + 1 and 3q + 2 As per the given question, if we take the square on both the sides, we get; x ²= (3q) ²= 9q ²= 3.3q ²Let 3q ²= m Therefore, x ²= 3m ………………….(1) x ²= (3q + 1) ²= (3q) ²+ 1 ²+ 2 × 3q × 1 = 9q ²+ 1 + 6q = 3(3q ²+ 2q) + 1 Substituting 3q ²+2q = m we get, x ²= 3m + 1 ……………………………. (2) x ²= (3q + 2) ²= (3q) ²+ 2 ²+ 2 × 3q × 2 = 9q ²+ 4 + 12q = 3(3q ²+ 4q + 1) + 1 Again, substituting 3q ²+ 4q + 1 = m, we get, x ²= 3m + 1…………………………… (3) Hence, from eq. 1, 2 and 3, we conclude that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

Q.2: Express each number as a product of its prime factors:

(i) 140

(ii) 156

(iii) 3825

(iv) 5005

(v) 7429

Solution:

(i) 140 Using the division of a number by prime numbers method, we can get the product of prime factors of 140. Therefore, 140 = 2 × 2 × 5 × 7 × 1 = 2 ²× 5 × 7 (ii) 156 Using the division of a number by prime numbers method, we can get the product of prime factors of 156. Hence, 156 = 2 × 2 × 13 × 3 = 2 ²× 13 × 3 (iii) 3825 Using the division of a number by prime numbers method, we can get the product of prime factors of 3825. Hence, 3825 = 3 × 3 × 5 × 5 × 17 = 3 ²× 5 ²× 17 (iv) 5005 Using the division of a number by prime numbers method, we can get the product of prime factors of 5005. Hence, 5005 = 5 × 7 × 11 × 13 = 5 × 7 × 11 × 13 (v) 7429 Using the division of a number by prime numbers method, we can get the product of prime factors of 7429. Hence, 7429 = 17 × 19 × 23 = 17 × 19 × 23

Q.3: Given that HCF (306, 657) = 9, find LCM (306, 657).

Solution:

As we know, HCF × LCM = Product of the two given numbers So, 9 × LCM = 306 × 657 LCM = (306 × 657)/9 = 22338 Therefore, LCM(306,657) = 22338

Q.4: Prove that 3 + 2√5 is irrational.

Solution:

Let 3 + 2√5 be a rational number. Then the co-primes x and y of the given rational number where (y ≠ 0) is such that: 3 + 2√5 = x/y Rearranging, we get, 2√5 = (x/y) – 3 √5 = 1/2[(x/y) – 3] Since x and y are integers, thus, 1/2[(x/y) – 3] is a rational number. Therefore, √5 is also a rational number. But this confronts the fact that √5 is irrational. Thus, our assumption that 3 + 2√5 is a rational number is wrong. Hence, 3 + 2√5 is irrational.

Q.5: 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

Solution:

Note: If the denominator has only factors of 2 and 5 or in the form of 2 ^m× 5 ⁿthen it has a terminating decimal expansion.

If the denominator has factors other than 2 and 5 then it has a non-terminating repeating decimal expansion. (i) 13/3125 Factoring the denominator, we get, 3125 = 5 × 5 × 5 × 5 × 5 = 5 ⁵Or = 2 ⁰× 5 ⁵Since the denominator is of the form 2 ^m× 5 ⁿthen, 13/3125 has a terminating decimal expansion. (ii) 17/8 Factoring the denominator, we get, 8 = 2× 2 × 2 = 2 ³Or = = 2 ³× 5 ⁰Since the denominator is of the form 2 ^m× 5 ⁿthen, 17/8 has a terminating decimal expansion. (iii) 64/455 Factoring the denominator, we get, 455 = 5 × 7 × 13 Since the denominator is not in the form of 2 ^m× 5 ⁿ, therefore 64/455 has a non-terminating repeating decimal expansion. (iv) 15/1600 Factoring the denominator, we get, 1600 = 2 ⁶× 5 ²Since the denominator is in the form of 2 ^m× 5 ⁿ, 15/1600 has a terminating decimal expansion.

Q.6: 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. . .

Solution:

(i) 43.123456789 Since it has a terminating decimal expansion, it is a rational number in the form of p/q and q has factors of 2 and 5 only. (ii) 0.120120012000120000. . . Since it has a non-terminating and non-repeating decimal expansion, it is an irrational number.

For More Extra Questions, Just download the PDF and start solving the questions.

Real Numbers Class 10 Extra Questions

Want more such Class 10 Maths Chapter wise extra Question? Check out below:-

CBSE Class 10 Mind Maps Book For 2026 Board Exams

CBSE Class 10 NCERT Exemplar Problems Mathematics

Tips to Solve Real Numbers Questions Quickly

To score full marks in this chapter, it's not enough to just know the formulas. You should also understand how to apply them smartly during the exam. Below are some helpful and detailed tips to solve real numbers important questions Class 10 accurately and on time:

Start HCF questions properly: Always begin with Euclid’s Division Lemma. Use the formula a = bq + r and keep applying it until the remainder becomes 0. This step-by-step method helps avoid mistakes in calculations.
Irrational number proof tip: To prove numbers like √2 or √3 are irrational, assume they are rational (like √2 = p/q, where p and q have no common factor), square both sides, and find a contradiction. This type of proof is common in exams.
Check decimal form of rational numbers: Look at the prime factorisation of the denominator. If it has only 2, only 5, or both, the decimal will terminate. If it has any other prime number, the decimal will be non-terminating and repeating.
Count trailing zeroes in factorials: To find how many zeroes are at the end of numbers like 100!, count the number of times 5 appears in its prime factors. This is an easy trick used in many exam questions.
Use prime factorisation smartly: In questions about the Fundamental Theorem of Arithmetic, write the full prime factorisation clearly. This helps in solving LCM, HCF, and other related questions easily.
Write all steps clearly: Never skip steps. Even if you know the answer, show how you got it. Marks are given for the correct method, especially in board exams.
Practise with a timer: While solving class 10th real numbers extra questions, use a timer and finish each within 2–3 minutes. This builds speed and helps you stay confident during exams.