Test of knowledge or test of taking tests?
If this is the general form of these exams I'd not relish sitting them! I'd be happy just to finish let alone pass.
In the interests of those who might, here is my reasoning:
Not Idle - both carburation and fuel-injection have mechanisms to ensure slow running w/o the engine stopping - idle jet and fuel control unit respectively will ensure a richer and so cooler mixture at idling revs (hence the pre-takeoff check that the engine does idle and not stop).
Not 1:9 - 1 part air to 9 parts fuel will not ignite! Funnily enough a ratio of 9 air to 1 fuel is the approximate stoichiometric ratio for burning ethanol (according to Google) but note the ratio inversion for this piece of knowledge that may become slightly less useless in future.
Cruise setting - Possibly correct answer except 1) this is not a ratio and 2) actual power settings are usually expressed as a certain RPM or RPM&MP at a particular EGT rich of peak i.e. the mixture is indeed leaned but not to highest or peak EGT.
1:1/1:15 - Most likely answer. 1) the number 15 is close to the key number for a stoichiometrically correct mixture for burning gasoline of 14.7 parts air and 1 part fuel by WEIGHT (volume differs according to temp and pres) and 2) if you do the algebra you get the ratio:
1:1/1:15 IET (1/1) / (1/15) IET (1/1) * (15/1) IET (15/1) IET 15:1
NB. IET means "is equivalent to".
Also, several text books mention the 1:15 ratio for efficient burning but not explicitly for peak EGT burning which anyway according to most engine manufacturer's operating manuals is probably not the most efficient for reaching TBO hence their recommendations of ROP for cooler and longer operation (lean of peak is a whole other discussion).
Answer: out of "cruise" or "1:15" I'd go for the second. Even if the CAA/JAA intended you to "see" the algebra I'd say it was a bit naughty as a "systems" question.
I usually start with AP3456 when looking for references. Highly recommended but anyone have others?