| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric integration |
| Type | Parametric area under curve |
| Difficulty | Standard +0.3 This is a straightforward parametric area question requiring standard techniques: finding axis intercepts by setting x=0 and y=0, then applying the parametric area formula ∫y(dx/dt)dt with clear limits. The algebra is routine (quadratic factorization, polynomial integration) with no conceptual surprises, making it slightly easier than average for C4. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(x = 0 \Rightarrow t^2 = 2, \quad t \geq 0 \Rightarrow t = \sqrt{2} \quad \therefore (0, 2 + \sqrt{2})\) | M1 A1 | |
| \(y = 0 \Rightarrow t(t+1) = 0, \quad t \geq 0 \Rightarrow t = 0 \quad \therefore (2, 0)\) | M1 A1 | |
| (b) \(\frac{dx}{dt} = -2t\) | M1 | |
| \(\text{area} = \int_0^{\sqrt{2}} t(t+1) \times (-2t) dt\) | A1 | |
| \(= \int_0^{\sqrt{2}} (2t^3 + 2t) dt\) | A1 | |
| \(= \left[\frac{1}{2}t^4 + t^2\right]_0^{\sqrt{2}}\) | M1 A1 | |
| \(= (2 + \frac{4}{3}\sqrt{2}) - (0) = 2 + \frac{4}{3}\sqrt{2}\) | M1 A1 | (10) |
**(a)** $x = 0 \Rightarrow t^2 = 2, \quad t \geq 0 \Rightarrow t = \sqrt{2} \quad \therefore (0, 2 + \sqrt{2})$ | M1 A1 |
$y = 0 \Rightarrow t(t+1) = 0, \quad t \geq 0 \Rightarrow t = 0 \quad \therefore (2, 0)$ | M1 A1 |
**(b)** $\frac{dx}{dt} = -2t$ | M1 |
$\text{area} = \int_0^{\sqrt{2}} t(t+1) \times (-2t) dt$ | A1 |
$= \int_0^{\sqrt{2}} (2t^3 + 2t) dt$ | A1 |
$= \left[\frac{1}{2}t^4 + t^2\right]_0^{\sqrt{2}}$ | M1 A1 |
$= (2 + \frac{4}{3}\sqrt{2}) - (0) = 2 + \frac{4}{3}\sqrt{2}$ | M1 A1 | (10)6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{922f404e-12d5-490b-9c8d-509f3a304c1e-10_438_700_255_518}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the curve with parametric equations
$$x = 2 - t ^ { 2 } , \quad y = t ( t + 1 ) , \quad t \geq 0$$
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the points where the curve meets the coordinate axes.
\item Find the exact area of the region bounded by the curve and the coordinate axes.\\
6. continued
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q6 [10]}}