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 5 of 17

1. Q81 CBSE 2026
   Prove ​that $\sqrt{5}$ is an irrational ​number.
   Show answer

   Assuming $\sqrt{5}=\frac{a}{b}$ with $a,b$ coprime and $b\ne 0$ leads to $5$ dividing both $a$ and $b$, contradicting that they are coprime. Hence the assumption is wrong and $\sqrt{5}$ is an irrational number.

   Step-by-step solution →
2. Q82 CBSE 2024
   Prove ​that $\sqrt{3}$ is ​an irrational number.
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   Assuming $\sqrt{3}=\frac{a}{b}$ with $a$, $b$ coprime integers and $b\ne 0$ gives $a^2=3b^2$, from which $3$ divides $a$ and then $3$ divides $b$. This contradicts that $a$ and $b$ are coprime, so the assumption is wrong and $\sqrt{3}$ is irrational. Hence proved.

   Step-by-step solution →
3. Q83 CBSE 2026
   Prove that $\sqrt{2}$ ​is an irrational ​number.
   Show answer

   Assuming $\sqrt{2}=\frac{p}{q}$ in lowest terms leads to $2$ dividing both $p$ and $q$, contradicting that they are coprime. Therefore $\sqrt{2}$ is irrational.

   Step-by-step solution →
4. Q84 CBSE 2026
   Prove that $\frac{2+3\sqrt{5}}{7}$ is ​an ​irrational number, ​given ​that $\sqrt{5}$ is an ​irrational number.
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   Assuming $\frac{2+3\sqrt{5}}{7}$ is rational forces $\sqrt{5}=\frac{7p-2q}{3q}$ to be rational, which contradicts the given fact that $\sqrt{5}$ is irrational. Therefore $\frac{2+3\sqrt{5}}{7}$ is irrational. Hence proved.

   Step-by-step solution →
5. Q85 CBSE 2026
   Given that $\sqrt{5}$ ​is an ​irrational number, prove ​that $3+2\sqrt{5}$ is also ​an ​irrational ​number.
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   Assuming $3+2\sqrt{5}=\frac{p}{q}$ with integers $p,q$ ($q\ne 0$) leads to $\sqrt{5}=\frac{p-3q}{2q}$, a rational number - contradicting the given fact that $\sqrt{5}$ is irrational. Hence the assumption is wrong and $3+2\sqrt{5}$ is an irrational number.

   Step-by-step solution →
6. Q86 CBSE 2026
   Neha claimed that there ​does not exist any irrational number ​between $1$ ​and $2$. Raunak claimed that $\sqrt{2}$ ​lies between $1$ and $2$ and $\sqrt{2}$ is an irrational number. Who do you think ​is ​correct ? ​Justify by proving either $\sqrt{2}$ as ​an irrational number or otherwise.
   Show answer

   Raunak is correct : $\sqrt{2}$ is an irrational number (proved by contradiction) and it lies between $1$ and $2$, since $1<2<4$ gives $1<\sqrt{2}<2$. Neha's claim is therefore false.

   Step-by-step solution →
7. Q87 CBSE 2020
   Prove ​that $\sqrt{2}$ ​is an ​irrational number.
   Show answer

   $\sqrt{2}$ is irrational. Assuming $\sqrt{2}=\frac{p}{q}$ in lowest terms forces $2$ to divide both $p$ and $q$, contradicting that they are coprime; hence no such fraction exists.

   Step-by-step solution →
8. Q88 CBSE 2023
   Prove ​that $\sqrt{5}$ is ​an irrational number.
   Show answer

   $\sqrt{5}$ is irrational: assuming $\sqrt{5}=\frac{a}{b}$ with $a$, $b$ coprime integers and $b\ne 0$ gives $5b^2=a^2$, so $5$ divides $a$; putting $a=5c$ gives $b^2=5c^2$, so $5$ divides $b$. Then $5$ is a common factor of $a$ and $b$, which contradicts that they are coprime. Hence proved.

