| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area under polynomial curve |
| Difficulty | Moderate -0.8 This is a straightforward C2 question with routine tasks: differentiation to show monotonicity (standard result f'(x) < 0), simple substitution to verify a point, and direct integration with given limits. All three parts are textbook exercises requiring only basic technique with no problem-solving or insight needed. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07o Increasing/decreasing: functions using sign of dy/dx1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f'(x) = -1 - 3x^2\) | M1 A1 | |
| \(x^2 \geq 0\) for all real \(x \Rightarrow -1 - 3x^2 \leq -1\) | M1 | |
| \(\therefore f'(x) < 0 \Rightarrow f(x)\) is decreasing for all values of \(x\) | A1 | |
| (b) \(f(1) = 2 - 1 - 1 = 0 \quad \therefore (1,0)\) on curve | B1 | |
| (c) \(= \int_0^1 (2 - x - x^3) \, dx\) | ||
| \(= [2x - \frac{1}{2}x^2 - \frac{1}{4}x^4]_0^1\) | M1 A1 | |
| \(= \left(2 - \frac{1}{2} - \frac{1}{4}\right) - (0) = \frac{5}{4}\) | M1 A1 | (9) |
**(a)** $f'(x) = -1 - 3x^2$ | M1 A1 |
$x^2 \geq 0$ for all real $x \Rightarrow -1 - 3x^2 \leq -1$ | M1 |
$\therefore f'(x) < 0 \Rightarrow f(x)$ is decreasing for all values of $x$ | A1 |
**(b)** $f(1) = 2 - 1 - 1 = 0 \quad \therefore (1,0)$ on curve | B1 |
**(c)** $= \int_0^1 (2 - x - x^3) \, dx$ | |
$= [2x - \frac{1}{2}x^2 - \frac{1}{4}x^4]_0^1$ | M1 A1 |
$= \left(2 - \frac{1}{2} - \frac{1}{4}\right) - (0) = \frac{5}{4}$ | M1 A1 | **(9)**4.
$$f ( x ) = 2 - x - x ^ { 3 }$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { f } ( x )$ is decreasing for all values of $x$.
\item Verify that the point $( 1,0 )$ lies on the curve $y = \mathrm { f } ( x )$.
\item Find the area of the region bounded by the curve $y = \mathrm { f } ( x )$ and the coordinate axes.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q4 [9]}}