Final answer:
Increasing the edge weights by the same value in a weighted graph with distinct positive weights does not change the minimum spanning tree or the shortest paths, since these are determined by the relative weights, not the absolute values.
Step-by-step explanation:
When every edge weight in a weighted connected undirected graph with distinct positive edge weights is increased by the same value, the minimum spanning tree (MST) does not change. This is because the structure of the MST is dependent on the relative weights of the edges, not their absolute values. Since every edge is increased by the same amount, the relative weights remain the same, thus preserving the MST.
However, the shortest path between any pair of vertices could theoretically change if the graph initially contained edges with the same weight, or if the graph's paths contained different numbers of edges. But given the information that the graph has distinct positive edge weights, increasing each by the same value will not affect the shortest paths because the added value will uniformly increase the path length, implying that the paths will retain their order of shortest to longest. Thus, the shortest paths remain unchanged too.
Therefore, the correct answer to the student's question is option (D): Both statements P and Q are true.