First of all yes the answers are unique. Occasionally the answers might look slightly different but there is only one answer to each question.

Regarding this is NOT a cosine (when transformed back).

Note we have

,

and it HAS to be not .

The correct way to approach this is to realise that is NOT in the tables therefore we might want to write it as a sum of simpler objects that might be in the tables… this means you must do partial fractions…

Therefore factorise the bottom (note so is a difference of two squares) as . Now we have

,

using two Rule Is… because it is two Rule Is you may use the Cover-Up Method to find and .

ALTERNATIVE METHOD — acceptable but I am not teaching it

I mentioned an alternative using the tables that are at the back of the notes.

You will see that in these tables there is

.

This is what is called HYPERBOLIC cosine and is defined by

where is the imaginary unit that satisfies … now you might think that this different but actually we have

,

so you do get the same answer.

We can see this directly by noting

so

.

Regards,

J.P.

Just on two questions 1.(iii) & 3.(c); my attempts are attached.

To me these look like laplace transformer of , but from your answers I’m thinking that you rearranged this function to be a laplace transform of some ? I can’t seem to see how to get this in terms of if that’s the way you worked it out?

Also is there some way of knowing what to look for in these questions, because some of these look like you could go down different routes . (Eg is there only one solution to each problem?)

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