| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Area between curve and line |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: (a) completing the square or differentiation to find a minimum, (b) integration to find area between curves with given intersection points, and (c) related rates using the chain rule. All parts are routine A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\left[\frac{dy}{dx}=\right]\frac{9}{2}x-12\ [=0]\) or \([y=]\frac{9}{4}\left\{\left(x-\frac{8}{3}\right)^2+\frac{8}{9}\right\}\) or \(\frac{9}{4}\left(x-\frac{8}{3}\right)^2+2\) | B1 | OE. Either \(\frac{dy}{dx}\) or a correct expression in completed square form. Allow unsimplified |
| \(x=\frac{24}{9}\) | B1 | OE. Condone 2.67 AWRT |
| \(y=2\) | B1 | CAO. Note: \(x=\frac{-b}{2a}=\frac{8}{3}\) B1; substitute \(\frac{8}{3}\) for \(x\) in \(y=\) B1; \(y=2\) B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([\text{Area} =] \int\left(18 - \frac{3}{8}x^2 - \left(\frac{9}{4}x^2 - 12x + 18\right)\right)dx\) | M1 | Intention to integrate and subtract areas (either way around). Can be two separate functions or combined. Using \(y^2\) scores 0/5 but condone inclusion of \(\pi\) except for the final mark. |
| Either separately: \(\left[18x\right] - \frac{3x^{\frac{7}{2}}}{8 \times \frac{7}{2}}\) , \(\left(\frac{9x^3}{4\times3} - \frac{12x^2}{2}\left[+18x\right]\right)\) | B1, B1 | One mark for correct integration of each curve, allow unsimplified. \(\left(\left[18x\right] - \frac{3}{28}x^{\frac{7}{2}}\right)\left(\frac{3}{4}x^3 - 6x^2\left[+18x\right]\right)\) or \(\left[18x\right] - \frac{3}{28}x^{\frac{7}{2}} - \frac{3}{4}x^3 + 6x^2\left[-18x\right]\) BUT condone sign errors that are only due to missing brackets. |
| Or combined: \(\left[18x\right] - \frac{3x^{\frac{7}{2}}}{8\times\frac{7}{2}} - \frac{9x^3}{4\times3} + \frac{12x^2}{2}\left[-18x\right]\) | ||
| \(= \left(-\frac{3}{28}\times4^{\frac{7}{2}} - \frac{3}{4}\times4^3 + 6\times4^2\right)\left[-(0)\right]\) | M1 | Clear substitution of 4 into at least one integrated expression (defined by at least one correct power) which can be unsimplified. |
| \(= \frac{240}{7}\) or 34.3 AWRT | A1 | SC: If all marks awarded except the final M1, SCB1 is available for the correct final answer. |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left[\frac{dy}{dx} =\right] \frac{-5\times3}{2\times8}x^{\frac{3}{2}}\left[= -\frac{15}{16}x^{\frac{3}{2}}\right]\) | B1 | Allow unsimplified. |
| \(\frac{dy}{dt} = \frac{dy}{dx}\times\frac{dx}{dt} \Rightarrow \frac{dy}{dt} = -\frac{15}{16}\times8\times2\) | M1 | Substitute \(x = 4\) into their \(\frac{dy}{dx}\) and multiply by 2. |
| \(-15\) | A1 | Accept decreasing [at/by] 15 |
| 3 | Note: If incorrect curve used, this is not a MR and only M1 mark is available. Expect \(\left(\frac{9(4)}{2} - 12\right)\times2\ [=12]\) |
## Question 11(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\left[\frac{dy}{dx}=\right]\frac{9}{2}x-12\ [=0]$ or $[y=]\frac{9}{4}\left\{\left(x-\frac{8}{3}\right)^2+\frac{8}{9}\right\}$ or $\frac{9}{4}\left(x-\frac{8}{3}\right)^2+2$ | B1 | OE. Either $\frac{dy}{dx}$ or a correct expression in completed square form. Allow unsimplified |
| $x=\frac{24}{9}$ | B1 | OE. Condone 2.67 AWRT |
