| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find tangent equation at parameter |
| Difficulty | Standard +0.3 This is a straightforward parametric differentiation question requiring the chain rule (dy/dx = (dy/dt)/(dx/dt)), followed by finding a tangent equation and its x-intercept. All steps are standard C4 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(\frac{dx}{dt} = 2\sin t \cos t\), \(\frac{dy}{dt} = 2\sec^2 t\) | B1 B1 |
| \(\frac{dy}{dx} = \frac{\sec^2 t}{\sin t \cos t}\left(-\frac{1}{\sin t \cos^3 t}\right)\) | M1 A1 | or equivalent |
| (b) | At \(t = \frac{\pi}{3}\): \(x = \frac{3}{4}\), \(y = 2\sqrt{3}\) | B1 |
| \(\frac{dy}{dx} = \frac{\sec^2 \frac{\pi}{3}}{\sin \frac{\pi}{3} \cos \frac{\pi}{3}} = \frac{16}{\sqrt{3}}\) | M1 A1 | |
| \(y - 2\sqrt{3} = \frac{16}{\sqrt{3}}\left(x - \frac{3}{4}\right)\) | M1 | |
| \(y = 0 \Rightarrow x = \frac{3}{8}\) | M1 A1 |
(a) | $\frac{dx}{dt} = 2\sin t \cos t$, $\frac{dy}{dt} = 2\sec^2 t$ | B1 B1 | |
| $\frac{dy}{dx} = \frac{\sec^2 t}{\sin t \cos t}\left(-\frac{1}{\sin t \cos^3 t}\right)$ | M1 A1 | or equivalent |
(b) | At $t = \frac{\pi}{3}$: $x = \frac{3}{4}$, $y = 2\sqrt{3}$ | B1 | |
| $\frac{dy}{dx} = \frac{\sec^2 \frac{\pi}{3}}{\sin \frac{\pi}{3} \cos \frac{\pi}{3}} = \frac{16}{\sqrt{3}}$ | M1 A1 | |
| $y - 2\sqrt{3} = \frac{16}{\sqrt{3}}\left(x - \frac{3}{4}\right)$ | M1 | |
| $y = 0 \Rightarrow x = \frac{3}{8}$ | M1 A1 | |
---4. A curve $C$ has parametric equations
$$x = \sin ^ { 2 } t , \quad y = 2 \tan t , \quad 0 \leqslant t < \frac { \pi } { 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
The tangent to $C$ at the point where $t = \frac { \pi } { 3 }$ cuts the $x$-axis at the point $P$.
\item Find the $x$-coordinate of $P$.\\
\section*{LU}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2010 Q4 [10]}}