Question

Two irrational number between 2 and 2.5 are:
(This questions has multiple correct options)
\[
  (a){\text{ }}\sqrt 5 {\text{ and }}\sqrt {2 \times \sqrt 5 } \\
  (b){\text{ }}\sqrt 5 {\text{ and }}\sqrt {2 \times 5} \\
  (c){\text{ }}\sqrt 5 {\text{ and }}\sqrt {2 \times \sqrt 7 } \\
  (d){\text{ none of these}} \\
 \]

Answer 1

Hint: In this question we have to find two irrational numbers between 2 and 2.5. Irrational numbers are those which cannot be expressed in the form of $\dfrac{a}{b}$ where $b \ne 0$. Use this definition to start with the question of finding irrational numbers between the given numbers.

Complete step-by-step answer:
Given numbers are 2 and 2.5.
Now we have to find out two irrational numbers between them,
As we know irrational numbers cannot be written in the form of $\dfrac{a}{b}$, where $b \ne 0$.
And we also know any number which is not a complete square and the square root of that number is an irrational number.
So, First Square the given number we have,
$ \Rightarrow {2^2}$ and ${\left( {2.5} \right)^2}$
$ \Rightarrow 4$ and $6.25$
Now take square root we have,
$ \Rightarrow \sqrt 4 $ and $\sqrt {6.25} $
Doing this we get the original number and 4 and 6.25 is a rational number from the above property.
Now the number greater than $\sqrt 4 $ and less than $\sqrt {6.25} $ is the required irrational numbers.
Now as we know the value of $\sqrt 5 $ and $\sqrt 7 $ are less than 3. So in option (A) and (C) 2 is multiplied by value less than 3. Then we get values less than $ \sqrt 6.25$.
So from the given options we can see clearly that options (A) and (C) are the required irrational numbers between 2 and 2.5.
Hence option (A) and (C) are correct.

Note: Whenever we face such types of problems it is always advisable to check the options that satisfy the required condition asked in the question using the basic definition of rational and irrational number. This approach helps to save a lot of time.