Correct Option (2)

The problem specifies that p and k are prime numbers, and the expression (p² + k) must also be a prime number less than 30.

First, identify all prime numbers less than 30: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}.

We systematically evaluate possible prime values for p:

Case 1: p = 2
- p² = 4. We need 4 + k to be a prime number less than 30.
- If k = 2, 4 + 2 = 6 (not prime).
- If k = 3, 4 + 3 = 7 (prime). This is a valid k.
- If k = 5, 4 + 5 = 9 (not prime).
- If k = 7, 4 + 7 = 11 (prime). This is a valid k.
- If k = 11, 4 + 11 = 15 (not prime).
- If k = 13, 4 + 13 = 17 (prime). This is a valid k.
- If k = 17, 4 + 17 = 21 (not prime).
- If k = 19, 4 + 19 = 23 (prime). This is a valid k.
- If k = 23, 4 + 23 = 27 (not prime).
- If k = 29, 4 + 29 = 33 (not prime, and > 30).
- For p = 2, the valid values for k are {3, 7, 13, 19}.
Case 2: p = 3
- p² = 9. We need 9 + k to be a prime number less than 30.
- If k = 2, 9 + 2 = 11 (prime). This is a valid k.
- If k = 3, 9 + 3 = 12 (not prime).
- If k = 5, 9 + 5 = 14 (not prime).
- If k = 7, 9 + 7 = 16 (not prime).
- If k = 11, 9 + 11 = 20 (not prime).
- If k = 13, 9 + 13 = 22 (not prime).
- If k = 17, 9 + 17 = 26 (not prime).
- If k = 19, 9 + 19 = 28 (not prime).
- If k = 23, 9 + 23 = 32 (not prime, and > 30).
- For p = 3, the valid value for k is {2}.
Case 3: p = 5
- p² = 25. We need 25 + k to be a prime number less than 30.
- If k = 2, 25 + 2 = 27 (not prime).
- If k = 3, 25 + 3 = 28 (not prime).
- Any other prime value for k (e.g., k=5) would result in 25 + k ≥ 30, exceeding the condition.
- For p = 5, there are no valid values for k.
Case 4: p ≥ 7
- If p = 7, p² = 49. Even with the smallest prime k = 2, 49 + 2 = 51, which is not less than 30.
- Therefore, no valid k exists for p ≥ 7.

Combining all unique valid values of k found from the above cases:

From p = 2: {3, 7, 13, 19}
From p = 3: {2}

The complete set of all possible unique values for k is {2, 3, 7, 13, 19}.

The total number of possible values of k is 5.

Incorrect Options:

Options 1 (4), 3 (6), and 4 (7) are incorrect. A thorough evaluation of prime numbers p and k, adhering to the specified conditions, yields precisely 5 unique values for k. These options do not represent the complete set of valid k values derived from the systematic analysis.