Chapter 1: Real Numbers

Important questions — the ones the board asked most (20)

1. Q1Short answer3 marksAsked: 2023, 2024, 2025, 2026

   Prove that sqrt(3) is an irrational number.

   

   Medha Ma’am

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   Model answer · Proving irrationality of roots

   1. 1.Let us assume, to the contrary, that √3 is a rational number. Then √3 = p/q, where p and q are integers, q ≠ 0, and p and q are coprime.0.5 marks
   2. 2.Squaring on both sides and rearranging, we get 3q² = p² So p² is divisible by 3. Since 3 is a prime, if 3 divides p², then 3 divides p. So p is divisible by 3. ----- (i)1 mark
   3. 3.Let p = 3a, where a is some integer. Substituting for p, we get 3q² = 9a², that is, q² = 3a² So q² is divisible by 3, and therefore q is divisible by 3. ----- (ii)1 mark
   4. 4.From (i) and (ii), p and q have at least 3 as a common factor. But this contradicts the fact that p and q are coprime. This contradiction has arisen because of our incorrect assumption that √3 is rational. Therefore, √3 is an irrational number.0.5 marks

   • In the first line, write that p and q are integers, q ≠ 0 and "p and q are coprime".
   • Write both steps, "p² is divisible by 3" and then "p is divisible by 3", and do the same for q² and q.
   • End with the contradiction, "this contradicts the fact that p and q are coprime", and then the conclusion that √3 is an irrational number.

   Marks split as in the board’s published marking scheme

2. Q2Short answer3 marksAsked: 2023, 2024

   Prove that 2 + 5sqrt(3) is an irrational number, if it is given that sqrt(3) is an irrational number.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 55 s

   Model answer · Proving numbers irrational

   1. 1.Let us assume, to the contrary, that x = 2 + 5√3 is a rational number. So x can be written as p/q, where p and q are integers and q ≠ 0.1 mark
   2. 2.Rearranging this equation: x − 2 = 5√3 √3 = (x − 2)/51 mark
   3. 3.Since x, 2 and 5 are rational, (x − 2)/5 is a rational number. So the RHS is a rational number, but the LHS, √3, is an irrational number. This contradiction has arisen because of our incorrect assumption, so our assumption is wrong. Hence 2 + 5√3 is an irrational number.1 mark

   • Begin with "Let us assume, to the contrary, that x = 2 + 5√3 is a rational number".
   • Write the rearranged form "√3 = (x − 2)/5" on its own line.
   • Say that the RHS is rational but the LHS √3 is irrational, then end with "Hence 2 + 5√3 is an irrational number".

   Marks split as in the board’s published marking scheme

3. Q3Very short answer2 marksAsked: 2023, 2024

   Find the HCF of 84 and 144 by prime factorisation method.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 40 s

   Model answer · HCF by prime factorisation

   1. 1.Given numbers: 84 and 144. By the prime factorisation method, each number is written as a product of primes. 84 = 2 × 2 × 3 × 70.5 marks
   2. 2.144 = 2 × 2 × 2 × 2 × 3 × 30.5 marks
   3. 3.The common prime factors of 84 and 144 are 2, 2 and 3. Therefore, HCF (84, 144) = 2 × 2 × 3 = 121 mark

   • Write 84 and 144 fully as products of primes: "84 = 2 × 2 × 3 × 7" and "144 = 2 × 2 × 2 × 2 × 3 × 3".
   • Multiply only the common prime factors and write the final line "HCF (84, 144) = 2 × 2 × 3 = 12".

   Marks split as in the board’s published marking scheme

4. Q4Very short answer2 marksAsked: 2023, 2024

   Find the LCM of 231 and 396 by prime factorisation method.

   How the model answer begins

   1. Given: the numbers 231 and 396. By the prime factorisation method, the prime factorisation of 231 is: 231 = 3 × 7 × 11

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5. Q5Very short answer2 marksAsked: 2023, 2026

   Prove that 2 - 5sqrt(3) is an irrational number given that sqrt(3) is irrational.

   How the model answer begins

   1. Let us assume, to the contrary, that 2 − 5√3 is a rational number. That is, we can find coprime integers a and b (b ≠ 0) such that 2 − 5√3 = a/b

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6. Q6Case-based4 marksAsked: 2023

   Khushi wants to organize her birthday party. Being health conscious, she decided to serve only fruits in her birthday party. She bought 36 apples and 60 bananas and decided to distribute fruits equally among all. Based on the above information, answer the following questions : (i) How many guests Khushi can invite at the most ? (ii) How many apples and bananas will each guest get ? (iii) (a) If Khushi decides to add 42 mangoes, how many guests Khushi can invite at the most ? OR (b) If the cost of 1 dozen bananas is Rs 60, cost of 1 apple is Rs 15 and cost of 1 mango is Rs 20, then how much amount will have to be spent to buy 60 bananas, 36 apples and 42 mangoes ?

