Class 10 Maths · Chapter 1

Real Numbers

336 Class 10 Maths questions on Real Numbers from the NCERT textbook, NCERT Exemplar and CBSE board papers 2018–2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 8 of 17

1. Q141 CBSE 2025
   Prove that $\sqrt{2}$ ​is an irrational ​number.
   Show answer

   Assuming $\sqrt{2}=\frac{p}{q}$ with $p$ and $q$ co-prime makes $2$ a common factor of $p$ and $q$, a contradiction; so the assumption is wrong, and $\sqrt{2}$ is irrational.

   Step-by-step solution →
2. Q142 CBSE 2025
   Prove that $\sqrt{3}$ ​is ​an irrational ​number.
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   $\sqrt{3}$ is irrational: assuming $\sqrt{3}=\frac{p}{q}$ with $p,q$ coprime gives $p^2=3q^2$, so $3$ divides $p$; then $q^2=3k^2$, so $3$ divides $q$, which contradicts that $p$ and $q$ are coprime. Hence proved.

   Step-by-step solution →
3. Q143 CBSE 2025
   Find ​which among the following ​numbers $a$, $b$ and $c$ ​is/are composite ​numbers: $a=7\times 11\times 13+13$; $b=6\times 5\times 4+4$; $c=7\times 13+6$.
   Show answer

   $a=1014=2\times 3\times 13^2$ and $b=124=2^2\times 31$ are composite numbers; $c=97$ is a prime number, so it is not composite.

   Step-by-step solution →
4. Q144 CBSE 2025
   Let $x$ and $y$ ​be two distinct ​prime numbers ​and $p=x^2{y}^3,q=x{y}^4,r=x^5{y}^2$. ​Find the HCF ​and LCM of $p$, $q$ and $r$. Further check ​if $\text{HCF}(p,q,r)\times \text{LCM}(p,q,r)=p\times q\times r$ or ​not.
   Show answer

   $\text{HCF}(p,q,r)=x{y}^2$ and $\text{LCM}(p,q,r)=x^5{y}^4$; since $\text{HCF}\times \text{LCM}=x^6{y}^6$ while $p\times q\times r=x^8{y}^9$, the relation does not hold.

   Step-by-step solution →
5. Q145 CBSE 2025
   The factor tree of ​a ​number $x$ is shown ​below. Find the ​values of $x$, $y$, $a$ ​and $b$. Hence, write the product of ​the prime ​factors of the ​number $x$ so obtained.
   Show answer

   $x=840$, $y=420$, $a=3$ and $b=7$. The product of the prime factors of $x$ is $840=2\times 2\times 2\times 3\times 5\times 7=2^3\times 3\times 5\times 7$.

   Step-by-step solution →
6. Q146 CBSE 2025
   State ​the Fundamental ​Theorem ​of ​Arithmetic ​and use it to ​find LCM ​of $36$ and $54$.
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   Fundamental Theorem of Arithmetic: every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. $36=2^2\times 3^2$ and $54=2\times 3^3$, so $\text{LCM}(36,54)=2^2\times 3^3=108$.

   Step-by-step solution →
7. Q147 CBSE 2025
   State true or false ​for each of the following statements ​and ​justify in ​each case: ​

   1. (i)$2\times 3\times 5\times 7+7$ is ​a composite number. ​
   2. (ii)$2\times 3\times 5\times 7+1$ is a composite number.
   Show answer

   (i) True: $2\times 3\times 5\times 7+7=7\times 31=217$ has the factor $7$, so it is composite; (ii) False: $2\times 3\times 5\times 7+1=211$ is prime, so it is not composite.

   Step-by-step solution →
8. Q148 CBSE 2025
   Let $p$, $q$ and $r$ be three distinct prime ​numbers. Check whether $p\cdot q\cdot r+q$ is ​a composite number or ​not. ​Further, give an ​example for $3$ distinct primes $p$, $q$, $r$ such that

   1. (i)$p\cdot q\cdot r+1$ is a composite ​number,
   2. (ii)$p\cdot q\cdot r+1$ ​is ​a prime number.
   Show answer

   $pqr+q=q(pr+1)$ is a composite number. (i) For $p=2$, $q=5$, $r=11$: $pqr+1=111=3\times 37$, which is composite; (ii) For $p=2$, $q=3$, $r=5$: $pqr+1=31$, which is prime.

   Step-by-step solution →
9. Q149 CBSE 2025
   Prove ​that $\left(5\sqrt{3}+\frac{2}{3}\right)$ is an ​irrational ​number, ​given ​that $\sqrt{3}$ ​is an ​irrational number.
   Show answer

   Assuming $5\sqrt{3}+\frac{2}{3}$ is rational makes $\sqrt{3}=\frac{3a-2b}{15b}$ rational, which contradicts the given fact. Hence $5\sqrt{3}+\frac{2}{3}$ is an irrational number.

