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 15 of 17

1. Q281 CBSE 2020
   $840$ ​can be ​expressed as a product of prime ​numbers as ​:

   1. (a)$2^2\times 6\times 5\times 7$
   2. (b)$2^3\times 3\times 5\times 7$
   3. (c)$2\times 3\times 4\times 5\times 7$
   4. (d)$3\times 5\times 7\times 8$
   Show answer

   (b) $2^3\times 3\times 5\times 7$

   Step-by-step solution →
2. Q282 CBSE 2020
   $2.\overline{35}$ is

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

   (b) a rational number

   Step-by-step solution →
3. Q283 CBSE 2020
   Which of ​the ​following rational numbers ​is expressible as ​a terminating decimal ​?

   1. (a)$\frac{124}{165}$
   2. (b)$\frac{131}{30}$
   3. (c)$\frac{2027}{625}$
   4. (d)$\frac{1625}{462}$
   Show answer

   (c) $\frac{2027}{625}$

   Step-by-step solution →
4. Q284 CBSE 2020
   Which ​of the ​following is ​the decimal ​expansion ​of an irrational ​number?

   1. (a)$3.14$
   2. (b)$3.3333\dots$
   3. (c)$6.010010001\dots$
   4. (d)$7.25$
   Show answer

   (c) $6.010010001\dots$

   Step-by-step solution →
5. Q285 CBSE 2020
   $\left(\frac{2+\sqrt{5}}{3}\right)$ ​is ​________ number.
   Show answer

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

   Step-by-step solution →
6. Q286 CBSE 2020
   Given ​that $\text{HCF}(135,225)=45$, ​find the $\text{LCM}(135,225)$.
   Show answer

   $\text{LCM}(135,225)=675$

   Step-by-step solution →
7. Q287 CBSE 2020
   Write ​one ​irrational ​number ​between $0.15$ and $0.21$.
   Show answer

   One irrational number is $0.1601600160001\dots$ (a non-terminating, non-recurring decimal between $0.15$ and $0.21$).

   Step-by-step solution →
8. Q288 CBSE 2020
   Find ​the ​HCF ​of $12$, $18$ and $30$.
   Show answer

   The HCF of $12$, $18$ and $30$ is $6$.

   Step-by-step solution →
9. Q289 CBSE 2020
   The LCM of ​two ​numbers is $182$ ​and their HCF is $13$. If one ​of the numbers is $26$, ​find the ​other.
   Show answer

   The other number is $91$.

   Step-by-step solution →
10. Q290 CBSE 2020
    After ​how many ​decimal ​places will the ​decimal representation ​of the rational ​number $\frac{229}{2^2\times 5^7}$ terminate ?
    Show answer

    It terminates after $7$ decimal places (the value is $0.0007328$).

    Step-by-step solution →
11. Q291 CBSE 2020
    Show that $5+2\sqrt{7}$ ​is ​an ​irrational number, where $\sqrt{7}$ ​is ​given to be an irrational number.
    Show answer

    $5+2\sqrt{7}$ is irrational, proved by contradiction using the given irrationality of $\sqrt{7}$.

    Step-by-step solution →
12. Q292 CBSE 2020
    Check ​whether $12^n$ can end ​with ​the ​digit $0$ ​for ​any natural number $n$.
    Show answer

    No. $12^n=2^{2n}\times 3^n$ has no factor of $5$, so it is never divisible by $10$; hence $12^n$ can never end with the digit $0$ for any natural number $n$.

    Step-by-step solution →
13. Q293 CBSE 2020
    Show that ​every positive even integer ​is of the ​form $2q$ and that ​every positive ​odd integer ​is of the form $2q+1$, ​where $q$ is some integer.
    Show answer

    Every positive integer is $2q$ or $2q+1$. The form $2q$ is even and $2q+1$ is odd, proving the statement.

    Step-by-step solution →
14. Q294 CBSE 2020
    Use Euclid Division Lemma to show ​that the ​square ​of any positive integer is either of the ​form $3q$ or $3q+1$ for some ​integer $q$.
    Show answer

    The square of any positive integer is of the form $3q$ (when $n$ is a multiple of $3$) or $3q+1$ (otherwise), for some integer $q$.

    Step-by-step solution →
15. Q295 CBSE 2020
    Given that $\sqrt{3}$ ​is ​irrational, ​prove that $5\sqrt{3}-2$ is an ​irrational ​number.
    Show answer

    The assumption that $5\sqrt{3}-2$ is rational forces $\sqrt{3}$ to be rational, contradicting the given. Hence $5\sqrt{3}-2$ is irrational.

    Step-by-step solution →
16. Q296 CBSE 2020
    There ​is a ​circular path around a sports field. Sonia takes $18$ minutes for one round ​of ​the field and Ravi takes $12$ minutes for the same. Suppose they both ​start from the ​same point and at ​the same time. After how many ​minutes will they meet again at the starting point?
    Show answer

    They will meet again at the starting point after $36$ minutes.

    Step-by-step solution →
17. Q297 CBSE 2020
    Assuming ​that $\sqrt{3}$ ​is an ​irrational number, ​prove that $5\sqrt{3}-7$ ​is an ​irrational ​number.
    Show answer

    Proved: $5\sqrt{3}-7$ is irrational (the assumption that it is rational forces $\sqrt{3}$ to be rational, a contradiction).

    Step-by-step solution →
18. Q298 CBSE 2020
    If ​the ​HCF of $65$ and $117$ ​is ​written as $65m-117$, ​then ​find the value of $m$.
    Show answer

    $m=2$

    Step-by-step solution →
19. Q299 CBSE 2020
    If $\sqrt{2}$ ​is given as ​an irrational number, ​then ​prove that $(7-2\sqrt{2})$ is ​an ​irrational ​number.
    Show answer

    Assuming $7-2\sqrt{2}$ rational forces $\sqrt{2}$ rational, a contradiction; hence $7-2\sqrt{2}$ is irrational.

    Step-by-step solution →
20. Q300 CBSE 2020
    Given ​that $\sqrt{3}$ ​is ​an irrational number, ​show that $(5+2\sqrt{3})$ ​is an ​irrational number.
    Show answer

    Assuming $5+2\sqrt{3}$ rational forces $\sqrt{3}$ to be rational, contradicting the given; hence $5+2\sqrt{3}$ is irrational.

    Step-by-step solution →