Answer: Jack omitted 5 questions.

Jack must have scored at least 87 points from his correct answers.
12 correct $\times$ 7 = 84 points
13 correct $\times$ 7 = 91 points
91 points $-$ (2 wrong $\times$ 2) = 87 points.
So Jack could have got 13 right, 2 wrong and omitted 5

Could he have got more than 13 correct?
14 correct $\times$ 7 = 98 points
98 points $-$ (number wrong $\times$ 2) cannot be 87

15 correct $\times$ 7 = 105 points
105 points $-$ (9 wrong $\times$ 2) = 87 points.
But 15 right and 9 wrong is not possible because there are only 20 questions.