I wanted to offer my thoughts in case my line of reasoning resonates with someone and helps them better understand.
Ok, so we need to know if a>0.
given,
1. a^3-a<0
i could factor out a from both terms but that may make it complicated, but I ma told that the left hand side is negative, so whatever the value of a, whether positive or negative, I ought to be able to say that a is bigger than a^3, otherwise a^3-a would not be less than 0.
so, a^3<a.
if something is cubed and it is still less than the original something, then something is funky, and not your regular positive whole numbers. it could be fraction, positive or negative, i don't know yet, it could be a negative whole number too i suppose, but i would need to test that. let me do that real quick. clearly, one of the possibilities has to be a positive fraction between 0 and 1 because then it would be say (1/2)^3<1/2. that statement holds. so a can be positive. i just need to check if it can be negative too. what if a is a negative fraction? a=-1/2 say. then -1/8<-1/2. that's not right, so a is not a negative fraction, could it be a negative whole number? instantly, i can see -2 cubed would be -8 is less than -2. so a could be a negative whole or a positive fraction. that doesn't conclusively answer the question. so crossing out statement 1, noting that a could be a positive fraction or negative whole. actually, i am gonna draw it on the number line. ok. circle the 0 to 1 portion and less than -1 portion. moving on.
2.1-a^2>0
ok, instantly i can see that 1 has to be greater than a^2 for their difference to be positive. so without worrying about what sign mumbojumbo, i can say a^2<1. well, i have learned from
mgmat advanced quant book that anytime i see a^2<1, i can simply write it as |a|<1 and that i can write it as -1<a<1. so a is between -1 and 1. so a could be positive or negative and doesn't conclusively answer the question. so insufficient.
BUT drawing this statement 2 relation on a number line and comparing it to the earlier number line i drew for statement 1, i can see that there is a common region from 0 to 1. so taken together, i can see that a lies between 0 and 1 which are all positive. and it sufficiently answers the original question, Yes, a>0.
Hope my line of reasoning is not a wrong way that still got the right answer, and I hope it at least makes someone follow and understand the solution to this problem.