Chapter 1: Real Numbers

VVI questions — the ones the board asked most (19)

1. Q1Short answer2 marksAsked: 2017, 2020, 2022

   Prove that 1/sqrt(2) is an irrational number.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 50 s

   Model answer · proof, irrationality of 1/√2

   1. 1.Let us assume, to the contrary, that 1/√2 is a rational number. So, we can find integers p and q (q ≠ 0) such that 1/√2 = p/q, where p and q are coprime.1 mark
   2. 2.Rearranging the equation 1/√2 = p/q, we get √2 = q/p. Since p and q are integers, q/p is rational, so √2 is rational. But this contradicts the fact that √2 is irrational. This contradiction has arisen because of our incorrect assumption that 1/√2 is rational. So, we conclude that 1/√2 is irrational.1 mark

   • Start by writing 'Let us assume, to the contrary, that 1/√2 is rational' and express it as p/q with q ≠ 0.
   • End with the word 'contradiction' and state clearly that this contradicts √2 being irrational.
   • Show the step √2 = q/p before concluding, since this is what proves the contradiction.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

2. Q2Short answer2 marksAsked: 2022, 2026

   Prove that 6 + sqrt(2) is an irrational number.

   

   Medha Ma’am

   AskMedha’s AI teacher

   Hear it from Medha Ma’am · 50 s

   Model answer · proof, irrational numbers

   1. 1.Let us assume, to the contrary, that 6 + √2 is a rational number. That is, we can find integers p and q (q ≠ 0) such that 6 + √2 = p/q.1 mark
   2. 2.Rearranging this equation, we get √2 = p/q - 6 = (p - 6q)/q. Since p and q are integers, (p - 6q)/q is a rational number, and so √2 is rational. But this contradicts the fact that √2 is irrational. This contradiction has arisen because of our incorrect assumption that 6 + √2 is rational. So, we conclude that 6 + √2 is irrational.1 mark

   • Start by writing 6 + √2 = p/q with p, q integers and q ≠ 0.
   • Rearrange to isolate √2 as (p - 6q)/q before stating the contradiction.
   • End by stating √2 is irrational, so the assumption is wrong and 6 + √2 is irrational.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

3. Q3Short answer2 marksAsked: 2025, 2026

   Using Euclid's division algorithm find the H.C.F. of 135 and 225.

   

   Medha Ma’am

   AskMedha’s AI teacher

   Hear it from Medha Ma’am · 60 s

   Model answer · Euclid's division algorithm, HCF

   1. 1.Since 225 is greater than 135, we apply Euclid's division lemma to 225 and 135 to get 225 = 135 × 1 + 90.1 mark
   2. 2.Since the remainder 90 is not zero, we apply the division lemma to 135 and 90 to get 135 = 90 × 1 + 45. Now we consider the new divisor 90 and the new remainder 45, and apply the division lemma to get 90 = 45 × 2 + 0. The remainder has now become zero, so our procedure stops. Since the divisor at this stage is 45, the HCF of 135 and 225 is 45.1 mark

   • Start the first line with '225 = 135 × 1 + 90', naming it as Euclid's division lemma.
   • Write both remaining steps '135 = 90 × 1 + 45' and '90 = 45 × 2 + 0' before stating the HCF.
   • End with the sentence stating the HCF of 135 and 225 is 45, since the remainder has become zero.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

4. Q4Short answer2 marksAsked: 2020, 2022

   Using Euclid's division algorithm, find the HCF of 252 and 1540.

   How the model answer begins

   1. By Euclid's division lemma, for two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b. Taking the larger integer 1540 as a and 252 as b, we apply the division lemma to 1540…

   The full answer from Medha Ma’am →
5. Q5Short answer2 marksAsked: 2017, 2023

   Find the H.C.F. and L.C.M. of 24 and 36 using prime factorisation.

   How the model answer begins

   1. The prime factorisation of 24 and 36 is found by expressing each number as a product of primes. 24 = 2 × 2 × 2 × 3 = 2³ × 3 36 = 2 × 2 × 3 × 3 = 2² × 3²

   The full answer from Medha Ma’am →
6. Q6Short answer2 marksAsked: 2025

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

   How the model answer begins

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

   The full answer from Medha Ma’am →
7. Q7Short answer2 marksAsked: 2026

   Find the largest number which divides 245 and 1037, leaving remainder 5 in each case.

   How the model answer begins

   1. The required number must divide 245 and 1037 leaving remainder 5 in each case. So subtracting the remainder 5 from the given numbers, we get 245 - 5 = 240 and 1037 - 5 = 1032. The required largest number is the HCF of…

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8. Q8Short answer2 marksAsked: 2026

   Without actual division, show that the rational number 77/210 is a non-terminating repeating decimal.

   How the model answer begins

   1. The rational number 77/210, when reduced to its simplest form by cancelling the common factor 7 from numerator and denominator, is 11/30.

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9. Q9Short answer2 marksAsked: 2026

   Simplify : (2sqrt(27) + 3sqrt(12)) / (4sqrt(3)).

   How the model answer begins

   1. We simplify 2√27 and 3√12 by breaking the numbers under the root into prime factors. Since 27 = 9 × 3, we get √27 = 3√3, so 2√27 = 2 × 3√3 = 6√3. Since 12 = 4 × 3, we get √12 = 2√3, so 3√12 = 3 × 2√3 = 6√3.

   The full answer from Medha Ma’am →
10. Q10Short answer2 marksAsked: 2022

    Determine whether the decimal expansion of 13/3125 is terminating or not.

