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 17 of 17

1. Q321 CBSE 2019
   On a morning walk, three ​persons step out together and their steps measure $30$ cm, $36$ cm and $40$ ​cm ​respectively. What is ​the ​minimum ​distance ​each should walk so that each can cover the ​same distance in complete steps?
   Show answer

   The minimum distance each should walk is $360$ cm.

   Step-by-step solution →
2. Q322 CBSE 2019
   Find ​the ​HCF of $612$ and $1314$ ​using ​prime factorisation.
   Show answer

   HCF of $612$ and $1314$ is $18$.

   Step-by-step solution →
3. Q323 CBSE 2019
   Show that any positive odd ​integer is of ​the ​form $6m+1$ ​or $6m+3$ ​or $6m+5$, where $m$ ​is some integer.
   Show answer

   Every positive odd integer is of the form $6m+1$, $6m+3$, or $6m+5$. Hence proved.

   Step-by-step solution →
4. Q324 CBSE 2019
   Use ​Euclid's ​division algorithm ​to ​find ​the ​HCF of $255$ and $867$.
   Show answer

   HCF$(255,867)=51$.

   Step-by-step solution →
5. Q325 CBSE 2019
   Prove that $2+5\sqrt{3}$ ​is an irrational ​number, ​given that $\sqrt{3}$ is ​an ​irrational number.
   Show answer

   $2+5\sqrt{3}$ is irrational (proved by contradiction using the irrationality of $\sqrt{3}$).

   Step-by-step solution →
6. Q326 CBSE 2019
   Using ​Euclid's Algorithm, ​find ​the ​HCF of $2048$ ​and $960$.
   Show answer

   HCF $(2048,960)=64$.

   Step-by-step solution →
7. Q327 CBSE 2019
   Find the ​largest number ​which on ​dividing $1251$, $9377$ ​and $15628$ ​leaves remainders $1$, $2$ and $3$ ​respectively.
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   The largest number is $625$.

   Step-by-step solution →
8. Q328 CBSE 2019
   Prove ​that $2+3\sqrt{3}$ is an irrational ​number when it ​is given that $\sqrt{3}$ ​is an ​irrational number.
   Show answer

   Assuming $2+3\sqrt{3}$ rational forces $\sqrt{3}=\frac{p-2q}{3q}$ to be rational, contradicting the given. Hence $2+3\sqrt{3}$ is irrational.

   Step-by-step solution →
9. Q329 CBSE 2018
   What ​is ​the HCF of smallest ​prime number and the ​smallest ​composite number?
   Show answer

   HCF $=2$.

   Step-by-step solution →
10. Q330 CBSE 2018
    Write whether the ​rational number $\frac{13}{3125}$ has ​a decimal ​expansion which ​is ​terminating or ​non-terminating repeating.
    Show answer

    $\frac{13}{3125}$ has a terminating decimal expansion (it equals $0.00416$), since $3125=5^5$ is of the form $2^m\times 5^n$.

    Step-by-step solution →
11. Q331 CBSE 2018
    What ​is the ​HCF of smallest prime ​number and ​the ​smallest ​composite ​number ?
    Show answer

    The HCF of the smallest prime ($2$) and the smallest composite ($4$) is $2$.

    Step-by-step solution →
12. Q332 CBSE 2018
    Using Euclid's ​Division Algorithm, find the ​HCF ​of $255$ ​and $867$.
    Show answer

    $\text{HCF}(255,867)=51$.

    Step-by-step solution →
13. Q333 CBSE 2018
    Given ​that $\sqrt{2}$ is ​irrational, ​prove that $(5+3\sqrt{2})$ ​is ​an ​irrational number.
    Show answer

    Assuming $5+3\sqrt{2}$ rational forces $\sqrt{2}=\frac{p-5q}{3q}$ to be rational, contradicting the given fact; hence $5+3\sqrt{2}$ is irrational.

    Step-by-step solution →
14. Q334 CBSE 2018
    Show ​that any ​positive odd integer is ​of ​the ​form $4q+1$ or $4q+3$ for some ​integer $q$.
    Show answer

    Every positive odd integer is of the form $4q+1$ or $4q+3$ for some integer $q$. Proved.

    Step-by-step solution →
15. Q335 CBSE 2018
    Find HCF and LCM of $404$ and $96$ and ​verify that ​HCF $\times$ LCM $=$ ​Product of ​the ​two ​given ​numbers.
    Show answer

    HCF $=4$, LCM $=9696$; and HCF $\times$ LCM $=4\times 9696=38784=404\times 96$, verifying the property.

    Step-by-step solution →
16. Q336 CBSE 2018
    Find HCF and ​LCM of 404 and ​96 and ​verify that $\text{HCF}\times \text{LCM}=$ ​Product of the two ​given ​numbers.
    Show answer

    $\text{HCF}=4$, $\text{LCM}=9696$, and $\text{HCF}\times \text{LCM}=38784=404\times 96$.

    Step-by-step solution →