   How the model answer begins

   1. (i) Given: 36 apples and 60 bananas are to be shared equally among all the guests. The largest number of guests is the HCF of 36 and 60. We find it by the prime factorisation method. 36 = 2² × 3² 60 = 2² × 3 × 5 HCF is…

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7. Q7Case-based4 marksAsked: 2023

   Khushi wants to organize her birthday party. Being health conscious, she decided to serve only fruits in her birthday party. She bought 36 apples and 60 bananas and decided to distribute fruits equally among all. Based on the above information, answer the following questions : (i) How many guests Khushi can invite at the most ? (1) (ii) How many apples and bananas will each guest get ? (1) (iii) (a) If Khushi decides to add 42 mangoes, how many guests Khushi can invite at the most ? (2) OR (b) If the cost of 1 dozen of bananas is ₹ 60, the cost of 1 apple is ₹ 15 and cost of 1 mango is ₹ 20, find the total amount spent on 60 bananas, 36 apples and 42 mangoes. (2)

   How the model answer begins

   1. (i) The greatest number of guests is the HCF of 36 and 60, so that every guest gets an equal share. 36 = 2² × 3² 60 = 2² × 3 × 5 HCF (36, 60) = 2² × 3 = 12 Khushi can invite at the most 12 guests.

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8. Q8Short answer3 marksAsked: 2025

   (a) Prove that sqrt(5) is an irrational number. OR (b) State the "Fundamental Theorem of Arithmetic" and use it to find LCM of 36 and 54.

9. Q9Short answer3 marksAsked: 2025

   (a) Prove that sqrt(2) is an irrational number. OR (b) Find which among the following numbers a, b and c is/are composite numbers. a = 7 * 11 * 13 + 13, b = 6 * 5 * 4 + 4, c = 7 * 13 + 6

   How the model answer begins

   1. (a) Let us assume, to the contrary, that √2 is rational. Then we can find integers p and q (q ≠ 0) such that √2 = p/q, where p and q are coprime.

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10. Q10Short answer3 marksAsked: 2025

    The traffic lights at three different road crossings change after every 45 seconds, 75 seconds and 60 seconds respectively. If they change together at 5.00 a.m., then at what time they will change together next ?

11. Q11Short answer3 marksAsked: 2025

    Find the smallest number which when increased by 20, is exactly divisible by 72, 90 and 150.

    How the model answer begins

    1. The smallest number which when increased by 20 is exactly divisible by 72, 90 and 150 is 1780. It is found as follows. Writing each number as a product of its prime factors (the prime factorisation method): 72 = 2 × 2 ×…

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12. Q12Short answer3 marksAsked: 2025

    Three measuring rods are of lengths 120 cm, 100 cm and 150 cm. Find the least length of a fence that can be measured an exact number of times, using any of the rods. How many times each rod will be used to measure the length of the fence ?

    How the model answer begins

    1. The least length of the fence is the LCM of the rod lengths 120 cm, 100 cm and 150 cm, because each rod must measure it an exact number of times. By the prime factorisation method: 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5…

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13. Q13Short answer3 marksAsked: 2025

    Three friends plan to go for a morning walk. They step off together and their steps measures 48 cm, 52 cm and 56 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps ten times ?

14. Q14Short answer3 marksAsked: 2025

    Given that sqrt(5) is an irrational number, prove that 2 + 3sqrt(5) is an irrational number.

    How the model answer begins

    1. Let us assume, to the contrary, that 2 + 3√5 is rational. Let 2 + 3√5 = a, where 'a' is a rational number.

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15. Q15Short answer3 marksAsked: 2024

    In a teachers' workshop, the number of teachers teaching French, Hindi and English are 48, 80 and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.

    How the model answer begins

    1. Given: there are 48 French teachers, 80 Hindi teachers and 144 English teachers. Every room has the same number of teachers, and all of them teach the same subject. Minimum number of rooms required means there should be…

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16. Q16Short answer3 marksAsked: 2024

    (a) Prove that sqrt(3) is an irrational number. OR (b) Prove that (sqrt(2) + sqrt(3))^2 is an irrational number, given that sqrt(6) is an irrational number.