   Step-by-step solution →
10. Q150 CBSE 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 will ​each ​rod be used to measure the length of the fence?
    Show answer

    The least length of the fence is $600$ cm ($6$ m). The $120$ cm rod is used $5$ times, the $100$ cm rod $6$ times and the $150$ cm rod $4$ times.

    Step-by-step solution →
11. Q151 CBSE 2025
    Prove that $\frac{1}{\sqrt{5}}$ is ​an ​irrational ​number.
    Show answer

    Assuming $\frac{1}{\sqrt{5}}=\frac{p}{q}$ with $p$, $q$ coprime leads to $5$ dividing both $p$ and $q$, which is a contradiction; hence $\frac{1}{\sqrt{5}}$ is irrational.

    Step-by-step solution →
12. Q152 CBSE 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?
    Show answer

    The traffic lights will change together next at $5:15$ a.m., that is, $900$ seconds ($15$ minutes) after $5:00$ a.m.

    Step-by-step solution →
13. Q153 CBSE 2025
    Ranjita, Neha and Salma start weaving sweaters ​at ​the same time for the children of ​an orphan ​home. They need 15, 18 and 20 days, respectively, to ​complete a sweater. After how many days will all ​of them start making a new sweater again? ​By that time how many sweaters will have been completed by them?
    Show answer

    All three start making a new sweater together again after $180$ days. By then $31$ sweaters have been completed: Ranjita $12$, Neha $10$ and Salma $9$.

    Step-by-step solution →
14. Q154 CBSE 2025
    The difference of the squares of ​two ​positive numbers is $45$. ​The ​square of the smaller number is $4$ times the greater ​number. ​Determine ​the numbers.
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    The numbers are $9$ (the greater) and $6$ (the smaller).

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15. Q155 CBSE 2025
    A Mathematics exhibition is being ​conducted ​in your school ​and one of your ​friends ​is making a model of a 'factor tree'. He has some difficulty and asks for your help in completing ​a quiz for the audience. Observe the following 'factor tree' ​and answer the following questions :

    1. (i)What is the value of $a$?[1]
    2. (ii)(a)What is the value of $b$?[2]
    3. (ii)(b)What is the value of $c$? (OR alternative to (ii)(a))[2]
    4. (iii)Write the prime factorisation of $24115$.[1]
    Show answer

    (i) $a=24115$; (ii)(a) $b=13$; (ii)(b) $c=53$; (iii) $24115=5\times 7\times 13\times 53$

    Step-by-step solution →
16. Q156 CBSE 2025
    The science department of a college is conducting ​an international seminar in which the number of participants in Physics, Chemistry and Biology ​are $65$, $91$ and $117$ respectively. The coordinator has made ​the arrangement ​such that in each room, ​the same number ​of participants are to be seated with all of them being in the same ​subject. Based on the information given above, answer ​the following questions:

    1. (i)Find the HCF of $65$, $91$ and $117$.[1]
    2. (ii)Find the LCM of $65$, $91$ and $117$.[1]
    3. (iii)(a)Find the minimum number of rooms required based on the above conditions.[2]
    4. (iii)(b)OR Find the minimum number of participants to be accommodated in each of the rooms.[2]
    Show answer

    (i) $\text{HCF}=13$; (ii) $\text{LCM}=4095$; (iii)(a) The minimum number of rooms is $21$ ($5$ for Physics, $7$ for Chemistry, $9$ for Biology); (iii)(b) Each room accommodates $13$ participants.

    Step-by-step solution →
17. Q157 CBSE 2024
    The total numbers ​of ​factors ​of ​a ​prime ​number ​is :

    1. (a)$1$
    2. (b)$0$
    3. (c)$2$
    4. (d)$3$
    Show answer

    (c) $2$

    Step-by-step solution →
18. Q158 CBSE 2024
    The ​LCM of 24, ​36 and 60 ​in ​terms of their ​prime factors is :

    1. (a)$2^2\times 3\times 5$
    2. (b)$2^3\times 3^2$
    3. (c)$2^3\times 3^2\times 5$
    4. (d)$2^3\times 3^3\times 5$
    Show answer

    (c) $2^3\times 3^2\times 5$

    Step-by-step solution →
19. Q159 CBSE 2024
    The HCF of ​the smallest $2$-digit number ​and ​the smallest composite number is ​:

    1. (a)$2$
    2. (b)$20$
    3. (c)$40$
    4. (d)$4$
    Show answer

    (a) $2$

    Step-by-step solution →
20. Q160 CBSE 2024
    HCF $\times$ ​LCM ​for the numbers ​40 and ​30 is ​:

    1. (a)12
    2. (b)120
    3. (c)1200
    4. (d)40
    Show answer

    (c) 1200

    Step-by-step solution →