| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area between curve and line |
| Difficulty | Standard +0.3 This is a standard C2 integration question with routine steps: verifying a root by substitution, finding a tangent equation using differentiation (including differentiating x^{-1/2}), and calculating an area between a curve and line. While it requires multiple techniques across 12 marks, each step follows textbook procedures without requiring problem-solving insight or novel approaches, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | 7.2 +C1 | 5 |
| Totals | 30 | 29 |
Question 10:
10 | 7.2 +C1 | 5 | 5 | 2 | 12
Totals | 30 | 29 | 5 | 5 | 6 | 75\includegraphics{figure_2}
Figure 2 shows part of the curve $C$ with equation
$$y = 9 - 2x - \frac{2}{\sqrt{x}}, \quad x > 0.$$
The point $A(1, 5)$ lies on $C$ and the curve crosses the $x$-axis at $B(b, 0)$, where $b$ is a constant and $b > 0$.
\begin{enumerate}[label=(\alph*)]
\item Verify that $b = 4$. [1]
\end{enumerate}
The tangent to $C$ at the point $A$ cuts the $x$-axis at the point $D$, as shown in Fig. 2.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that an equation of the tangent to $C$ at $A$ is $y + x = 6$. [4]
\item Find the coordinates of the point $D$. [1]
\end{enumerate}
The shaded region $R$, shown in Fig. 2, is bounded by $C$, the line $AD$ and the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Use integration to find the area of $R$. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q10 [12]}}