| 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 application of dy/dx = (dy/dθ)/(dx/dθ), followed by solving for when dx/dθ = 0. The trigonometric differentiation and simplification are routine, and finding where the tangent is vertical is a standard technique. Slightly above average difficulty due to the double angle terms and algebraic manipulation required, but no novel insight needed. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks |
|---|---|
| 4(i) | dx dy |
| Answer | Marks |
|---|---|
| dθ dθ | B1 |
| Use dy/dx = dy/dθ ÷ dx/dθ | M1 |
| Answer | Marks |
|---|---|
| dx 2cosθ+2cos2θ | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 4(ii) | Equate denominator to zero and use any correct double angle formula | M1* |
| Obtain correct 3-term quadratic in cos θ in any form | A1 | |
| Solve for θ | depM1* |
| Answer | Marks |
|---|---|
| 2 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(i) ---
4(i) | dx dy
Obtain =2cosθ+2cos2θ or = −2sinθ−2sin2θ
dθ dθ | B1
Use dy/dx = dy/dθ ÷ dx/dθ | M1
dy 2sinθ+2sin2θ
Obtain correct in any form, e.g. –
dx 2cosθ+2cos2θ | A1
3
--- 4(ii) ---
4(ii) | Equate denominator to zero and use any correct double angle formula | M1*
Obtain correct 3-term quadratic in cos θ in any form | A1
Solve for θ | depM1*
1
Obtain x = 3 3/2 and y = , or exact equivalents
2 | A1
4
Question | Answer | Marks | GuidanceThe parametric equations of a curve are
$$x = 2\sin\theta + \sin 2\theta, \quad y = 2\cos\theta + \cos 2\theta,$$
where $0 < \theta < \pi$.
\begin{enumerate}[label=(\roman*)]
\item Obtain an expression for $\frac{dy}{dx}$ in terms of $\theta$. [3]
\item Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the $y$-axis. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q4 [7]}}