    How the model answer begins

    1. The prime factorization of the denominator 3125 is obtained as follows: 3125 = 5 × 5 × 5 × 5 × 5 = 5^5. So the denominator 3125 can be written as 5^5.

    The full answer from Medha Ma’am →
11. Q11Short answer2 marksAsked: 2022

    Express 0.0875 in the form of p/q where there is no common factor in p and q.

    How the model answer begins

    1. 0.0875 can be written as 875/10000, since the number has four digits after the decimal point, so 0.0875 = 875/10000.

    The full answer from Medha Ma’am →
12. Q12Short answer2 marksAsked: 2022

    Express 1.8 (bar over 8) as a fraction in simplest form.

    How the model answer begins

    1. Let x = 1.8888... and call this equation 1.

    The full answer from Medha Ma’am →
13. Q13Short answer2 marksAsked: 2023

    Find the H.C.F. and L.C.M. of 24 and 36 using prime factorisation.

    How the model answer begins

    1. The prime factorisation of 24 and 36 is found as follows. 24 = 2 × 2 × 2 × 3 36 = 2 × 2 × 3 × 3 So, 24 = 2^3 × 3^1 and 36 = 2^2 × 3^2.

    The full answer from Medha Ma’am →
14. Q14Short answer2 marksAsked: 2021

    Prove that 2 + sqrt(2) is an irrational number.

    How the model answer begins

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

    The full answer from Medha Ma’am →
15. Q15Short answer2 marksAsked: 2021

    Write 2.4 bar in the simplest form of a fraction.

    How the model answer begins

    1. Let x = 2.444..., where the digit 4 repeats indefinitely after the decimal point.

    The full answer from Medha Ma’am →
16. Q16Short answer2 marksAsked: 2021

    Find the H.C.F. of 255 and 867 using Euclid's division algorithm.

    How the model answer begins

    1. By Euclid's division lemma, for positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b. Since 867 > 255, we apply this lemma to 867 and 255. We get 867 = 255 × 3 + 102. Since the…

    The full answer from Medha Ma’am →
17. Q17Short answer2 marksAsked: 2021

    Prove that 5 + √3 is an irrational number.

    How the model answer begins

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

    The full answer from Medha Ma’am →
18. Q18Short answer2 marksAsked: 2021

    Using Euclid's division algorithm, find the H.C.F. of 231 and 396.

    How the model answer begins

    1. Since 396 is greater than 231, we apply Euclid's division lemma to 396 and 231, to get 396 = 231 × 1 + 165.

    The full answer from Medha Ma’am →
19. Q19Short answer2 marksAsked: 2021

    Convert 0.12̄ into the form p/q.

    How the model answer begins

    1. Let x = 0.1222... (here the digit 2 repeats after the decimal point).

    The full answer from Medha Ma’am →

Objective questions with the board’s key (10)

1. 1. Bihar Board 2026

   According to Euclid's division lemma for any two given positive integers a and b, there exist unique integers q and r satisfying the equation a = bq + r where

   • (A) 0 <= b < r
   • (B) 0 <= r < b ✓
   • (C) 0 >= b > r
   • (D) 0 >= r >= b

   Answer: (B)

2. 2. Bihar Board 2026

   According to Euclid's division algorithm the HCF of 867 and 255 is

   • (A) 13
   • (B) 15
   • (C) 17
   • (D) 51 ✓

   Answer: (D)

3. 3. Bihar Board 2026

   Product of prime factors of 7429 is

   • (A) 17 x 19 x 23 ✓
   • (B) 13 x 17 x 19
   • (C) 13 x 19 x 23
   • (D) 17 x 19 x 29

   Answer: (A)

4. 4. Bihar Board 2026

   If c and d are the LCM and HCF of two numbers a and b respectively then

   • (A) a + b = c + d
   • (B) a - b = c - d
   • (C) ab = cd ✓
   • (D) ad = bc

   Answer: (C)

5. 5. Bihar Board 2026

   Which of the following numbers is a rational number ?

   • (A) sqrt(2)
   • (B) sqrt(3)
   • (C) sqrt(4) ✓
   • (D) sqrt(5)

   Answer: (C)

6. 6. Bihar Board 2026

   The number whose generalised form is p/q where p and q are integers and q!=0 is said to be

   • (A) even number
   • (B) odd number
   • (C) rational number ✓
   • (D) irrational number

   Answer: (C)

7. 7. Bihar Board 2026

   The number whose decimal expansion is neither terminating nor repeating is said to be

   • (A) irrational number ✓
   • (B) rational number
   • (C) prime number
   • (D) integer

   Answer: (A)

8. 8. Bihar Board 2026

   (3 - 2sqrt(5))^2 =

   • (A) 29 - 12sqrt(5) ✓
   • (B) 29 + 12sqrt(5)
   • (C) 29
   • (D) - 11

   Answer: (A)

9. 9. Bihar Board 2026

   Which of the following fractions is expressible as a terminating decimal?

   • (A) 19/105
   • (B) 67/210
   • (C) 13/88
   • (D) 3/160 ✓

   Answer: (D)

10. 10. Bihar Board 2026

    If in division algorithm a = bq + r, a = 72, b = 9 then r =

    • (A) 8
    • (B) -8
    • (C) 0 ✓
    • (D) 1

    Answer: (C)

Questions parents and students ask

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

In the Bihar Board papers of 2021–2026, this chapter carried about 6 marks and about 10 questions a year; by marks it ranks 10 in Maths.

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

The 19 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 19 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

BSEB does not publish a marking scheme. Each answer is written point by point on a structure derived from past papers and checked by a second, independent model; it is never called the board’s official scheme.