Each of the questions given below consists of a statement and/or a question…

Concept: In Data Sufficiency, a statement is sufficient only when it forces, by itself, one single, definite answer to the question asked. A statement that is automatically true for every value already allowed by the question's own conditions adds no real constraint and can never be sufficient, however true it reads. Also useful here: whenever an equation reduces the quantity under a square root to a perfect-square multiple of an integer variable (such as 4y2), the square root is guaranteed to be a whole number for every integer value of that variable.

Apply this to each statement independently:

1. Statement I alone: x and y are already given to be integers, so x2 + y2 is automatically a whole number for every such pair - Statement I adds nothing beyond what is already known. Two pairs that both satisfy it give different outcomes: x = 1, y = 1 gives √(12 + 12) = √2 (not a whole number), while x = 3, y = 4 gives √(32 + 42) = √25 = 5 (a whole number). Since the same statement is compatible with both a 'yes' and a 'no', Statement I alone cannot decide the question.

2. Statement II alone: x2 – 3y2 = 0 gives x2 = 3y2. Substituting into the expression asked about: x2 + y2 = 3y2 + y2 = 4y2, so √(x2 + y2) = √(4y2) = 2|y|. Because y is an integer, 2|y| is a whole number for every integer y that satisfies the equation - Statement II alone therefore forces a single, definite 'yes' answer.

Cross-check: Statement II settles the question purely through its own algebra, with no need to borrow anything from Statement I; and Statement I, being automatically true for any integers, adds nothing even when read alongside Statement II. So the data in Statement II alone is sufficient, while Statement I alone is not sufficient — option (b).