| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Show gradient expression then find coordinates |
| Difficulty | Standard +0.3 This is a standard parametric differentiation question requiring the chain rule (dy/dx = (dy/dt)/(dx/dt)), basic trigonometric identities (sin 2t, cos 2t derivatives, and double angle formula for simplification), and routine application to find stationary points. The 'show that' format and straightforward algebraic manipulation make it slightly easier than average, though it requires careful execution across multiple parts. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(12\sin t \cos t\) or equivalent for \(\frac{dx}{dt}\) | B1 | |
| Obtain \(4\cos 2t - 6\sin 2t\) or equivalent for \(\frac{dy}{dt}\) | B1 | |
| Obtain expression for \(\frac{dy}{dx}\) in terms of \(t\) | M1 | |
| Use \(2\sin t \cos t = \sin 2t\) | A1 | |
| Confirm given answer \(\frac{dy}{dx} = \frac{2}{3}\cot 2t - 1\) with no errors seen | A1 | |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(\tan 2t = \frac{2}{3}\) | B1 | |
| Obtain \(t = 0.294\) | B1 | |
| Obtain \(t = 1.865\) | B1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt solution of \(2\sin 2t + 3\cos 2t = 0\) at least as far as \(\tan 2t = \ldots\) | M1 | |
| Obtain \(\tan 2t = -\frac{3}{2}\) or equivalent | A1 | |
| Substitute to obtain \(-\frac{13}{9}\) | A1 | |
| Total: 3 |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $12\sin t \cos t$ or equivalent for $\frac{dx}{dt}$ | B1 | |
| Obtain $4\cos 2t - 6\sin 2t$ or equivalent for $\frac{dy}{dt}$ | B1 | |
| Obtain expression for $\frac{dy}{dx}$ in terms of $t$ | M1 | |
| Use $2\sin t \cos t = \sin 2t$ | A1 | |
| Confirm given answer $\frac{dy}{dx} = \frac{2}{3}\cot 2t - 1$ with no errors seen | A1 | |
| **Total: 5** | | |
---
## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\tan 2t = \frac{2}{3}$ | B1 | |
| Obtain $t = 0.294$ | B1 | |
| Obtain $t = 1.865$ | B1 | |
| **Total: 3** | | |
---
## Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt solution of $2\sin 2t + 3\cos 2t = 0$ at least as far as $\tan 2t = \ldots$ | M1 | |
| Obtain $\tan 2t = -\frac{3}{2}$ or equivalent | A1 | |
| Substitute to obtain $-\frac{13}{9}$ | A1 | |
| **Total: 3** | | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{77672e56-a268-47b8-ab8b-cd84b4b3de4f-10_551_689_258_726}
The parametric equations of a curve are
$$x = 6 \sin ^ { 2 } t , \quad y = 2 \sin 2 t + 3 \cos 2 t$$
for $0 \leqslant t < \pi$. The curve crosses the $x$-axis at points $B$ and $D$ and the stationary points are $A$ and $C$, as shown in the diagram.\\
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 3 } \cot 2 t - 1$.\\
(ii) Find the values of $t$ at $A$ and $C$, giving each answer correct to 3 decimal places.\\
(iii) Find the value of the gradient of the curve at $B$.\\
\hfill \mbox{\textit{CAIE P2 Q7 [11]}}