would you expect the infrared spectrum of ethyl ethanote to show absorption at around 1740cm-1?

Question

i have to give a reason for that and suggest the trough(if formed) is likley to be broadened.

Answer 1

Yes!!!!

The keto - group (C=O)in the ester linkage will produce a very strong absorption at approximately 1740 cm^-1. An absorption in the range 1680 - 1750 cm^-1 is very characteristic of the keto group (C=O) in any molecule. It is usually shown as a very narrow 'spike'.
The C-O part of the ester produces an absorption in the range 1000 - 1300 cm^-1, but this usually is a series of absorptions, and not a broadening.
The hydroxy group (O-H) in alcohols produces are broad absorption in the range 3230 - 3550 cm^-1
The hydroxy group (O-H) in acids produces a very characteristic broad absorption in the range 2500 -3000 cm^-1.
As ethyl ethanoate is an ester it does not have any hydroxy groups present, so it will not produce a broad signal.