DP Mathematics: Applications and Interpretation Questionbank

Find the maximum possible value of $e$.

Find an expression for $e\text{'}$ in terms of $e$.

Given that the complement of $G$ is also planar and connected, find the possible values of $e$.

$H$ is a simple graph with $v$ vertices and $e$ edges.

Given that both $H$ and its complement are planar and connected, find the maximum possible value of $v$.

Use Prim’s algorithm, starting at vertex $\text{B}$, to find the weight of the minimum spanning tree of the remaining graph. You should indicate clearly the order in which the algorithm selects each edge.

Hence, for the case where $x<9$, find a lower bound for Daniel’s travel time, in terms of $x$.

Write down a Hamiltonian cycle in $W$.

$p$.

The tourist office in the city has received complaints about the lack of cleanliness of some routes between the attractions. Corinne, the office manager, decides to inspect all the routes between all the attractions, starting and finishing at $\text{H}$. The sum of the weights of all the edges in graph $W$ is $(92+x)$.

Corinne inspects all the routes as quickly as possible and takes $2$ hours.

Find the value of $x$ during Corinne’s inspection.

$q$.

Find the least value of $x$ for which the edge $\text{AF}$ will definitely not be used by Daniel.

$r$.

Use the nearest neighbour algorithm to find two possible cycles.

Find the best upper bound for Daniel’s travel time.

Hence state the value of the upper bound for Daniel’s travel time for the value of $x$ found in part (e)(i).

Write down an expression for the position vector $\mathit{r}$ of the ship, $t$ hours after $1:00 \text{pm}$.

Find the time at which the bearing of the ship from the harbour is $045˚$.

Write down the number of days it takes for the pollution to reach the last town.

Find which town is last to be polluted. Justify your answer.

By using Kruskal’s algorithm, find the minimum spanning tree for $S$, showing clearly the order in which edges are added.

Hence find the minimum installation cost for the cables that would allow all the buildings to be part of the computer network.

State why a path that forms a Hamiltonian cycle does not always form an Eulerian circuit.

Write down the minimum number of edges in $G$.

Find the maximum number of edges in $G$.

Find the maximum number of edges in $G$, given that $G$ contains an Eulerian circuit.

Explain why $2e=kf$.

Find the value of $f$ when $v=9$ and $k=3$.

Find the possible values of $f$ when $v=13$.