Calculating a "killer question" for a total test score all -
I'm trying to figure out if this can be done at all, and if so, how; unfortunately, my skill set isn't up to the task, and I'm hoping to find an answer here - or at least guidance on how to approach it.
I need to score a 10-question test that has weighted questions on a pass/fail basis, with 70% as the cut score. So far, so good - sum the total difficulties, multiply by 0.7, then check to see if the candidate's score is above or below (the framework we are using does all this automatically.) However, I now want to add another condition: if the candidate does a certain thing (something I can test for), I need the score to always be a fail (below 70%.)
Changing the scoring framework to do this would be problematic in lots of ways - but treating the condition test as a hidden 11th question would be trivial. The challenge, then, is this:
Given 10 questions with associated difficulty scores (as below), is it possible to calculate a difficulty score for question #11 such that failing it will produce a score below 70% in all cases, but passing it will result in the same pass/fail rate as if q#11 wasn't present? The weight can't be negative.
Hopefully, I've managed to phrase this in a coherent manner; please feel free to ask any questions you need to clarify.
Example of question weights:


*

*2

*3

*2

*2

*3

*2

*1

*3

*2

*3

*???

 A: No it can't be done.
For simplicity we scale all the scores such that the original ones sum to 1. Now we define x to be the weight of question 11 so the new total score is $1 + x$. Then for a student who gets all questions but 11 right to still fail we need 
$$1 < 0.7(1+x) \iff x > 3/7 \approx 0.43$$
but for a student with 60% on the first 10 questions and question 11 right to also fail we need
$$\frac{0.6+x}{1+x} < 0.7 \iff x < 1/3 \approx 0.33$$
a contradiction.
Remark: This assumes it to be possible to score somewhere between 58% and 70% on the original test, which is certainly true for the scores you posted: Answering only the first 6 questions correctly will get you 60%
A: Note:  in what follows I am treating $70$ as an absolute score. I believe the conditions of the post can not be satisfied as written, but if you let $70$ be an absolute score instead of a proportion of the total you can do it.
Let question $11$ be worth $31$ and let the other ten questions sum to $55.7$ (so $5.57$ each).  
Then if you miss $11$ your maximum score is $69$ so you are sure to fail.
In order to pass you need question $11$ plus $70\%$ of the rest.  Thus the rest has to have value $$100\times \left(1-\frac {.31}{.7}\right)\approx 55.714$$  It follows that, having gotten $11$ right you need to pass the test in the ordinary way.
Note:  with these weights, your maximum score is $86.7$.  Not sure if that's a problem.
