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 11 of 17

1. Q201 CBSE 2024
   Prove that $\sqrt{3}$ ​is ​an irrational ​number.
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   Assume $\sqrt{3}=\frac{a}{b}$ with $a$, $b$ coprime integers and $b\ne 0$. Then $3b^2=a^2$, so $3$ divides $a$; writing $a=3c$ gives $b^2=3c^2$, so $3$ divides $b$. Thus $3$ is a common factor of $a$ and $b$, which contradicts that they are coprime. Hence $\sqrt{3}$ is irrational.

   Step-by-step solution →
2. Q202 CBSE 2024
   Prove ​that ${\left(\sqrt{2}+\sqrt{3}\right)}^2$ is an ​irrational number, ​given that $\sqrt{6}$ is an ​irrational number.
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   $(\sqrt{2}+\sqrt{3}{)}^2=5+2\sqrt{6}$. If it were rational, say $\frac{p}{q}$, then $\sqrt{6}=\frac{p-5q}{2q}$ would be rational, which contradicts the given fact that $\sqrt{6}$ is irrational. Hence $(\sqrt{2}+\sqrt{3}{)}^2$ is irrational. Hence proved.

   Step-by-step solution →
3. Q203 CBSE 2024
   Find ​the values of $x$, $y$, $z$ and $w$ in the ​following factor ​tree. ​Also, ​write the prime ​factorisation of $x$.
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   $w=91$, $z=273$, $y=1638$ and $x=3276$. The prime factorisation of $x$ is $3276=2^2\times 3^2\times 7\times 13$.

   Step-by-step solution →
4. Q204 CBSE 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.
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   The minimum number of rooms required is $17$, with $16$ teachers in each room ($3$ for French, $5$ for Hindi, $9$ for English).

   Step-by-step solution →
5. Q205 CBSE 2024
   Prove that $3+2\sqrt{5}$ ​is an ​irrational ​number, ​given that $\sqrt{5}$ ​is an ​irrational ​number.
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   Assuming $3+2\sqrt{5}=\frac{a}{b}$ with integers $a,b$ and $b\ne 0$ gives $\sqrt{5}=\frac{a-3b}{2b}$, which is rational. This contradicts the given fact that $\sqrt{5}$ is irrational, so $3+2\sqrt{5}$ is irrational. Hence proved.

   Step-by-step solution →
6. Q206 CBSE 2024
   Prove that $7-3\sqrt{5}$ ​is an ​irrational number, ​given ​that $\sqrt{5}$ is ​an ​irrational ​number.
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   Assuming $7-3\sqrt{5}=\frac{a}{b}$ gives $\sqrt{5}=\frac{7b-a}{3b}$, a rational number, which contradicts the fact that $\sqrt{5}$ is irrational. Hence $7-3\sqrt{5}$ is irrational.

   Step-by-step solution →
7. Q207 CBSE 2024
   Prove that $\frac{2-\sqrt{3}}{5}$ is ​an irrational ​number, ​given that $\sqrt{3}$ ​is an irrational number.
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   Assuming $\frac{2-\sqrt{3}}{5}=\frac{p}{q}$ gives $\sqrt{3}=\frac{2q-5p}{q}$, which is rational — a contradiction. Hence $\frac{2-\sqrt{3}}{5}$ is irrational.

   Step-by-step solution →
8. Q208 CBSE 2024
   Prove that $2+5\sqrt{3}$ ​is ​an ​irrational number, if ​it is ​given that $\sqrt{3}$ is ​an irrational ​number.
   Show answer

   $2+5\sqrt{3}$ is irrational. Assuming $2+5\sqrt{3}=\frac{a}{b}$ (integers $a$, $b$, $b\ne 0$) gives $\sqrt{3}=\frac{a-2b}{5b}$, which would be rational, contradicting the given fact that $\sqrt{3}$ is irrational. Hence proved.

