| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area involving fractional powers |
| Difficulty | Moderate -0.3 This is a straightforward C2 integration question requiring finding an x-intercept by solving a simple equation, then integrating a power function with fractional exponent. Both parts are standard textbook exercises with no problem-solving insight needed, making it slightly easier than average but not trivial due to the fractional power manipulation. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(4x^3 - x = 0\) | ||
| \(x^{\frac{1}{4}}(4 - x^{\frac{3}{4}}) = 0\) | M1 | |
| \(x^{\frac{1}{4}} = 0\) (at \(O\)) or \(x^{\frac{3}{4}} = 4\) | M1 | |
| \(x \geq 0 \therefore x = (\sqrt[4]{4})^3 = 8, a = 8\) | A1 | |
| (b) \(= \int_0^8 (4x^{\frac{1}{4}} - x) \, dx\) | M1 A2 | |
| \(= [3x^{\frac{4}{3}} - \frac{1}{2}x^2]_0^8\) | M1 A2 | |
| \(= (48 - 32) - (0) = 16\) | M1 A1 | (8 marks) |
(a) $4x^3 - x = 0$ | |
$x^{\frac{1}{4}}(4 - x^{\frac{3}{4}}) = 0$ | M1 |
$x^{\frac{1}{4}} = 0$ (at $O$) or $x^{\frac{3}{4}} = 4$ | M1 |
$x \geq 0 \therefore x = (\sqrt[4]{4})^3 = 8, a = 8$ | A1 |
(b) $= \int_0^8 (4x^{\frac{1}{4}} - x) \, dx$ | M1 A2 |
$= [3x^{\frac{4}{3}} - \frac{1}{2}x^2]_0^8$ | M1 A2 |
$= (48 - 32) - (0) = 16$ | M1 A1 | (8 marks)5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{824da843-3ea1-4a83-9170-d0082b2e8d1c-3_458_862_906_511}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows the curve with equation $y = 4 x ^ { \frac { 1 } { 3 } } - x , x \geq 0$.\\
The curve meets the $x$-axis at the origin and at the point $A$ with coordinates $( a , 0 )$.
\begin{enumerate}[label=(\alph*)]
\item Show that $a = 8$.
\item Find the area of the finite region bounded by the curve and the positive $x$-axis.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q5 [8]}}