In my workbook I need to fill in the time signatures and in this example I have used 4/2 as the time signature
as you can see there is a whole note across the bar, is this allowed in 4/2 time? shouldnt you need to use a tie across the middle?
It's a bit visually strange, but permissible. In general, the convention is to clearly delineate the half-measure in 4/2 (or 2n>0/X) time. But when the rhythm is straightforward, as it is here, using a whole-note can be easier to read.
More and more, we see dots written out as such - with no easily seen demarcation of the middle of a bar in even numbered timing. As in 4/4, crotchet, minim, crotchet, instead of crotchet, crotchet tied to crotchet, crotchet.
I guess when there is a complex rhythm, there's a need for that 'rule' to be in place, but in simplistic cases, as yours, there's little to be gained. After all, the whole point of writing out dots is to make it easy for anyone trying to read said dots. Maybe at the stage someone is at when tackling questions at this level, adhering to the 'rules' would be better. Again, questioning the question!
I think I might spot one potential confusion. Ian Stewart talks about "lies-to-children," over-simplifications that are necessary to scaffold an understanding, but that are not the full truth. One of those that we were probably all told was "This is a quarter note. It is one beat," and "This is a whole note. It's called that because it takes up a whole measure." The truth is, of course, that any duration can be a beat, and that a whole note is simply 4 x a quarter note. In your example, your whole note is not going "across" a bar line; it simply occupies the second and third of your half-note beats.