| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area under polynomial curve |
| Difficulty | Moderate -0.3 This is a straightforward C2 question testing basic coordinate geometry, differentiation for tangent equations, and integration with respect to y. All parts are routine applications of standard techniques with no problem-solving insight required. The multi-part structure and 12 marks elevate it slightly above trivial, but it remains easier than average for A-level. |
| Spec | 1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks |
|---|---|
| \(A: y = 1\) | B1 |
| \(B: y = 4\) | |
| (1 mark) |
| Answer | Marks |
|---|---|
| \(\frac{dy}{dx} = \frac{2x}{25} = \frac{2}{5}\) where \(x = 5\) | M1 A1 |
| Tangent: \(y - 1 = \frac{2}{5}(x - 5)\) (or \(5y = 2x - 5\)) | M1 A1 |
| (4 marks) |
| Answer | Marks |
|---|---|
| \(x = 5y^{\frac{1}{2}}\) | B1 B1 |
| (2 marks) |
| Answer | Marks |
|---|---|
| Integrate: \(\frac{5y^{\frac{3}{2}}}{\frac{3}{2}} = \frac{10y^{\frac{3}{2}}}{3}\) | M1 A1ft |
| \(\left[y^4 - 1\right]_1 = \left(\frac{10 \times 4^{\frac{3}{2}}}{3}\right) - \left(\frac{10 \times 1^{\frac{3}{2}}}{3}\right) = \frac{70}{3}\) | M1 A1, A1 |
| \((23\frac{1}{3}, 23.3)\) | |
| (5 marks) | |
| (12 marks total) |
## (a)
$A: y = 1$ | B1 |
$B: y = 4$ | |
| (1 mark) |
## (b)
$\frac{dy}{dx} = \frac{2x}{25} = \frac{2}{5}$ where $x = 5$ | M1 A1 |
Tangent: $y - 1 = \frac{2}{5}(x - 5)$ (or $5y = 2x - 5$) | M1 A1 |
| (4 marks) |
## (c)
$x = 5y^{\frac{1}{2}}$ | B1 B1 |
| (2 marks) |
## (d)
Integrate: $\frac{5y^{\frac{3}{2}}}{\frac{3}{2}} = \frac{10y^{\frac{3}{2}}}{3}$ | M1 A1ft |
$\left[y^4 - 1\right]_1 = \left(\frac{10 \times 4^{\frac{3}{2}}}{3}\right) - \left(\frac{10 \times 1^{\frac{3}{2}}}{3}\right) = \frac{70}{3}$ | M1 A1, A1 |
$(23\frac{1}{3}, 23.3)$ | |
| (5 marks) |
| **(12 marks total)** |
---\includegraphics{figure_1}
The curve $C$, shown in Fig. 1, represents the graph of $y = \frac{x^2}{25}$, $x \geq 0$.
The points $A$ and $B$ on the curve $C$ have $x$-coordinates $5$ and $10$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Write down the $y$-coordinates of $A$ and $B$. [1]
\item Find an equation of the tangent to $C$ at $A$. [4]
\end{enumerate}
The finite region $R$ is enclosed by $C$, the $y$-axis and the lines through $A$ and $B$ parallel to the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item For points $(x, y)$ on $C$, express $x$ in terms of $y$. [2]
\item Use integration to find the area of $R$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q8 [12]}}