Explanation:

- Originally, Set A contains the integers: 27, 28, 30, 32, and 33.

- First, calculate the sum of Set A:

\( 27 + 28 + 30 + 32 + 33 = 150 \).

- The average of Set A is:

\( \frac{150}{5} = 30 \).

- With integer \( k \) included, the average needs to increase by 30%:

\( 30 + (0.30 \times 30) = 39 \).

- The new average with \( k \) is:

\( \frac{150 + k}{6} = 39 \).

- Solving for \( k \):

\( 150 + k = 39 \times 6 \).

\( 150 + k = 234 \).

\( k = 234 - 150 = 84 \).

- Correct answer: Option 3 - 84