| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Gradient condition leads to trig equation |
| Difficulty | Standard +0.3 This is a straightforward parametric differentiation question requiring dy/dx = (dy/dθ)/(dx/dθ), solving a trigonometric equation in a restricted domain, and substituting to find a gradient. The steps are standard and methodical with no novel insight required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(\frac{dx}{d\theta} = -6\sin 2\theta\) and \(\frac{dy}{d\theta} = 4\cos\theta\) | B1 | |
| Use \(\frac{dy}{dx} = \frac{dy}{d\theta} / \frac{dx}{d\theta}\) and equate to 2 | M1 | |
| Use \(\sin 2\theta = 2\sin\theta\cos\theta\) and attempt value of \(\sin\theta\) | M1 | |
| Obtain \(\sin\theta = -\frac{1}{6}\) | A1 | |
| Obtain \(\theta = 3.31\) only | A1 | AWRT; and no second answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Using \(x = 3(2\cos^2\theta - 1)\), \(x = 3(1 - 2\sin^2\theta)\) or \(x = 3(\cos^2\theta - \sin^2\theta)\) to obtain \(\frac{dx}{d\theta} = a\sin\theta\cos\theta\) | M1 | |
| Obtain \(\left(\frac{dy}{dx} =\right) \frac{4\cos\theta}{-12\sin\theta\cos\theta} = 2\) | A1 | |
| Attempt value of \(\sin\theta\) | M1 | |
| Obtain \(\sin\theta = -\frac{1}{6}\) | A1 | |
| Obtain \(\theta = 3.31\) only in the given range | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(9\cos 2\theta + 4\sin\theta = 0\) and use identity to obtain quadratic in \(\sin\theta\) | M1 | |
| Obtain \(18\sin^2\theta - 4\sin\theta - 9 = 0\) | A1 | OE |
| Attempt solution to find negative value of \(\sin\theta\) | DM1 | |
| Obtain \(\sin\theta = -0.604...\) | A1 | Or \(\frac{2-\sqrt{166}}{18}\), \(\theta = 3.79...\) |
| Substitute value of \(\sin\theta\) (or *their* \(\theta\) between \(\pi\) and \(\frac{3}{2}\pi\)) in expression for first derivative | M1 | |
| Obtain \(0.551\) | A1 | AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Cartesian equation of curve \(1 - \dfrac{y^2}{8} = \dfrac{x}{3}\) oe | M1 | Must be a complete method, allow unsimplified. |
| Intersection of line and curve \(27x^2 + 8x - 24 = 0\) oe | M1 | |
| \(x = 0.8062...\) | A1 | |
| \(\theta = 3.791...\) | A1 | |
| Substitute value of \(\theta\) (or *their* \(\theta\) between \(\pi\) and \(\frac{3}{2}\pi\)) in expression for first derivative | M1 | |
| Obtain \(0.551\) | A1 | AWRT |
| 6 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\frac{dx}{d\theta} = -6\sin 2\theta$ and $\frac{dy}{d\theta} = 4\cos\theta$ | B1 | |
| Use $\frac{dy}{dx} = \frac{dy}{d\theta} / \frac{dx}{d\theta}$ and equate to 2 | M1 | |
| Use $\sin 2\theta = 2\sin\theta\cos\theta$ and attempt value of $\sin\theta$ | M1 | |
| Obtain $\sin\theta = -\frac{1}{6}$ | A1 | |
| Obtain $\theta = 3.31$ only | A1 | AWRT; and no second answer |
**Alternative method:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Using $x = 3(2\cos^2\theta - 1)$, $x = 3(1 - 2\sin^2\theta)$ or $x = 3(\cos^2\theta - \sin^2\theta)$ to obtain $\frac{dx}{d\theta} = a\sin\theta\cos\theta$ | M1 | |
| Obtain $\left(\frac{dy}{dx} =\right) \frac{4\cos\theta}{-12\sin\theta\cos\theta} = 2$ | A1 | |
| Attempt value of $\sin\theta$ | M1 | |
| Obtain $\sin\theta = -\frac{1}{6}$ | A1 | |
| Obtain $\theta = 3.31$ only in the given range | A1 | |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $9\cos 2\theta + 4\sin\theta = 0$ and use identity to obtain quadratic in $\sin\theta$ | M1 | |
| Obtain $18\sin^2\theta - 4\sin\theta - 9 = 0$ | A1 | OE |
| Attempt solution to find negative value of $\sin\theta$ | DM1 | |
| Obtain $\sin\theta = -0.604...$ | A1 | Or $\frac{2-\sqrt{166}}{18}$, $\theta = 3.79...$ |
| Substitute value of $\sin\theta$ (or *their* $\theta$ between $\pi$ and $\frac{3}{2}\pi$) in expression for first derivative | M1 | |
| Obtain $0.551$ | A1 | AWRT |
## Question 7(b) [Alternative Method]:
| Answer | Mark | Guidance |
|--------|------|----------|
| Cartesian equation of curve $1 - \dfrac{y^2}{8} = \dfrac{x}{3}$ oe | M1 | Must be a complete method, allow unsimplified. |
| Intersection of line and curve $27x^2 + 8x - 24 = 0$ oe | M1 | |
| $x = 0.8062...$ | A1 | |
| $\theta = 3.791...$ | A1 | |
| Substitute value of $\theta$ (or *their* $\theta$ between $\pi$ and $\frac{3}{2}\pi$) in expression for first derivative | M1 | |
| Obtain $0.551$ | A1 | AWRT |
| | **6** | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{68f4b2dc-a05d-4061-aaf0-de15cfe186a9-10_657_792_269_664}
The diagram shows the curve with parametric equations
$$x = 3 \cos 2 \theta , \quad y = 4 \sin \theta ,$$
for $\pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi$. Points $P$ and $Q$ lie on the curve. The gradient of the curve at $P$ is 2 . The straight line $3 x + y = 0$ meets the curve at $Q$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $\theta$ at $P$, giving your answer correct to 3 significant figures.
\item Find the gradient of the curve at $Q$, giving your answer correct to 3 significant figures.\\
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 P2 2022 Q7 [11]}}