Class 10 Maths · Chapter 1

Real Numbers

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

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 4 of 5

1. Q61 CBSE 2026
   If HCF ​of $66$ and $99$ is ​expressible ​in ​the ​form of $55m-132$, ​then the ​value of $m$ is :

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

   (d) $3$

   Step-by-step solution →
2. Q62 CBSE 2026
   $7\times 29\times 23+1$ is :

   1. (a)a prime number.
   2. (b)divisible by $23$.
   3. (c)an odd number.
   4. (d)a composite number.
   Show answer

   (d) a composite number.

   Step-by-step solution →
3. Q63 CBSE 2026
   Directions : ​In ​the following question, a statement ​of Assertion (A) is followed by ​a ​statement of ​Reason (R). Choose ​the correct option.

   Assertion (A): $(\sqrt{3}+\sqrt{5})$ is an irrational number.

   Reason (R): Sum of any two irrational numbers is always irrational.

   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

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

   Step-by-step solution →
4. Q64 CBSE 2026
   In the ​question below, a statement of Assertion ​(A) is followed by ​a statement of ​Reason ​(R). ​Choose ​the correct ​option.

   Assertion (A): $\text{HCF}(36m^2,18m)=18m$, where $m$ is a prime number.

   Reason (R): The HCF of two numbers is always less than or equal to the smaller number.

   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

   (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).

   Step-by-step solution →
5. Q65 CBSE 2026
   Direction : In ​the ​following ​question, a statement of ​Assertion ​(A) ​is followed by a statement of Reason ​(R). Choose the ​correct option.

   Assertion (A): $4^n$ can not end with the digit zero.

   Reason (R): Prime factorisation of $4^n$ is unique.

   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

   (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).

   Step-by-step solution →
6. Q66 CBSE 2026
   Assertion (A): $7\times 2+3$ is a composite number.

   Reason (R): A composite number has more than two factors.

   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. Q67 CBSE 2026
   Select the ​correct ​answer ​from the codes ​(a), (b), ​(c) and ​(d) as ​given below.

   Assertion (A): $(\sqrt{2}+\sqrt{3})$ is an irrational number.

   Reason (R): The sum of two irrational numbers is always an irrational number.

   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

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

   Step-by-step solution →
8. Q68 CBSE 2026
   Assertion (A): $(\sqrt{3}+1{)}^2$ is a rational number.

   Reason (R): $(\sqrt{3}{)}^2=3$ is a rational number.

   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

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

   Step-by-step solution →
9. Q69 CBSE 2026
   Find the smallest $5$-digit ​number ​exactly divisible ​by $24$ ​and $36$.
   Show answer

   The smallest $5$-digit number divisible by both $24$ and $36$ is $10008$.

   Step-by-step solution →
10. Q70 CBSE 2026
    Find the ​length ​of the plank ​that can be used to measure ​the lengths $4$ m $20$ cm and $5$ m $4$ ​cm ​exactly, in the least time.
    Show answer

    The required length of the plank is $84$ cm (that is, $0.84$ m).

    Step-by-step solution →
11. Q71 CBSE 2026
    Prove that $2-5\sqrt{3}$ ​is an irrational ​number, ​given that $\sqrt{3}$ ​is ​irrational.
    Show answer

    Assuming $2-5\sqrt{3}=\frac{p}{q}$ (integers $p,q$, $q\ne 0$) leads to $\sqrt{3}=\frac{2q-p}{5q}$, which is rational and contradicts the given that $\sqrt{3}$ is irrational. Hence $2-5\sqrt{3}$ is irrational.

    Step-by-step solution →
12. Q72 CBSE 2026
    Two ​numbers are in the ratio $3:5$ ​and ​their LCM ​is $180$. ​Find ​the HCF of these two ​numbers.
    Show answer

    The HCF of the two numbers is $12$.

    Step-by-step solution →
13. Q73 CBSE 2026
    Find ​the ​H.C.F. ​and L.C.M. of $1530$ ​and $2040$.
    Show answer

    HCF $=510$ and LCM $=6120$.

    Step-by-step solution →
14. Q74 CBSE 2026
    Prove that $2+3\sqrt{5}$ is ​an irrational ​number, given ​that $\sqrt{5}$ is ​an irrational number.
    Show answer

    $2+3\sqrt{5}$ is irrational. Assuming it equals $\frac{a}{b}$ forces $\sqrt{5}=\frac{a-2b}{3b}$ to be rational, which contradicts the given that $\sqrt{5}$ is irrational.

    Step-by-step solution →
15. Q75 CBSE 2026
    If ​the HCF ​of $210$ and $55$ ​is ​expressed ​as $210\times 5+55m$, then ​find the value ​of $m$.
    Show answer

    The value of $m$ is $-19$.

    Step-by-step solution →
16. Q76 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 →
17. Q77 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 →
18. Q78 CBSE 2026
    Prove that $\frac{2+3\sqrt{5}}{7}$ is ​an ​irrational number, ​given ​that $\sqrt{5}$ is an ​irrational number.
    Show answer

    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 →
19. Q79 CBSE 2026
    Given that $\sqrt{5}$ ​is an ​irrational number, prove ​that $3+2\sqrt{5}$ is also ​an ​irrational ​number.
    Show answer

    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 →
20. Q80 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 →