| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Combined region areas |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring sketching a cubic with given roots, identifying intercepts, and computing area using definite integration. The integration involves expanding a cubic and applying standard techniques with clear limits from the roots. While it requires multiple steps, each component is routine A-level material with no novel problem-solving required. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Cubic curve, correct orientation | B1 | |
| cuts \(x\)-axis twice to left of \(O\) and once to right | B1 | Allow cubic of incorrect orientation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((-3a, 0)\), \((-a, 0)\), \((b, 0)\) all shown correctly | B1 | allow \(-3a\), \(-a\), \(b\) marked on \(x\)-axis |
| \((0, -3a^2b)\) shown correctly | B1 | allow \(-3a^2b\) marked on \(y\)-axis |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_{-3}^{-1}(x+3)(x+1)(x-4)\,dx\) and \(\int_{-1}^{4}(x^3-13x-12)\,dx\) | M1 | State or imply integrating \(f(x)\) with any limits or none. Allow incorrect expansion |
| \(\left[\frac{x^4}{4} - \frac{13x^2}{2} - 12x\right]_{-3}^{-1} = 8\) | A1 | A1 for either answer or for \(\frac{375}{4}\). BC Allow A1 for \(\pm\frac{343}{4}\) |
| \(\left[\frac{x^4}{4} - \frac{13x^2}{2} - 12x\right]_{-1}^{4} = -\frac{375}{4}\) or \(-93.75\) | A1 | |
| \(8 - (-93.75)\) or \(8 + 93.75\) | M1 | Attempt subtract their integrals with correct limits, one +ve, one -ve i.e. \(I_1 - I_2\) or \(I_1 + (-I_2)\) or \(I_1 + |
| \(\frac{407}{4}\) or \(101.75\) or \(102\) (3 sf) | A1 |
## Question 5:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Cubic curve, correct orientation | B1 | |
| cuts $x$-axis twice to left of $O$ and once to right | B1 | Allow cubic of incorrect orientation |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(-3a, 0)$, $(-a, 0)$, $(b, 0)$ all shown correctly | B1 | allow $-3a$, $-a$, $b$ marked on $x$-axis |
| $(0, -3a^2b)$ shown correctly | B1 | allow $-3a^2b$ marked on $y$-axis |
### Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_{-3}^{-1}(x+3)(x+1)(x-4)\,dx$ and $\int_{-1}^{4}(x^3-13x-12)\,dx$ | M1 | State or imply integrating $f(x)$ with any limits or none. Allow incorrect expansion |
| $\left[\frac{x^4}{4} - \frac{13x^2}{2} - 12x\right]_{-3}^{-1} = 8$ | A1 | A1 for either answer or for $\frac{375}{4}$. **BC** Allow A1 for $\pm\frac{343}{4}$ |
| $\left[\frac{x^4}{4} - \frac{13x^2}{2} - 12x\right]_{-1}^{4} = -\frac{375}{4}$ or $-93.75$ | A1 | |
| $8 - (-93.75)$ or $8 + 93.75$ | M1 | Attempt subtract their integrals with correct limits, one +ve, one -ve i.e. $I_1 - I_2$ or $I_1 + (-I_2)$ or $I_1 + |I_2|$, dep $I_2$ being -ve |
| $\frac{407}{4}$ or $101.75$ or $102$ (3 sf) | A1 | |
---5 The function f is defined by $\mathrm { f } ( x ) = ( x + a ) ( x + 3 a ) ( x - b )$ where $a$ and $b$ are positive integers.
\begin{enumerate}[label=(\alph*)]
\item On the axes in the Printed Answer Booklet, sketch the curve $y = \mathrm { f } ( x )$.
\item On your sketch show, in terms of $a$ and $b$, the coordinates of the points where the curve meets the axes.
It is now given that $a = 1$ and $b = 4$.
\item Find the total area enclosed between the curve $y = \mathrm { f } ( x )$ and the $x$-axis.
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q5 [8]}}