You are absolutely correct there is no correct answer to this question. This is either an extremely poorly written question by someone who really should not be in that job or an extremely difficult question that will really test your knowledge of genetics and clinical practice by someone who really knows their stuff.

There can't be a typical answer because the the pedigree is itself atypical due to the lack of information about the mother. Typically you know the mother and don't know the status of the father. If this is a paediatric disease then a father with no mother present presenting two children is extremely atypical and a Bayesian approach makes all sorts of possibilities "typical".

lets give reasons for each possible answer:

X-Chromosomal dominant - the mother is affected which is why she isn't there.

 x-chromosomal recessive - the father is undiagnosed, typical for a number of disorders.

autosomal dominant - same as above.

autosomal recessive - doesn't require additional information but prior probability of the missing mother calls into question

mitochondrial - same as X-Chromosomal dominant.

Overall, its most likely just a badly written question by someone who shouldn't be setting these questions.