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Class 10 Maths Chapter 1: Real Numbers — Important Questions & Sample Paper

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Reviewed by qpaper's CBSE curriculum team · Edited by Mohit · Updated June 2026

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Yes — this page has 44+ original Class 10 Mathematics Chapter 1 (“Real Numbers”) important questions with answers (Multiple Choice (MCQ), Assertion–Reason, Short Answer, Short Answer, Long Answer, Case Study). Practise them free, or generate a full CBSE board-pattern sample paper (80 marks) and export it to PDF or Word — in English & Hindi, for 2026-27.

Chapter 1 of Class 10 CBSE Mathematics, 'Real Numbers', revisits the number system to build a deeper understanding of rational and irrational numbers. The chapter begins with Euclid's division lemma, a fundamental concept used primarily to compute the Highest Common Factor (HCF) of two positive integers. It then proceeds to the Fundamental Theorem of Arithmetic, which states that every composite number can be uniquely expressed as a product of primes. This theorem forms the backbone of several applications, including finding HCF and Least Common Multiple (LCM) via prime factorisation, and verifying the relationship HCF × LCM = product of the two numbers. A significant portion of the chapter is devoted to irrational numbers, where students learn to prove the irrationality of numbers like √2, √3, and their combinations (e.g., 3+2√5). The decimal expansions of rational numbers are explored, with clear conditions for terminating and non‑terminating repeating decimals. Typical exam questions include proving irrationality, determining after how many decimal places a rational number terminates, word problems on HCF and LCM (such as packing items or arranging objects), and proofs involving properties of integers. Mastering this chapter not only builds algebraic reasoning but also enhances problem‑solving skills critical for higher mathematics.

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Mathematics — Real Numbers

Class 10Time: 3 hrsMax Marks: 80

SECTION A

  1. 1.

    Two numbers are in the ratio 3:4 and their LCM is 72. The sum of the numbers is:

    (a) 42(b) 36(c) 84(d) 21
    1
  2. 2.

    Which of the following rational numbers has a non-terminating repeating decimal expansion?

    (a) 7/50(b) 11/25(c) 33/120(d) 53/240
    1
  3. 3.

    If p is a prime number, then √p is:

    (a) always rational(b) always irrational(c) rational only if p=2(d) sometimes rational
    1

+ 41 more questions in the full paper

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Marks distribution & blueprint

In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question typeMarks eachIn our bank
Multiple Choice (MCQ)1 mark13
Assertion–Reason1 mark6
Short Answer2 marks8
Short Answer3 marks6
Long Answer5 marks5
Case Study4 marks6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

Real, exam-style questions to practise and revise — each with its answer. Generate a full paper for unlimited more.

  1. Q1. Two numbers are in the ratio 3:4 and their LCM is 72. The sum of the numbers is:

    1 mark
    Multiple Choice (MCQ)
    (A) 42(B) 36(C) 84(D) 21
    ▸ Answer

    42

  2. Q2. Which of the following rational numbers has a non-terminating repeating decimal expansion?

    1 mark
    Multiple Choice (MCQ)
    (A) 7/50(B) 11/25(C) 33/120(D) 53/240
    ▸ Answer

    53/240

  3. Q3. If p is a prime number, then √p is:

    1 mark
    Multiple Choice (MCQ)
    (A) always rational(B) always irrational(C) rational only if p=2(D) sometimes rational
    ▸ Answer

    always irrational

  4. Q4. The HCF of 405 and 2520 is:

    1 mark
    Multiple Choice (MCQ)
    (A) 45(B) 60(C) 75(D) 90
    ▸ Answer

    45

  5. Q5. Assertion (A): The number 5 - √3 is irrational. Reason (R): The difference of a rational number and an irrational number is an irrational number.

    1 mark
    Assertion–Reason
    (A) Both A and R are true and R is the correct explanation of A.(B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
    ▸ Answer

    Both A and R are true and R is the correct explanation of A.

  6. Q6. Prove that √5 is irrational.

    2 marks
    Short Answer
    ▸ Answer

    √5 is irrational.

  7. Q7. Prove that the square of any positive integer is of the form 5q, 5q+1, or 5q+4, for some integer q.

    2 marks
    Short Answer
    ▸ Answer

    The square of any positive integer is of the form 5q, 5q+1, or 5q+4.

  8. Q8. Prove that √3 is irrational.

    3 marks
    Short Answer
    ▸ Answer

    √3 is irrational.

  9. Q9. Prove that one and only one out of n, n+2, n+4 is divisible by 3, where n is any positive integer.

    3 marks
    Short Answer
    ▸ Answer

    Exactly one of n, n+2, n+4 is divisible by 3.

  10. Q10. Show that the square of any positive integer cannot be expressed in the form 5q + 2 or 5q + 3, where q is an integer.

    5 marks
    Long Answer
    ▸ Answer

    The square is always of the form 5q, 5q+1, or 5q+4, never 5q+2 or 5q+3.

  11. Q11. Prove that √2 + √3 is an irrational number.

    5 marks
    Long Answer
    ▸ Answer

    Therefore, √2 + √3 is irrational.

  12. Q12. Sara is checking whether certain rational numbers have terminating decimal expansions. She knows that a reduced fraction a/b has a terminating decimal if and only if b has no prime factors other than 2 and 5. She considers the number 27/400.

    4 marks
    Case Study
    1. (i) Express 400 as a product of its prime factors.2 marks
    2. (ii) Does 27/400 have a terminating decimal expansion? Justify your answer.2 marks
    ▸ Answer

    Prime factorization: 400 = 2^4 × 5^2. The fraction 27/400 terminates because its denominator has only 2 and 5 as prime factors.

Frequently asked questions

What is the condition for a rational number to have a terminating decimal expansion?

A rational number p/q (in lowest terms) has a terminating decimal expansion if the prime factorisation of q is of the form 2ⁿ5ᵐ, where n and m are non‑negative integers.

How is the irrationality of numbers like 3+2√5 proven?

Assume contrary that 3+2√5 is rational, say equal to a/b. Then √5 = (a/b − 3)/2, making √5 rational, which contradicts the known fact that √5 is irrational. Hence, 3+2√5 is irrational.

What is the relationship between HCF, LCM, and the product of two numbers?

For any two positive integers a and b, the product of their HCF and LCM equals the product of the numbers: HCF(a,b) × LCM(a,b) = a × b. This is useful for verification and solving problems.

How can prime factorisation be used to find HCF and LCM?

For HCF, take the product of the smallest powers of all common prime factors. For LCM, take the product of the greatest powers of all prime factors appearing in the numbers. This method is efficient and illustrative of the Fundamental Theorem of Arithmetic.

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