By defining ‘n’ as a ratio of integers, the question data eliminates the possibility of ‘n’ being an irrational number (in simpler terms, a root). That’s valuable information, especially when you are interpreting information like the one given in statement 1.
From statement 1 alone, \(n^2\) is an integer.
This is where the question data is valuable. Had it not been told to us that n=\(\frac{p}{q}\), p and q being non-zero integers, interpreting statement 1 would take a different turn.
Because, \(n^2\) being an integer does not automatically make ‘n’ an integer; ‘n’ could be a root also.
However, in this question, we know that ‘n’ is a rational number and hence cannot be an irrational root. Therefore, if \(n^2\) is an integer, it necessarily means that n is an integer.
Statement 1 alone is sufficient to answer the question with a definite YES. Answer options B, C and E can be eliminated. Possible answer options are A or D.
From statement 2 alone, \(\frac{(2n+4)}{2}\) is an integer.
Upon simplification, we obtain n+2 as an integer. This is possible only if n is an integer itself.
Statement 2 alone is sufficient to answer the question with a definite YES. Answer option A can be eliminated.
The correct answer option is D.
Hope that helps!
_________________