| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| 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 application of dy/dx = (dy/dθ)/(dx/dθ). Part (i) involves routine differentiation and substitution. Part (ii) requires recognizing that a vertical tangent means dx/dθ = 0 and solving a trigonometric equation using the double angle formula. While it requires multiple techniques, all steps are standard A-level procedures with no novel insight needed. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(\frac{dx}{d\theta} = -4\sin 2\theta + 3\cos\theta\) | B1 | B1 may be implied |
| Use \(\frac{dy}{dx} = \frac{dy}{d\theta} \big/ \frac{dx}{d\theta}\) in terms of \(\theta\) or with 1 already substituted | M1 | |
| Obtain or imply \(\dfrac{dy}{dx} = \dfrac{-3\sin\theta}{-4\sin 2\theta + 3\cos\theta}\) | A1 | |
| Substitute 1 to obtain 1.25 | A1 | Or greater accuracy \(1.252013\ldots\) |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Equate denominator of first derivative to zero | M1 | |
| Use \(\sin 2\theta = 2\sin\theta\cos\theta\) | A1 | |
| Obtain \(\sin\theta = \dfrac{3}{8}\) | A1 | |
| Total | 3 |
## Question 5(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $\frac{dx}{d\theta} = -4\sin 2\theta + 3\cos\theta$ | B1 | B1 may be implied |
| Use $\frac{dy}{dx} = \frac{dy}{d\theta} \big/ \frac{dx}{d\theta}$ in terms of $\theta$ or with 1 already substituted | M1 | |
| Obtain or imply $\dfrac{dy}{dx} = \dfrac{-3\sin\theta}{-4\sin 2\theta + 3\cos\theta}$ | A1 | |
| Substitute 1 to obtain 1.25 | A1 | Or greater accuracy $1.252013\ldots$ |
| **Total** | **4** | |
---
## Question 5(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Equate denominator of first derivative to zero | M1 | |
| Use $\sin 2\theta = 2\sin\theta\cos\theta$ | A1 | |
| Obtain $\sin\theta = \dfrac{3}{8}$ | A1 | |
| **Total** | **3** | |
---5 The parametric equations of a curve are
$$x = 2 \cos 2 \theta + 3 \sin \theta , \quad y = 3 \cos \theta$$
for $0 < \theta < \frac { 1 } { 2 } \pi$.\\
(i) Find the gradient of the curve at the point for which $\theta = 1$ radian.\\
(ii) Find the value of $\sin \theta$ at the point on the curve where the tangent is parallel to the $y$-axis.\\
\hfill \mbox{\textit{CAIE P2 2018 Q5 [7]}}