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 7 of 17

1. Q121 CBSE 2025
   Assertion (A): $4^n$ ends with digit $0$ for some natural number $n$.

   Reason (R): For a number $x$ having $2$ and $5$ as its prime factors, $x^n$ always ends with digit $0$ for every natural number $n$.

   1. (a)Both A and R are true and R is the correct explanation of A.
   2. (b)Both A and R are true but R is not the correct explanation of A.
   3. (c)A is true but R is false.
   4. (d)A is false but R is true.
   Show answer

   (d) A is false but R is true.

   Step-by-step solution →
2. Q122 CBSE 2025
   Directions: In the following question, a statement of ​Assertion (A) ​is ​followed ​by ​a statement ​of Reason (R). Choose ​the correct option.

   Assertion (A): If the HCF of two numbers is $16$ and their product is $3072$, then their LCM is $162$.

   Reason (R): If $a$ and $b$ are two positive integers, then HCF $\times$ LCM $=a\times b$.

   1. (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
   2. (b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
   3. (c)Assertion (A) is true but Reason (R) is false.
   4. (d)Assertion (A) is false but Reason (R) is true.
   Show answer

   (d) Assertion (A) is false but Reason (R) is true.

   Step-by-step solution →
3. Q123 CBSE 2025
   Assertion (A): $(a+\sqrt{b})(a-\sqrt{b})$ ​is a ​rational ​number, where $a$ and $b$ ​are positive integers. Reason (R): ​The product ​of two irrationals is always rational. Choose ​the correct ​option.

   Assertion (A): $(a+\sqrt{b})(a-\sqrt{b})$ is a rational number, where $a$ and $b$ are positive integers.

   Reason (R): The product of two irrationals is always rational.

   1. (a)Both A and R are true, and R is the correct explanation of A.
   2. (b)Both A and R are true, but R is not the correct explanation of A.
   3. (c)A is true but R is false.
   4. (d)A is false but R is true.
   Show answer

   (c) A is true but R is false.

   Step-by-step solution →
4. Q124 CBSE 2025
   Assertion (A): The prime numbers which divide 36 also divide 6.

   Reason (R): Any number which divides $p^2$ also divides $p$.

   1. (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
   2. (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
   3. (c)Assertion (A) is true, but Reason (R) is false.
   4. (d)Assertion (A) is false, but Reason (R) is true.
   Show answer

   (c) Assertion (A) is true, but Reason (R) is false.

   Step-by-step solution →
5. Q125 CBSE 2025
   Assertion (A): For any two natural numbers $a$ and $b$, the HCF of $a$ and $b$ is a factor of the LCM of $a$ and $b$.

   Reason (R): HCF of any two natural numbers divides both the numbers.

   1. (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
   2. (b)Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
   3. (c)Assertion (A) is true but Reason (R) is false.
   4. (d)Assertion (A) is false but Reason (R) is true.
   Show answer

   (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

   Step-by-step solution →
6. Q126 CBSE 2025
   Assertion (A): For any ​two ​prime numbers $p$ ​and $q$, their ​HCF is $1$ ​and LCM is $p+q$. Reason (R): For any two natural numbers, $\text{HCF}\times \text{LCM}=$ product ​of the numbers.

   Assertion (A): For any two prime numbers $p$ and $q$, their HCF is $1$ and LCM is $p+q$.

   Reason (R): For any two natural numbers, $\text{HCF}\times \text{LCM}=$ product of numbers.

   1. (a)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
   2. (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
   3. (c)Assertion (A) is true, but Reason (R) is false.
   4. (d)Assertion (A) is false, but Reason (R) is true.
   Show answer

   (d) Assertion (A) is false, but Reason (R) is true.

   Step-by-step solution →
7. Q127 CBSE 2025
   In a school, there ​are two Sections A and ​B of Class X. There are $48$ students in Section A and $36$ students in Section B. Determine the minimum number of books required for ​their class ​library so that ​they can be distributed equally among the students of Section A ​or ​that of Section ​B.
   Show answer

   The minimum number of books required is $144$.

   Step-by-step solution →
8. Q128 CBSE 2025
   Find ​the ​greatest number which divides $285$ ​and $1249$ leaving ​remainders $9$ ​and $7$ ​respectively.
   Show answer

   The greatest number is $138$.

