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Jack must get at least $13$ questions correct in order to score enough points. $13 \times 7 = 91$, so these questions score him $91$ points. Then he needs two wrong answers to reduce his score to $87$, meaning $5$ questions are left unanswered.

If he gets more than $13$ questions correct, marks can only be deducted in twos. Therefore he must get an odd number of marks from the correct questions, so must score at least $15 \times 7 = 105$ points from them. This means he needs at least $9$ incorrect questions also, a total of at least $15+9=24$. However, there are only $20$ questions in the test, so this cannot happen.

Therefore, the only way to get $87$ is to get $13$ questions correct and $2$ wrong, leaving $5$ questions unattempted.

This problem is taken from the UKMT Mathematical Challenges.