| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find parameter value given gradient condition |
| Difficulty | Moderate -0.3 Part (i) is a straightforward application of the chain rule for parametric differentiation (dy/dx = (dy/dt)/(dx/dt)) with standard derivatives of ln, sin, and cos. Part (ii) requires solving sec t = 3, which is routine trigonometry. This is slightly easier than average as it involves direct application of well-practiced techniques with no conceptual challenges or multi-step problem-solving. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(\frac{dx}{dt} = 1 - \sin t\) | B1 | |
| Use chain rule to find the derivative of \(y\) | M1 | |
| Obtain \(\frac{dy}{dt} = \frac{-\cos t}{1 + \sin t}\), or equivalent | A1 | |
| Use \(\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dx}{dt}\) | M1 | |
| Obtain the given answer correctly | A1 | [5] |
| (ii) State or imply \(t = \cos^{-1}\left(\frac{1}{x}\right)\) | B1 | |
| Obtain answers \(x = 1.56\) and \(x = -0.898\) | B1 + B1 | [3] |
**(i)** State $\frac{dx}{dt} = 1 - \sin t$ | B1 |
Use chain rule to find the derivative of $y$ | M1 |
Obtain $\frac{dy}{dt} = \frac{-\cos t}{1 + \sin t}$, or equivalent | A1 |
Use $\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dx}{dt}$ | M1 |
Obtain the given answer correctly | A1 | [5]
**(ii)** State or imply $t = \cos^{-1}\left(\frac{1}{x}\right)$ | B1 |
Obtain answers $x = 1.56$ and $x = -0.898$ | B1 + B1 | [3]
---4 The parametric equations of a curve are
$$x = t + \cos t , \quad y = \ln ( 1 + \sin t )$$
where $- \frac { 1 } { 2 } \pi < t < \frac { 1 } { 2 } \pi$.\\
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec t$.\\
(ii) Hence find the $x$-coordinates of the points on the curve at which the gradient is equal to 3 . Give your answers correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P3 2016 Q4 [8]}}