Let $G$ be a weighted connected undirected graph with distinct positive edge weights. If every edge weight is increased by the same value, then which of the following statements is/are TRUE? P: Minimum spanning tree of $G$ does not change Q: Shortest path between any pair of vertices does not change

• Statement P is TRUE: Every spanning tree of a connected undirected graph with $|V|$ vertices contains exactly $|V| - 1$ edges. If a constant value $c > 0$ is added to every edge weight, the total weight of every spanning tree increases by the exact same amount, $(|V| - 1)c$. Therefore, the spanning tree that had the minimum weight initially will still have the minimum weight. Additionally, algorithms like Kruskal's depend purely on the relative sorted order of edge weights, which remains unchanged when adding a constant $c$.

• Statement Q is FALSE: Different paths between a pair of vertices can have different numbers of edges. If path $P_1$ has $k_1$ edges and path $P_2$ has $k_2$ edges with $k_1 < k_2$, adding $c$ to each edge increases the weight of $P_1$ by $k_1 c$ and $P_2$ by $k_2 c$. A path with more edges incurs a larger penalty, which can cause a previously longer path with fewer edges to become the new shortest path.

Therefore, only statement P is TRUE, which corresponds to option (A).