The question is to proove that $d(Adx + Bdy) =( \frac{\partial B}{\partial x}- \frac{\partial A}{\partial y} ) dx \wedge dy $
There is an archive especificly for that book that is in the called "Road to Reality Legacy" and that exercise is indeed one the most commented exercised in all the boook in that website.
The discution is in http://legacy.roadtoreality.info/archive/viewtopic.php%3Ff=19&t=143.html
but, Roger Penrose, in his book haven´t especified wich kind of objets are the A,B and the question, after I have readed the discution in that site is:
i) what kind of objets most A,B to bee?, differential forms?, functions?, differentiable functions?, ... ii) all is about, why $ d(Adx) = dA \wedge dx $ ?, what is the way to formalize and answer this?, only this? what is the relation betwwen the product of A with dx (Adx) , and the wedge product of dA with dx ( $ dA \wedge dx $) ?
Thank you