| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Integrate after simplifying a quotient |
| Difficulty | Moderate -0.3 This is a straightforward C2 question testing algebraic manipulation, integration of powers, and definite integration. Part (a) requires simple rearrangement, part (b) involves rewriting as separate terms and applying standard power rules, and part (c) is routine evaluation of a definite integral. All techniques are standard with no conceptual challenges, making it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{1-8x^3}{x^2} = 0 \Rightarrow 1 - 8x^3 = 0\) | M1 | |
| \(x^3 = \frac{1}{8}\) | ||
| \(x = \frac{1}{2}\) | M1 A1 | |
| (b) \(f(x) = x^2 - 8x\) | ||
| \(\int f(x) \, dx = \int (x^2 - 8x) \, dx\) | ||
| \(= -x^{-1} - 4x^2 + c\) | M1 A2 | |
| (c) \(= -[-x^{-1} - 4x^2]_1^2\) | M1 | |
| \(= -((-\frac{1}{2} - 16) - (-2 - 1)) = 13\frac{1}{2}\) | M1 A1 | (9 marks) |
**(a)** $\frac{1-8x^3}{x^2} = 0 \Rightarrow 1 - 8x^3 = 0$ | M1 |
$x^3 = \frac{1}{8}$ |
$x = \frac{1}{2}$ | M1 A1 |
**(b)** $f(x) = x^2 - 8x$ |
$\int f(x) \, dx = \int (x^2 - 8x) \, dx$ |
$= -x^{-1} - 4x^2 + c$ | M1 A2 |
**(c)** $= -[-x^{-1} - 4x^2]_1^2$ | M1 |
$= -((-\frac{1}{2} - 16) - (-2 - 1)) = 13\frac{1}{2}$ | M1 A1 | (9 marks)\includegraphics{figure_3}
Figure 3 shows part of the curve $y = \text{f}(x)$ where
$$\text{f}(x) = \frac{1 - 8x^3}{x^2}, \quad x \neq 0.$$
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $\text{f}(x) = 0$. [3]
\item Find $\int \text{f}(x) \, dx$. [3]
\item Find the area of the shaded region bounded by the curve $y = \text{f}(x)$, the $x$-axis and the line $x = 2$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q7 [9]}}