=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Thus, look at condition 1).
Assume r and r+1 are roots of the equation x^2 + px + q = 0. It tells us that p = -25 and q = 156 for the following reason, which is exactly what we are looking for.
(r + 1)^2 – r^2 = r^2 + 2r + 1 – r^2 = 2r + 1 = 25 or r = 12.
Then we have x^2 + px + q = (x - r)(x -(r + 1)) = x^2 – (r + r + 1)x + r(r + 1) = x^2 – (2r+1)x + r(r+1) and we have p = -2r-1 = -25 and q = r(r+1) = 12*13 = 156.

The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 1) ALONE is sufficient.

Condition 2)
If 1 and 2 are roots of the equation, then we have (x - 1)(x - 2) = x^2 - 3x + 2 = x^2 + px + q, p = -3 and q = 2.
If 2 and 3 are roots of the equation, then we have (x - 2)(x - 3) = x^2 - 5x + 6 = x^2 + px + q, p = -5 and q = 6.
The answer is not unique, and the condition is not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Therefore, A is the correct answer.
Answer: A