| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Tangent parallel to axis condition |
| Difficulty | Standard +0.3 This is a straightforward parametric differentiation question requiring standard techniques: finding dy/dθ and dx/dθ, then dy/dx = (dy/dθ)/(dx/dθ). Part (ii) requires setting dx/dθ = 0 and solving a trigonometric equation. While it involves multiple steps, all techniques are routine for P3 level 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 |
| Obtain \(\frac{dx}{d\theta} = 2\cos\theta + 2\cos 2\theta\) or \(\frac{dy}{d\theta} = -2\sin\theta - 2\sin 2\theta\) | B1 | |
| Use \(dy/dx = dy/d\theta \div dx/d\theta\) | M1 | |
| Obtain correct \(\frac{dy}{dx}\) in any form, e.g. \(-\frac{2\sin\theta + 2\sin 2\theta}{2\cos\theta + 2\cos 2\theta}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Equate denominator to zero and use any correct double angle formula | M1* | |
| Obtain correct 3-term quadratic in \(\cos\theta\) in any form | A1 | |
| Solve for \(\theta\) | depM1* | |
| Obtain \(x = 3\sqrt{3}/2\) and \(y = \frac{1}{2}\), or exact equivalents | A1 |
## Question 4(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $\frac{dx}{d\theta} = 2\cos\theta + 2\cos 2\theta$ or $\frac{dy}{d\theta} = -2\sin\theta - 2\sin 2\theta$ | B1 | |
| Use $dy/dx = dy/d\theta \div dx/d\theta$ | M1 | |
| Obtain correct $\frac{dy}{dx}$ in any form, e.g. $-\frac{2\sin\theta + 2\sin 2\theta}{2\cos\theta + 2\cos 2\theta}$ | A1 | |
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## Question 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Equate denominator to zero and use any correct double angle formula | M1* | |
| Obtain correct 3-term quadratic in $\cos\theta$ in any form | A1 | |
| Solve for $\theta$ | depM1* | |
| Obtain $x = 3\sqrt{3}/2$ and $y = \frac{1}{2}$, or exact equivalents | A1 | |
---4 The parametric equations of a curve are
$$x = 2 \sin \theta + \sin 2 \theta , \quad y = 2 \cos \theta + \cos 2 \theta$$
where $0 < \theta < \pi$.\\
(i) Obtain an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$.\\
(ii) Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the $y$-axis.\\
\hfill \mbox{\textit{CAIE P3 2018 Q4 [7]}}