Question

Two irrational numbers between \[2\] and \[2.5\] are:
A). \[\sqrt 5 \] and \[\sqrt {2 \times \sqrt 5 } \]
B). \[\sqrt 5 \] and \[\sqrt {2 \times 5} \]
C). \[\sqrt 5 \] and \[\sqrt {2 \times \sqrt 7 } \]
D). None of these

Answer 1

Hint: In the given question, we have been given two rational numbers. It has been given that there are two irrational numbers between the two numbers. We have to choose the correct option fulfilling the condition. The numbers given in the option are in the form of square root. To do solve this question, we are going to convert the given numbers as the same form as the options, make the required amends to their forms, compare it with the options, and find the answer.

Formula used:
We are going to use the formula of square root, which is:
\[a = \sqrt {{a^2}} \]

Complete step by step solution:
We have to find the irrational numbers between the two numbers, which are:
\[2\] and \[2.5\]
Now, \[2 = \sqrt {2 \times 2} = \sqrt 4 \] and \[2.5 = \sqrt {2.5 \times 2.5} = \sqrt {6.25} \]
So, the two irrational numbers need to lie between \[\sqrt 4 \] and \[\sqrt {6.25} \], i.e., the numbers must be greater than \[\sqrt 4 \] and smaller than \[\sqrt {6.25} \].
For the lower limit, all the options fulfil the situation – the lower limit of all options is \[\sqrt 5 \], which is greater than \[\sqrt 4 \].
Hence, we need to find the option(s) whose upper limit is less than \[\sqrt {6.25} \].
Now, if we notice, all the options are of the form \[\sqrt {2 \times m} \], where \[m\] is a number.
We are going to find for this condition.
Now, we can discard the second option because it is equal to \[\sqrt {2 \times 5} = \sqrt {10} \] which is clearly bigger than \[\sqrt {6.25} \].
This now leaves us with two options. We can further build a general form for the two options.
The two options are of the form \[\sqrt {2 \times \sqrt n } \], where \[n\] is a number.
Let us find the upper limit in form of this construct:
\[\sqrt {6.25} = \sqrt {2 \times \sqrt n } \]
Squaring both sides,
\[6.25 = 2 \times \sqrt n \]
Taking \[2\] to the other side,
\[\dfrac{{6.25}}{2} = \sqrt n \Rightarrow \sqrt n = 3.125\]
Again, squaring both sides,
\[n = {\left( {3.125} \right)^2} < {\left( {3.2} \right)^2} < 10.24\]
So, the number should be less than \[10.24\].
And in the two options, both the numbers are less than \[10.24\].
Hence, the correct options are A and C.

Note: In the given question, we were given two numbers, and we were given that there were two irrational numbers between the given numbers, and we had to choose them from the options. We did that by converting the given numbers to the form of the numbers in options, found the common out of the options, compared them, and found the answer.