Calculating a "killer question" for a total test score

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%

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