Question

how to find an infinite language is regular or not is there any other method than drawing DFA as DFA made by us may or may not be correct how to ensure DFA is  correct and language is regular

Answer 1

It is rather easy to check whether an infinite language is non-regular or not. Here are a few tips:

Check whether any alphabet /string comparisons are being done. If yes, then non-regular.

Check whether any counting process is involved that requires a stack to remember alphabets. If yes, then non-regular.

Check whether any non-linear powers are involved in any alphabet. If yes, then non-regular

A sure method is to draw the DFA/NFA and write the correct regular expression. If this is possible, then regular.

To check whether a DFA is correct or not is to ensure that it accepts strings $a\in L$ and rejects strings $a\notin L$.

Comments

it is not regular

we have to compare m and n for m= n^2

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It is regular and actually even finite. This is the issue of using "short cuts". All finite languages are regular and this is a basic stuff. Since $n < 1000$ we do not have an infinite language possibility and whatever comparison is there between $m$ and $n$ is insignificant. Like this it is possible to make a case for any short cut- so unless you know why you do it, never use short cuts for exams like GATE where hardly any question is repeated.

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yaa I missed the finite part