| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2023 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric integration |
| Type | Parametric normal then bounded area |
| Difficulty | Challenging +1.2 This is a standard parametric calculus question requiring finding a normal equation (routine differentiation of parametric equations) and calculating area under a parametric curve. While it involves multiple steps and careful algebra with trigonometric identities, the techniques are well-practiced at A-level Further Maths. The area calculation requires subtracting the triangle area from the parametric integral, which adds modest complexity but follows standard patterns. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| At \(t = \frac{\pi}{4}\), \(P = \left(\frac{1}{2}, 2\right)\) | B1 | Correct coordinates for \(P\) stated or implied by working |
| \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2\sec^2 t}{2\sin t \cos t} = 4\) when \(t = \frac{\pi}{4}\) | M1 A1 | M1: Attempts \(\frac{dy/dt}{dx/dt}\) at \(t=\frac{\pi}{4}\); condone poor differentiation. A1: Correct \(\frac{dy}{dx}=4\) following correct differentiation |
| Equation of \(l\): \(y - 2 = -\frac{1}{4}\left(x - \frac{1}{2}\right) \Rightarrow 8y - 16 = -2x + 1 \Rightarrow 8y + 2x = 17\)* | dM1 A1* cso | dM1: Attempts normal equation at \(t=\frac{\pi}{4}\), dependent on previous M. A1*: cso, correct proof leading to \(8y+2x=17\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\int y\frac{dx}{dt}\,dt = \int 2\tan t \times 2\sin t \cos t\, dt\) | M1 | Attempts \(\int y\frac{dx}{dt}\,dt = \int 2\tan t \times 2\sin t \cos t\, dt\) with correct \(\frac{dx}{dt}\), condoning coefficient slips |
| \(= \int 4\sin^2 t\, dt\) | A1 | Correct simplified form |
| \(= \int 2 - 2\cos 2t\, dt = 2t - \sin 2t\) | dM1 A1 | dM1: Uses \(\cos 2t = \pm 1 \pm 2\sin^2 t\) and integrates to form \(\pm at \pm b\sin 2t\) |
| Total area \(= \left[2t - \sin 2t\right]_0^{\pi/4} + \frac{1}{2}\times 8 \times 2 = \frac{\pi}{2} - 1 + 8 = \frac{\pi}{2} + 7\) | M1 A1 | M1: Full method finding sum of parametric integral and triangle area with correct limits. A1: \(\frac{\pi}{2}+7\) |
## Question 8(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| At $t = \frac{\pi}{4}$, $P = \left(\frac{1}{2}, 2\right)$ | **B1** | Correct coordinates for $P$ stated or implied by working |
| $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2\sec^2 t}{2\sin t \cos t} = 4$ when $t = \frac{\pi}{4}$ | **M1 A1** | M1: Attempts $\frac{dy/dt}{dx/dt}$ at $t=\frac{\pi}{4}$; condone poor differentiation. A1: Correct $\frac{dy}{dx}=4$ following correct differentiation |
| Equation of $l$: $y - 2 = -\frac{1}{4}\left(x - \frac{1}{2}\right) \Rightarrow 8y - 16 = -2x + 1 \Rightarrow 8y + 2x = 17$* | **dM1 A1* cso** | dM1: Attempts normal equation at $t=\frac{\pi}{4}$, dependent on previous M. A1*: cso, correct proof leading to $8y+2x=17$ |
## Question 8(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\int y\frac{dx}{dt}\,dt = \int 2\tan t \times 2\sin t \cos t\, dt$ | **M1** | Attempts $\int y\frac{dx}{dt}\,dt = \int 2\tan t \times 2\sin t \cos t\, dt$ with correct $\frac{dx}{dt}$, condoning coefficient slips |
| $= \int 4\sin^2 t\, dt$ | **A1** | Correct simplified form |
| $= \int 2 - 2\cos 2t\, dt = 2t - \sin 2t$ | **dM1 A1** | dM1: Uses $\cos 2t = \pm 1 \pm 2\sin^2 t$ and integrates to form $\pm at \pm b\sin 2t$ |
| Total area $= \left[2t - \sin 2t\right]_0^{\pi/4} + \frac{1}{2}\times 8 \times 2 = \frac{\pi}{2} - 1 + 8 = \frac{\pi}{2} + 7$ | **M1 A1** | M1: Full method finding sum of parametric integral and triangle area with correct limits. A1: $\frac{\pi}{2}+7$ |
---8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c46ca445-cf59-4664-931e-add9f2f81851-26_582_773_255_648}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
\section*{In this question you must show all stages of your working.}
\section*{Solutions relying entirely on calculator technology are not acceptable.}
A curve $C$ has parametric equations
$$x = \sin ^ { 2 } t \quad y = 2 \tan t \quad 0 \leqslant t < \frac { \pi } { 2 }$$
The point $P$ with parameter $t = \frac { \pi } { 4 }$ lies on $C$.\\
The line $l$ is the normal to $C$ at $P$, as shown in Figure 3.
\begin{enumerate}[label=(\alph*)]
\item Show, using calculus, that an equation for $l$ is
$$8 y + 2 x = 17$$
The region $S$, shown shaded in Figure 3, is bounded by $C , l$ and the $x$-axis.
\item Find, using calculus, the exact area of $S$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P4 2023 Q8 [11]}}