17. Q17Short answer3 marksAsked: 2024

    Prove that 5 sqrt(2) - 3 is an irrational number, given that sqrt(2) is an irrational number.

    How the model answer begins

    1. Let us assume, to the contrary, that x = 5√2 − 3 is a rational number. Then x = a/b, where a and b are coprime integers and b ≠ 0.

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18. Q18Short answer3 marksAsked: 2024

    Find LCM and HCF of two numbers 336 and 54, using prime-factorisation method.

    How the model answer begins

    1. The prime factorisation of 336 and 54 is found by expressing each number as a product of primes. 336 = 2 × 2 × 2 × 2 × 3 × 7 = 2^4 × 3 × 7. 54 = 2 × 3 × 3 × 3 = 2 × 3^3.

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19. Q19Short answer3 marksAsked: 2024

    Find the HCF and LCM of 96 and 404, using prime-factorisation method.

    How the model answer begins

    1. The HCF of 96 and 404 is 4 and their LCM is 9696. Using the prime factorisation method, we write each number as a product of primes: 96 = 2 × 2 × 2 × 2 × 2 × 3 = 2⁵ × 3 404 = 2 × 2 × 101 = 2² × 101

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20. Q20Short answer3 marksAsked: 2024

    Check whether the number 4^n can end with digit 0 for any natural number n. Give reasons for your answer.

    How the model answer begins

    1. No, 4^n cannot end with the digit zero for any natural number n. A number ending with zero is divisible by 5, so its prime factorisation must contain the prime 5. But 4^n = (2^2)^n = 2^2n, so the only prime factor of…

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48 more questions of this chapter, each with its model answer, in the AskMedha app.

Objective questions with the board’s key (10)

1. 1. CBSE 2026

   The HCF of 960 and 432 is :

   • (A) 48 ✓
   • (B) 54
   • (C) 72
   • (D) 36

   Answer: (A)

2. 2. CBSE 2026

   The natural number 2 is :

   • (A) a prime number ✓
   • (B) a composite number
   • (C) prime as well as composite
   • (D) neither prime nor composite

   Answer: (A)

3. 3. CBSE 2026

   For any natural number n, 6^n ends with the digit :

   • (A) 0
   • (B) 6 ✓
   • (C) 3
   • (D) 2

   Answer: (B)

4. 4. CBSE 2026

   The LCM of 960 and 240 is :

   • (A) 960 ✓
   • (B) 240
   • (C) 60
   • (D) 15

   Answer: (A)

5. 5. CBSE 2026

   For any natural number n, the unit digit of 5^n is :

   • (A) 0
   • (B) 5 ✓
   • (C) 3
   • (D) 2

   Answer: (B)

6. 6. CBSE 2026

   There are two sections A and B of Grade X. There are 28 students in Section A and 30 students in Section B. What is the minimum number of books you will acquire for the class library so that they can be distributed equally among students of Section A or Section B ?

   • (A) 144
   • (B) 2
   • (C) 420 ✓
   • (D) 272

   Answer: (C)

7. 7. CBSE 2026

   (3 * 11 * 13 + 3) is :

   • (A) a prime number
   • (B) divisible by 13
   • (C) a composite number ✓
   • (D) an odd number

   Answer: (C)

8. 8. CBSE 2026

   If HCF (850, 325) is 25, then LCM (850, 325) is :

   • (A) 442
   • (B) 11050 ✓
   • (C) 8450
   • (D) 2210

   Answer: (B)

9. 9. CBSE 2026

   7 * 29 * 23 + 1 is :

   • (A) a prime number.
   • (B) divisible by 23.
   • (C) an odd number.
   • (D) a composite number. ✓

   Answer: (D)

10. 10. CBSE 2026

    Assertion (A) : 4^n can not end with the digit zero. Reason (R) : Prime factorisation of 4^n is unique.

    • (A) Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
    • (B) Both, Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A). ✓
    • (C) Assertion (A) is true, but Reason (R) is false.
    • (D) Assertion (A) is false, but Reason (R) is true.

    Answer: (B)

Questions parents and students ask

How many marks does "Real Numbers" carry in CBSE Class 10 Maths?

In the CBSE papers of 2023–2026, this chapter carried about 6 marks and about 3 questions a year; by marks it ranks 5 in Maths.

Which are the most important (VVI) questions of "Real Numbers"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 68 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

Yes. CBSE publishes its marking schemes; every model answer here is written point by point to the marks the board’s own scheme gives, and checked by a second, independent model.