UPSC 2016 CDS I Math Paper

Let m = 2 and n = 3.

m

n

+

n

m

=

2

3

+

3

2

=

13

16

,which is notan integer.
Hence, 1 is incorrect
mn(

m

n

+

n

m

)(m2+n2)−1
=mn(

m2+n2

mn

)(

1

m2+n2

)
= 1, which is an integer for all distinct natural numbers m and n.
Hence, 2 is correct.
Let m = 2 and n = 3.

mn

m2+n2

=

2×3

22+32

=

6

13

,which is not an integer. Hence 3 is incorrect.