   Step-by-step solution →
9. Q89 CBSE 2026
   Find ​the greatest number less ​than $10,000$ ​which is exactly divisible ​by $48$, $60$ ​and $65$.
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   The greatest number is $9360$.

   Step-by-step solution →
10. Q90 CBSE 2026
    The ​dimensions of a window are $156$ cm $\times$ $216$ cm. Arjun wants to put grill on the window creating ​complete squares of maximum size. Determine the ​side ​length ​of ​the ​square and hence find the number of squares formed.
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    The side of each square is $12$ cm and the number of squares formed is $234$.

    Step-by-step solution →
11. Q91 CBSE 2025
    If $x=a{b}^3$ and $y=a^{3}b$, ​where $a$ and $b$ ​are ​prime ​numbers, ​then $[\text{HCF}(x,y)-\text{LCM}(x,y)]$ ​is equal to:

    1. (a)$1-a^3{b}^3$
    2. (b)$ab(1-ab)$
    3. (c)$ab-a^4{b}^4$
    4. (d)$ab(1-ab)(1+ab)$
    Show answer

    (d) $ab(1-ab)(1+ab)$

    Step-by-step solution →
12. Q92 CBSE 2025
    The ratio ​of ​the HCF ​to the LCM ​of $7$, $21$ ​and $28$ ​is:

    1. (a)$1:4$
    2. (b)$3:4$
    3. (c)$1:8$
    4. (d)$1:12$
    Show answer

    (d) $1:12$

    Step-by-step solution →
13. Q93 CBSE 2025
    If $p=2^3\times 3^2\times 5$ and $q=2^2\times 3^3$, ​then ​the LCM ​of $p$ ​and $q$ is ​:

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

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

    Step-by-step solution →
14. Q94 CBSE 2025
    The greatest possible speed ​at which ​a ​man can walk $78$ m and $104$ m in ​exact ​number of ​minutes, is:

    1. (a)$17$ m/min
    2. (b)$13$ m/min
    3. (c)$26$ m/min
    4. (d)$39$ m/min
    Show answer

    (c) $26$ m/min

    Step-by-step solution →
15. Q95 CBSE 2025
    $(2-5\sqrt{3}{)}^2$ is

    1. (a)a negative integer
    2. (b)an irrational number
    3. (c)a rational number
    4. (d)a positive integer
    Show answer

    (b) an irrational number

    Step-by-step solution →
16. Q96 CBSE 2025
    $\sqrt{0.4}$ is a/an

    1. (a)natural number
    2. (b)integer
    3. (c)rational number
    4. (d)irrational number
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    (d) irrational number

    Step-by-step solution →
17. Q97 CBSE 2025
    If ​HCF $(x,20)=2$ ​and ​LCM $(x,20)=60$, ​then ​value ​of $x$ is :

    1. (a)$3$
    2. (b)$6$
    3. (c)$20$
    4. (d)$10$
    Show answer

    (b) $6$

    Step-by-step solution →
18. Q98 CBSE 2025
    The ​sum ​of ​first five prime numbers is

    1. (a)$18$
    2. (b)$26$
    3. (c)$28$
    4. (d)$39$
    Show answer

    (c) $28$

    Step-by-step solution →
19. Q99 CBSE 2025
    The value ​of ​(HCF $-$ LCM) ​for the two ​numbers ​3 ​and ​5 is :

    1. (a)2
    2. (b)4
    3. (c)14
    4. (d)$-14$
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    (d) $-14$

    Step-by-step solution →
20. Q100 CBSE 2025
    ${\left(\sqrt{3}+2\right)}^2+{\left(\sqrt{3}-2\right)}^2$ ​is ​a/an

    1. (a)positive rational number
    2. (b)negative rational number
    3. (c)positive irrational number
    4. (d)negative irrational number
    Show answer

    (a) positive rational number

    Step-by-step solution →