| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Connected Rates of Change |
| Type | Chain rule with three variables |
| Difficulty | Moderate -0.3 This is a straightforward connected rates of change question requiring basic chain rule application (dB/dt ÷ dF/dt), simple differentiation of polynomial, trigonometric, and exponential functions, and substitution at t=4. The interpretation and modeling critique parts are routine. Slightly easier than average due to direct application of standard techniques with no problem-solving insight required. |
| Spec | 1.07b Gradient as rate of change: dy/dx notation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Response | Mark | Guidance |
| The rate of increase in the number of bees regarding the increase in the number of plants | B1 | BOD 'regarding' |
| Rate of bees with respect to flowers | B0 | no 'change' |
| Rate of change in number of bees in terms of the number of flowers | B1 | BOD 'in terms of' |
| The rate of growth in the number of bees when the number of plants increases over time | B1 | — |
| The rate of change of flowers according to the number of bees | B0 | wrong way around |
| The rate of increase of bees over the rate of increase in wildflowers | B0 | not 'over' – suggests fraction |
| The rate of change of number of bees compared to number of plants | B1 | BOD 'number' |
| Rate at which the number of bees increases as the number of plants increase | B1 | Includes 'rate' and 'increases' |
| The change in number of bees in respect to the number of flowers | B0 | no 'rate' |
| The rate of increase in number of bees in accordance to the number of plants | B1 | BOD 'in accordance' |
| How the number of bees vary with the number of flowers | B0 | not rate of change |
| The rate at which the bees to flowers ratio is changing | B1 | BOD 'ratio' |
| Rate of growth of bees depending on the number of flowers | B1 | — |
| Rate of change of bees per wildflower plant | B1 | — |
| Rate of change of the number of bees in terms of flowers | B1 | — |
| The rate of change between the bees and the plants | B0 | no dependency implied |
| The rate of change in number of bees against the change in plants | B1 | — |
| The rate of change of bees compared to flowers | B1 | — |
| Rate of change of the number of bees as the number of flowers vary | B1 | — |
| The rate at which the number of bees increase with the number of plants | B1 | BOD 'with' |
## Question 9(a): Exemplar Responses
| Response | Mark | Guidance |
|---|---|---|
| The rate of increase in the number of bees regarding the increase in the number of plants | **B1** | BOD 'regarding' |
| Rate of bees with respect to flowers | **B0** | no 'change' |
| Rate of change in number of bees in terms of the number of flowers | **B1** | BOD 'in terms of' |
| The rate of growth in the number of bees when the number of plants increases over time | **B1** | — |
| The rate of change of flowers according to the number of bees | **B0** | wrong way around |
| The rate of increase of bees over the rate of increase in wildflowers | **B0** | not 'over' – suggests fraction |
| The rate of change of number of bees compared to number of plants | **B1** | BOD 'number' |
| Rate at which the number of bees increases as the number of plants increase | **B1** | Includes 'rate' and 'increases' |
| The change in number of bees in respect to the number of flowers | **B0** | no 'rate' |
| The rate of increase in number of bees in accordance to the number of plants | **B1** | BOD 'in accordance' |
| How the number of bees vary with the number of flowers | **B0** | not rate of change |
| The rate at which the bees to flowers ratio is changing | **B1** | BOD 'ratio' |
| Rate of growth of bees depending on the number of flowers | **B1** | — |
| Rate of change of bees per wildflower plant | **B1** | — |
| Rate of change of the number of bees in terms of flowers | **B1** | — |
| The rate of change between the bees and the plants | **B0** | no dependency implied |
| The rate of change in number of bees against the change in plants | **B1** | — |
| The rate of change of bees compared to flowers | **B1** | — |
| Rate of change of the number of bees as the number of flowers vary | **B1** | — |
| The rate at which the number of bees increase with the number of plants | **B1** | BOD 'with' |9 Conservationists are studying how the number of bees in a wildflower meadow varies according to the number of wildflower plants. The study takes place over a series of weeks in the summer. A model is suggested for the number of bees, $B$, and the number of wildflower plants, $F$, at time $t$ weeks after the start of the study.
In the model $B = 20 + 2 t + \cos 3 t$ and $F = 50 \mathrm { e } ^ { 0.1 t }$.
The model assumes that $B$ and $F$ can be treated as continuous variables.
\begin{enumerate}[label=(\alph*)]
\item State the meaning of $\frac { \mathrm { d } B } { \mathrm {~d} F }$.
\item Determine $\frac { \mathrm { d } B } { \mathrm {~d} F }$ when $t = 4$.
\item Suggest a reason why this model may not be valid for values of $t$ greater than 12 .
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2023 Q9 [6]}}