Let's evaluate the sets x and y. The set x is defined as x = {n\(^2\)+1:n is a positive integer and 1 \(\leq\) n \(\leq\) 5}. This means that x contains the elements obtained by squaring each positive integer from 1 to 5, and then adding 1 to the result. So, x = {2, 5, 10, 17, 26}.

The set y is defined as y = {5n:n is a positive integer and 1 \(\leq\) n \(\leq\) 5}. This means that y contains the elements obtained by multiplying each positive integer from 1 to 5 by 5. So, y = {5, 10, 15, 20, 25}.

The intersection of two sets is the set of elements that are common to both sets. So, x \(\cap\) y is the set of elements that are in both x and y. From the above evaluations of x and y, we can see that the only elements common to both sets are 5 and 10. So, x \(\cap\) y = {5, 10}.