Consider the following two statements about the function $$$f\left( x \right) = \left| x \right|:$$$

$$P.\,\,f\left( x \right)$$ is continuous for all real values of $$x$$
$$Q.\,\,f\left( x \right)$$ is differentiable for all real values of $$x$$

Which of the following is True?

The height of a binary tree is the maximum number of edges in any root to leaf path. The maximum number of nodes in a binary tree of height $$h$$ is:

Consider a weighted undirected graph with positive edge weights and let $$uv$$ be an edge in the graph. It is known that the shortest path from the source vertex $$s$$ to $$u$$ has weight 53 and the shortest path from $$s$$ to $$v$$ has weighted 65. Which one of the following statements is always true?

Let $$G$$ be the non-planar graph with minimum possible number of edges. Then $$G$$ has