| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2022 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Parametric curve crosses axis, find gradient there |
| Difficulty | Standard +0.3 This is a straightforward parametric differentiation question requiring standard techniques: finding where y=0, computing dy/dx using the chain rule, and determining domain/range from given parameter bounds. All steps are routine for P4 level with no novel problem-solving required, making it slightly easier than average. |
| 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 |
\includegraphics{figure_3}
Figure 3 shows a sketch of the curve $C$ with parametric equations
$$x = 1 + 3\tan t, \quad y = 2\cos 2t, \quad -\frac{\pi}{6} \leq t \leq \frac{\pi}{3}$$
The curve crosses the $x$-axis at point $P$, as shown in Figure 3.
\begin{enumerate}[label=(\alph*)]
\item Find the equation of the tangent to $C$ at $P$, writing your answer in the form $y = mx + c$, where $m$ and $c$ are constants to be found. [5]
\end{enumerate}
The curve $C$ has equation $y = f(x)$, where $f$ is a function with domain $\left[k, 1 + 3\sqrt{3}\right]$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the exact value of the constant $k$. [1]
\item Find the range of $f$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel P4 2022 Q6 [8]}}