What constitutes "valid" depends on many things (though I bet you've been given one blanket 'rule' which in many cases will be too strict and in some cases too weak, depending on your needs). In this case the requirement of 6dp accuracy preclude using approximations unless the sample sizes get very large indeed.
So obviously the idea is that one should apply the binomial directly.
You'd need to make some assumptions - which are necessary to apply the binomial.
Then you simply compute the actual binomial probabilities in question.
That is, there's no need to try to apply the normal approximation itself, you can just compute the required probability. Which marks on the test correspond to passing? What is the probability of getting a result among those values?
Some comments about the form of the question (not the OP's question, the one set for the OP to do):
Seriously - who needs probability to 6dp accuracy?
Even if you did need it, when are the assumptions (like independence, constant probability of success) ever satisfied closely enough to actually get that level of accuracy?
We're talking about an extreme tail probability. Even small amounts of dependence or heterogeneity of probability and the calculation could be out by orders of magnitude (i.e. not even close to one significant figure of accuracy)
This sort of question implies a ludicrously false level of accuracy to our models.
On a real-world problem of this kind we'd usually be lucky to get more than a couple of figures of accuracy, because we don't really have independence, etc. The shadow of George Box's famous maxim (in abbreviated form, 'all models are wrong; some are useful') is ever present.