| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Curve-Line Intersection Area |
| Difficulty | Standard +0.3 This is a multi-part question covering standard C3 techniques: differentiation using quotient rule, finding stationary points, area between curves via integration, and understanding transformations. While it has multiple parts (8 marks total), each step follows routine procedures with clear guidance ('show that', 'verify', 'hence'). The transformation insight in part (iv) is slightly clever but heavily scaffolded by part (iii). Overall, this is slightly easier than average due to the structured guidance and standard techniques required. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Area is the same as that found in part (ii) | M1 | award M1 for ± ans to 8(ii) (unless zero) |
| \(4\ln 4 - 7\frac{1}{2}\) | A1 cao | need not justify the change of sign |
**(iv)**
Area is the same as that found in part (ii) | M1 | award M1 for ± ans to 8(ii) (unless zero)
$4\ln 4 - 7\frac{1}{2}$ | A1 cao | need not justify the change of sign
**Total: [2]**
---8 Fig. 8 shows the line $y = 1$ and the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = \frac { ( x - 2 ) ^ { 2 } } { x }$. The curve touches the $x$-axis at $\mathrm { P } ( 2,0 )$ and has another turning point at the point Q .
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{955bebfb-04a3-4cd9-a33e-a8ba4b73e2ba-4_960_1472_450_285}
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\caption{Fig. 8}
\end{center}
\end{figure}
(i) Show that $\mathrm { f } ^ { \prime } ( x ) = 1 - \frac { 4 } { x ^ { 2 } }$, and find $\mathrm { f } ^ { \prime \prime } ( x )$.
Hence find the coordinates of Q and, using $\mathrm { f } ^ { \prime \prime } ( x )$, verify that it is a maximum point.\\
(ii) Verify that the line $y = 1$ meets the curve $y = \mathrm { f } ( x )$ at the points with $x$-coordinates 1 and 4 . Hence find the exact area of the shaded region enclosed by the line and the curve.
The curve $y = \mathrm { f } ( x )$ is now transformed by a translation with vector $\binom { - 1 } { - 1 }$. The resulting curve has equation $y = \mathrm { g } ( x )$.\\
(iii) Show that $\mathrm { g } ( x ) = \frac { x ^ { 2 } - 3 x } { x + 1 }$.\\
(iv) Without further calculation, write down the value of $\int _ { 0 } ^ { 3 } \mathrm {~g} ( x ) \mathrm { d } x$, justifying your answer.
\hfill \mbox{\textit{OCR MEI C3 2015 Q8 [18]}}