All 4 should be correct in my opinion.
(1) w is not in Span(v1,v2,v3). This means that w cannot be written as a linear combination of v1,v2,v3. {v1,v2,v3} is also given as a LI set. This means that v1 is non-zero vector, and, v2 doesn't belong to Span{v1}, and, v3 doesn't belong to Span{v1,v2} (The general theorem is that a set is LD, if and only if, some vector can be written as a linear combination of preceding vectors - source: David C Lay 1.7, Theorem 7). Since w is not in Span{v1,v2,v3}, using the same theorem, we can infer that {v1,v2,v3,w} must be LI.
(3) Since w is not in Span(v1,v2,v3), w naturally can't be zero vector, since zero vector belongs to Span(v1,v2,v3). So {w} is LI.
(2) Suppose {v1,v2,w} was LD. Then using theorem stated in (1), there are 3 possibilities. First, v1 could be zero vector. This is not possible, since {v1,v2,v3} is a LI set. Second option, it could be that v2 is in Span(v1), but this contradicts that {v1,v2,v3} is LI. Third and Final Possibility, w could be in Span(v1,v2). But if that is the case, it should also be in Span(v1,v2,v3), which contradicts the premise. So by contradiction, {v1,v2,w} must be LI.
(4) Has same reasoning as (2).