   Step-by-step solution →
9. Q209 CBSE 2024
   Prove ​that $(3+2\sqrt{5})$ is an irrational ​number, given ​that $\sqrt{5}$ is ​an ​irrational ​number.
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   Assuming $3+2\sqrt{5}=\frac{a}{b}$ (integers $a,b$, $b\ne 0$) gives $\sqrt{5}=\frac{a-3b}{2b}$, which is rational. This contradicts the given fact that $\sqrt{5}$ is irrational, so $3+2\sqrt{5}$ is irrational. Hence proved.

   Step-by-step solution →
10. Q210 CBSE 2024
    A school has invited ​42 ​Mathematics teachers, ​56 Physics teachers and 70 Chemistry teachers to attend a Science workshop. Find the ​minimum number of tables required, if the same number of teachers ​are to sit at a table and each table is ​occupied by teachers of the same ​subject.
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    Minimum number of tables $=12$ ($14$ teachers at each table: $3$ Mathematics, $4$ Physics and $5$ Chemistry tables).

    Step-by-step solution →
11. Q211 CBSE 2024
    Teaching Mathematics ​through activities is a powerful approach that enhances ​students' ​understanding and engagement. Keeping this in mind, Ms. Mukta planned a ​prime number game for class $5$ students. She announces the number $2$ in her class and asked the ​first student to multiply it by a prime number and then pass it to ​second student. Second ​student also multiplied it by a prime ​number and passed it to third student. In this way by multiplying to a prime number, the last student got $173250$. Now, Mukta asks some questions as given below to the students :

    1. (i)What is the least prime number used by students?[1]
    2. (ii)(a)How many students are in the class?[2]
    3. (ii)(b)OR What is the highest prime number used by students?[2]
    4. (iii)Which prime number has been used maximum times?[1]
    Show answer

    (i) The least prime used by the students is $3$; (ii)(a) There are $7$ students in the class; OR (ii)(b) The highest prime used by the students is $11$; (iii) The prime $5$ was used the maximum number of times ($3$ times).

    Step-by-step solution →
12. Q212 CBSE 2023
    The greatest ​number ​which ​divides ​both 30 and 80, leaving ​remainder 2 ​and 3 ​respectively, is ​:

    1. (a)10
    2. (b)7
    3. (c)11
    4. (d)14
    Show answer

    (b) 7

    Step-by-step solution →
13. Q213 CBSE 2023
    If two positive ​integers $x$ ​and $y$ ​are written as $x=a^3{b}^2$ and $y=a{b}^3$, ​where $a$ and $b$ are prime numbers, ​then their ​HCF $(x,y)$ is ​:

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

    (b) $a{b}^2$

    Step-by-step solution →
14. Q214 CBSE 2023
    If $p^2=\frac{32}{50}$, then $p$ ​is a/an

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

    (c) rational number

    Step-by-step solution →
15. Q215 CBSE 2023
    The ​prime factorisation of ​natural ​number $288$ ​is

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

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

    Step-by-step solution →
16. Q216 CBSE 2023
    The ratio of HCF ​to ​LCM ​of the ​least composite number ​and ​the least ​prime ​number is :

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

    (a) $1:2$

    Step-by-step solution →
17. Q217 CBSE 2023
    The ​greatest number ​which divides both $83$ ​and $138$, ​leaving remainders $5$ ​and $8$ ​respectively, ​is:

    1. (a)$13$
    2. (b)$65$
    3. (c)$26$
    4. (d)$39$
    Show answer

    (c) $26$

    Step-by-step solution →
18. Q218 CBSE 2023
    If $a$ and $b$ ​are ​two consecutive ​natural numbers, ​then $\text{HCF}(a,b)$ ​is ​:

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

    (d) $1$

    Step-by-step solution →
19. Q219 CBSE 2023
    The ​product of ​the ​HCF ​and LCM of two ​numbers 50 and ​20 is ​:

    1. (a)10
    2. (b)100
    3. (c)1000
    4. (d)20
    Show answer

    (c) 1000

    Step-by-step solution →
20. Q220 CBSE 2023
    The ratio ​of HCF ​and LCM ​of the ​least prime number ​and the least ​composite ​number, is ​:

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

    (a) $1:2$

    Step-by-step solution →