| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Show gradient expression then find coordinates |
| Difficulty | Moderate -0.3 This is a straightforward parametric differentiation question requiring standard techniques: finding dy/dx using the chain rule, locating stationary points by setting dy/dx = 0, and finding specific coordinates. The 'show that' part involves routine application of dx/dt and dy/dt with the double angle formula. While it requires multiple steps across three parts, each individual technique is standard A-level material with no novel problem-solving required. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State \(\frac{dx}{dt} = \sin t\) and \(\frac{dy}{dt} = -6\sin 2t\) | B1 | |
| Use \(\sin 2t = 2\sin t\cos t\) | B1 | |
| Form expression for \(\frac{dy}{dx}\) in terms of \(t\) | M1 | |
| Confirm \(-12\cos t\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Identify \(\frac{1}{2}\pi\) as value of \(t\) | B1 | |
| Obtain \((2, -2)\) | B1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Identify \(\cos 2t = -\frac{1}{3}\) | B1 | |
| Attempt to find value of \(t\) (or of \(\cos t\)) for at least one of the two points | M1 | |
| Obtain 0.955 (or \(\frac{1}{\sqrt{3}}\)) or 2.186 (or \(-\frac{1}{\sqrt{3}}\)) | A1 | |
| Obtain \(-\frac{12}{\sqrt{3}}\) or \(-4\sqrt{3}\) or \(-6.93\) and \(\frac{12}{\sqrt{3}}\) or \(4\sqrt{3}\) or \(6.93\) | A1 | [4] |
## Question 7:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State $\frac{dx}{dt} = \sin t$ and $\frac{dy}{dt} = -6\sin 2t$ | B1 | |
| Use $\sin 2t = 2\sin t\cos t$ | B1 | |
| Form expression for $\frac{dy}{dx}$ in terms of $t$ | M1 | |
| Confirm $-12\cos t$ | A1 | [4] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Identify $\frac{1}{2}\pi$ as value of $t$ | B1 | |
| Obtain $(2, -2)$ | B1 | [2] |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Identify $\cos 2t = -\frac{1}{3}$ | B1 | |
| Attempt to find value of $t$ (or of $\cos t$) for at least one of the two points | M1 | |
| Obtain 0.955 (or $\frac{1}{\sqrt{3}}$) or 2.186 (or $-\frac{1}{\sqrt{3}}$) | A1 | |
| Obtain $-\frac{12}{\sqrt{3}}$ or $-4\sqrt{3}$ or $-6.93$ and $\frac{12}{\sqrt{3}}$ or $4\sqrt{3}$ or $6.93$ | A1 | [4] |7\\
\includegraphics[max width=\textwidth, alt={}, center]{a07e6d2f-ded1-4c62-957b-41fb94b46a2d-3_423_837_1352_651}
The diagram shows the curve with parametric equations
$$x = 2 - \cos t , \quad y = 1 + 3 \cos 2 t$$
for $0 < t < \pi$. The minimum point is $M$ and the curve crosses the $x$-axis at points $P$ and $Q$.\\
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - 12 \cos t$.\\
(ii) Find the coordinates of $M$.\\
(iii) Find the gradient of the curve at $P$ and at $Q$.
\hfill \mbox{\textit{CAIE P2 2016 Q7 [10]}}