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The three different irrational numbers between the rational numbers $57 a n d 911$ are:

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0.74074007400074000074…….
0.7240724007240007250000……..
0.808008000800008….

0.74074007400074000074…….
0.7260726007260007260000……..
0.808008000800008….

0.74074007400074000074…….
0.7340734007340007350000……..
0.808008000800008….

0.74074007400074000074…….
0.7240724007240007250000……..
0.606006000600006….

answer is A.

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Detailed Solution

Given the numbers are

57 a n d 911

.
Decimal expansion of Irrational numbers:
The decimal expansion of irrational numbers is non terminating non-recurring (repeating).
Divide

57

and

911

to find irrational numbers:

7 0.714285 5049 - - 107 - - 3028 - - 2014 - - 6056 - - 4035 - - 5 - -

Thus ,

57 = 0 . 714285 ¯

.
And

1 1 0.8181 ... 9088 - - 2011 - - 9088 - - 9088 - - 2 - -

------
Thus,

911 = 0 . 81 ¯

.
That means the required irrational number will lie between

0 . 714285 ¯

and

0 . 81 ¯

.
Now, write the three irrational number between

0 . 714285 ¯

and

0 . 81 ¯

are:
0.74074007400074000074…….
0.7240724007240007250000……..
0.808008000800008….
Hence, the three different irrational numbers between the rational numbers

57

and

911

are:
0.74074007400074000074…….
0.7240724007240007250000……..
0.808008000800008….
Therefore, the correct option is 1.

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