   Step-by-step solution →
9. Q129 CBSE 2025
   Show that $45^n$ cannot end with the digit $0$, $n$ ​being ​a natural number. Write ​the prime number '$a$' which on multiplying with $45^n$ ​makes the product ​end ​with the ​digit $0$.
   Show answer

   $45^n=3^{2n}\times 5^n$ has no factor $2$, so it is not divisible by $10$ and cannot end with the digit $0$. The required prime is $a=2$, since $2\times 45^n$ contains the factor $2\times 5=10$.

   Step-by-step solution →
10. Q130 CBSE 2025
    Check whether $15^n\times 2^n$, $n$ ​being ​a ​natural number, ends ​with ​the digit ​zero.
    Show answer

    Yes. $15^n\times 2^n=3^n\times 5^n\times 2^n=3^n\times 10^n$; its prime factorisation contains both $2$ and $5$, so it is divisible by $10$ and ends with the digit $0$ for every natural number $n$.

    Step-by-step solution →
11. Q131 CBSE 2025
    Find ​HCF and ​LCM ​of $35$ ​and $55$ ​and verify ​your answer.
    Show answer

    The HCF of $35$ and $55$ is $5$ and the LCM is $385$; this is verified, since $\text{HCF}\times \text{LCM}=5\times 385=1925=35\times 55$.

    Step-by-step solution →
12. Q132 CBSE 2025
    If LCM of 51 and 85 ​can be ​expressed in ​the form of $6z-9$, then ​find ​the value of $z$.
    Show answer

    $z=44$

    Step-by-step solution →
13. Q133 CBSE 2025
    Find ​the ​smallest number ​which is ​divisible by both $644$ ​and $462$.
    Show answer

    The smallest number divisible by both $644$ and $462$ is their LCM, $21252$.

    Step-by-step solution →
14. Q134 CBSE 2025
    Two numbers are in the ratio $4:5$ and their ​HCF ​is $11$. ​Find ​the LCM ​of ​these numbers.
    Show answer

    The numbers are $44$ and $55$, and their LCM is $220$.

    Step-by-step solution →
15. Q135 CBSE 2025
    Using ​prime ​factorisation, ​find ​the HCF ​of $180$, $140$ ​and $210$.
    Show answer

    HCF of $180$, $140$ and $210$ $=2\times 5=10$.

    Step-by-step solution →
16. Q136 CBSE 2025
    Prove ​that $\frac{3+2\sqrt{5}}{7}$ is an irrational number, where, it is ​given ​that $\sqrt{5}$ ​is ​an irrational ​number.
    Show answer

    $\frac{3+2\sqrt{5}}{7}$ is irrational: assuming it equals $\frac{p}{q}$ gives $\sqrt{5}=\frac{7p-3q}{2q}$, a rational number, which contradicts the given fact that $\sqrt{5}$ is irrational. Hence proved.

    Step-by-step solution →
17. Q137 CBSE 2025
    Prove that $\sqrt{5}$ is ​an irrational ​number.
    Show answer

    Assuming $\sqrt{5}=\frac{p}{q}$ in lowest terms makes $5$ divide both $p$ and $q$, a contradiction; hence $\sqrt{5}$ is irrational.

    Step-by-step solution →
18. Q138 CBSE 2025
    Prove that $\frac{\sqrt{3}+5}{2}$ ​is an ​irrational ​number, where ​it ​is given that $\sqrt{3}$ ​is ​irrational.
    Show answer

    $\frac{\sqrt{3}+5}{2}$ is an irrational number. Hence proved.

    Step-by-step solution →
19. Q139 CBSE 2025
    Prove ​that $\sqrt{5}$ ​is an ​irrational number.
    Show answer

    Assuming $\sqrt{5}=\frac{p}{q}$ with $p$ and $q$ coprime forces $5$ to divide both $p$ and $q$, a contradiction. Hence our assumption is wrong and $\sqrt{5}$ is an irrational number.

    Step-by-step solution →
20. Q140 CBSE 2023
    Prove that $\sqrt{3}$ is ​an irrational ​number.
    Show answer

    Assuming $\sqrt{3}=\frac{a}{b}$ in lowest terms forces $3$ to divide both $a$ and $b$, a contradiction. Hence $\sqrt{3}$ is irrational. Hence proved.

    Step-by-step solution →