| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Properties of specific curves |
| Difficulty | Challenging +1.2 This is a structured multi-part parametric differentiation question requiring chain rule application, tangent equation derivation, and geometric proof using trigonometric identities. While it involves several steps and coordinate geometry, each part follows standard A-level techniques with clear guidance ('show that', 'hence'), making it moderately above average but not requiring novel insight. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| EITHER State \(\frac{dx}{dt} = -3a\cos^2 t \sin t\) or \(\frac{dy}{dt} = 3a\sin^2 t \cos t\), or equivalent | B1 | |
| Use \(\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\) | M1 | |
| OR State \(\frac{2}{3}x^{-\frac{1}{3}}dx\) or \(\frac{2}{3}y^{-\frac{1}{3}}dy\) as differentials of \(x^{\frac{2}{3}}\) or \(y^{\frac{2}{3}}\) respectively, or equivalent | B1 | |
| Obtain \(\frac{dy}{dx}\) in terms of \(t\), having taken differential of a constant to be zero | M1 | |
| Obtain \(\frac{dy}{dx}\) in any correct form | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Form the equation of the tangent | M1 | |
| Obtain the equation in any correct form | A1 | |
| Obtain the given answer | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State the \(x\)-coordinate of \(X\) or the \(y\)-coordinate of \(Y\) in any correct form | B1 | |
| Obtain the given answer with no errors seen | B1 |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **EITHER** State $\frac{dx}{dt} = -3a\cos^2 t \sin t$ or $\frac{dy}{dt} = 3a\sin^2 t \cos t$, or equivalent | B1 | |
| Use $\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$ | M1 | |
| **OR** State $\frac{2}{3}x^{-\frac{1}{3}}dx$ or $\frac{2}{3}y^{-\frac{1}{3}}dy$ as differentials of $x^{\frac{2}{3}}$ or $y^{\frac{2}{3}}$ respectively, or equivalent | B1 | |
| Obtain $\frac{dy}{dx}$ in terms of $t$, having taken differential of a constant to be zero | M1 | |
| Obtain $\frac{dy}{dx}$ in any correct form | A1 | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Form the equation of the tangent | M1 | |
| Obtain the equation in any correct form | A1 | |
| Obtain the given answer | A1 | |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State the $x$-coordinate of $X$ or the $y$-coordinate of $Y$ in any correct form | B1 | |
| Obtain the given answer with no errors seen | B1 | |
---6 The parametric equations of a curve are
$$x = a \cos ^ { 3 } t , \quad y = a \sin ^ { 3 } t$$
where $a$ is a positive constant and $0 < t < \frac { 1 } { 2 } \pi$.\\
(i) Express $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.\\
(ii) Show that the equation of the tangent to the curve at the point with parameter $t$ is
$$x \sin t + y \cos t = a \sin t \cos t$$
(iii) Hence show that, if this tangent meets the $x$-axis at $X$ and the $y$-axis at $Y$, then the length of $X Y$ is always equal to $a$.
\hfill \mbox{\textit{CAIE P3 2009 Q6 [8]}}