| $y=2$ | B1 | CAO. Note: $x=\frac{-b}{2a}=\frac{8}{3}$ B1; substitute $\frac{8}{3}$ for $x$ in $y=$ B1; $y=2$ B1 |
## Question 11(b):
**Note:** For 11(b) look for working to be marked on page 19 or annotate it as BP or SEEN
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[\text{Area} =] \int\left(18 - \frac{3}{8}x^2 - \left(\frac{9}{4}x^2 - 12x + 18\right)\right)dx$ | M1 | Intention to integrate **and** subtract areas (either way around). Can be two separate functions or combined. Using $y^2$ scores 0/5 but condone inclusion of $\pi$ except for the final mark. |
| Either separately: $\left[18x\right] - \frac{3x^{\frac{7}{2}}}{8 \times \frac{7}{2}}$ , $\left(\frac{9x^3}{4\times3} - \frac{12x^2}{2}\left[+18x\right]\right)$ | B1, B1 | One mark for correct integration of each curve, allow unsimplified. $\left(\left[18x\right] - \frac{3}{28}x^{\frac{7}{2}}\right)\left(\frac{3}{4}x^3 - 6x^2\left[+18x\right]\right)$ or $\left[18x\right] - \frac{3}{28}x^{\frac{7}{2}} - \frac{3}{4}x^3 + 6x^2\left[-18x\right]$ **BUT** condone sign errors that are only due to missing brackets. |
| Or combined: $\left[18x\right] - \frac{3x^{\frac{7}{2}}}{8\times\frac{7}{2}} - \frac{9x^3}{4\times3} + \frac{12x^2}{2}\left[-18x\right]$ | | |
| $= \left(-\frac{3}{28}\times4^{\frac{7}{2}} - \frac{3}{4}\times4^3 + 6\times4^2\right)\left[-(0)\right]$ | M1 | Clear substitution of 4 into at least one integrated expression (defined by at least one correct power) which can be unsimplified. |
| $= \frac{240}{7}$ or **34.3** AWRT | A1 | **SC:** If all marks awarded except the final M1, SCB1 is available for the correct final answer. |
| | **5** | |
---
## Question 11(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[\frac{dy}{dx} =\right] \frac{-5\times3}{2\times8}x^{\frac{3}{2}}\left[= -\frac{15}{16}x^{\frac{3}{2}}\right]$ | B1 | Allow unsimplified. |
| $\frac{dy}{dt} = \frac{dy}{dx}\times\frac{dx}{dt} \Rightarrow \frac{dy}{dt} = -\frac{15}{16}\times8\times2$ | M1 | Substitute $x = 4$ into their $\frac{dy}{dx}$ and multiply by 2. |
| $-15$ | A1 | Accept decreasing [at/by] 15 |
| | **3** | **Note:** If incorrect curve used, this is not a MR and only M1 mark is available. Expect $\left(\frac{9(4)}{2} - 12\right)\times2\ [=12]$ |11
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the minimum point of the curve $y = \frac { 9 } { 4 } x ^ { 2 } - 12 x + 18$.\\
\includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-18_675_901_1270_612}
The diagram shows the curves with equations $y = \frac { 9 } { 4 } x ^ { 2 } - 12 x + 18$ and $y = 18 - \frac { 3 } { 8 } x ^ { \frac { 5 } { 2 } }$. The curves intersect at the points $( 0,18 )$ and $( 4,6 )$.
\item Find the area of the shaded region.
\item A point $P$ is moving along the curve $y = 18 - \frac { 3 } { 8 } x ^ { \frac { 5 } { 2 } }$ in such a way that the $x$-coordinate of $P$ is increasing at a constant rate of 2 units per second.
Find the rate at which the $y$-coordinate of $P$ is changing when $x = 4$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q11 [11]}}