| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Surface area of revolution: Cartesian curve |
| Difficulty | Challenging +1.2 Part (a) requires applying the surface area of revolution formula and simplifying using differentiation and algebra—standard technique for Further Maths. Part (b) involves numerical integration and percentage error analysis, which is routine application. The question is methodical rather than conceptually challenging, though it's above average difficulty due to being Further Maths content with multi-step working. |
| Spec | 8.06b Arc length and surface area: of revolution, cartesian or parametric |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | 1 + d y 2 = 1 + ( 12 (1 + x 2 ) − 12 .2 x )2 |
| Answer | Marks |
|---|---|
| = 2 π 1 + 2 x 2 d x | M1 |
| Answer | Marks |
|---|---|
| [4] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | d y |
| Answer | Marks |
|---|---|
| (b) | Finding a numerical answer for A (BC) and two of |
| Answer | Marks |
|---|---|
| (yes) | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.1b |
| 1.1 | Note: A = 7.987 649 … |
Question 2:
2 | (a) | 1 + d y 2 = 1 + ( 12 (1 + x 2 ) − 12 .2 x )2
d x
x 2 1 + 2 x 2
= 1 + or
1 + x 2 1 + x 2
1 1+2x2
A = 2π 1+ x2 dx
1+ x2
0
10 ( )
= 2 π 1 + 2 x 2 d x | M1
A1
M1
A1
[4] | 1.1
1.1
1.1
1.1 | d y
Including attempt at using the Chain Rule
d x
Use of SA formula with their appropriate terms
i.e. k = 2
(b) | Finding a numerical answer for A (BC) and two of
{7.92, awrt 7.95, awrt 8.03, 8.08}
Comparison of correct numbers and correct conclusion
(yes) | M1
A1
[2] | 3.1b
1.1 | Note: A = 7.987 649 …
{8⋅0.99,𝐴⋅0.995,𝐴⋅1.005,8⋅1.01}
CAO from fully correct working2 A curve has equation $\mathrm { y } = \sqrt { 1 + \mathrm { x } ^ { 2 } }$, for $0 \leqslant x \leqslant 1$, where both the $x$ - and $y$-units are in cm. The area of the surface generated when this curve is rotated fully about the $x$-axis is $A \mathrm {~cm} ^ { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { A } = 2 \pi \int _ { 0 } ^ { 1 } \sqrt { 1 + \mathrm { kx } ^ { 2 } } \mathrm { dx }$ for some integer $k$ to be determined.
A small component for a car is produced in the shape of this surface. The curved surface area of the component must be $8 \mathrm {~cm} ^ { 2 }$, accurate to within one percent. The engineering process produces such components with a curved surface area accurate to within one half of one percent.
\item Determine whether all components produced will be suitable for use in the car.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2023 Q2 [6]}}