| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Session | Specimen |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume requiring substitution or integration by parts |
| Difficulty | Standard +0.3 This is a standard Further Pure question testing routine calculus techniques: implicit differentiation of inverse trig (bookwork), mean value of a function (straightforward integration using arctan), and volume of revolution (standard method but requires integration by parts or substitution for 1/(1+x²)²). All techniques are well-practiced at this level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.08d Volumes of revolution: about x and y axes4.08e Mean value of function: using integral4.08g Derivatives: inverse trig and hyperbolic functions |
| Answer | Marks | Guidance |
|---|---|---|
| 12 | (i) | DR |
| Answer | Marks |
|---|---|
| dx (cid:11) 1(cid:14)tan2y (cid:12) | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| m | Must be seen |
| Answer | Marks | Guidance |
|---|---|---|
| 12 | (ii) | DR |
| Answer | Marks |
|---|---|
| 4 | c |
| Answer | Marks |
|---|---|
| [3] | i |
| Answer | Marks |
|---|---|
| 1.1 | M1 for arctan must be seen |
| Answer | Marks | Guidance |
|---|---|---|
| 12 | (iii) | DR |
| Answer | Marks |
|---|---|
| 4 | M1 |
| Answer | Marks |
|---|---|
| [7] | 1.2 |
| Answer | Marks |
|---|---|
| 1.1 | Must be seen |
Question 12:
12 | (i) | DR
dy
tany(cid:32)x so sec2y (cid:32)1
dx
dy
(cid:11) 1(cid:14)tan2y (cid:12) (cid:32)1
dx
dy
(cid:11) 1(cid:14)x2(cid:12)
(cid:32)1
dx
dy 1
(cid:32) AG
dx (cid:11) 1(cid:14)tan2y (cid:12) | M1
A1
A1
[3] | 1.1
1.2
2.1
m | Must be seen
Evidence of
sec2 y(cid:32)1(cid:14)tan2 y must be
n
seen
Use x(cid:32) tany to obtain
e
given answer
12 | (ii) | DR
e
1
(cid:62)arctanx(cid:64)1
1(cid:16)(cid:11)(cid:16)1(cid:12) (cid:16)1
p
(cid:83)
4 | c
M1
A1
A1
[3] | i
1.1
2.1
1.1 | M1 for arctan must be seen
A0 for a decimal answer
12 | (iii) | DR
1
(cid:180) 1
(cid:83)(cid:181) dx
(cid:11) 1(cid:14)x2(cid:12)2
(cid:182)
(cid:16)1
1
(cid:180) 1
(cid:83)(cid:181) sec2udu
(cid:182) (1(cid:14)tan2u)2
(cid:16)1
(cid:83)
(cid:83)(cid:179)4 cos2u du
(cid:83)
(cid:16)
4
(cid:83)
(cid:83)(cid:179)4 1(cid:11)1 (cid:14) cos2u(cid:12)du
(cid:83)2
(cid:16)
4
1(cid:167) 1 (cid:183)
(cid:168)u(cid:14) sin2u(cid:184) e
2(cid:169) 2 (cid:185)
(cid:83)
(cid:11)(cid:83)(cid:14)2(cid:12)
4 | M1
M1
A1
A1
M1
c
A1
A1
[7] | 1.2
3.1a
2.1
m
1.1
3.1a
i
2.1
1.1 | Must be seen
Substitute x(cid:32) tanu,
n
dx(cid:32) sec2udu
One intermediate step
e
required
Integrand
New limits
Must be seen
Integrate
oe, but A0 for a decimal
answerIn this question you must show detailed reasoning.
\begin{enumerate}[label=(\roman*)]
\item Given that $y = \arctan x$, show that $\frac{dy}{dx} = \frac{1}{1+x^2}$. [3]
\end{enumerate}
Fig. 12 shows the curve $y = \frac{1}{1+x^2}$.
\includegraphics{figure_12}
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find, in exact form, the mean value of the function $f(x) = \frac{1}{1+x^2}$ for $-1 \leq x \leq 1$. [3]
\item The region bounded by the curve, the $x$-axis, and the lines $x = 1$ and $x = -1$ is rotated through $2\pi$ radians about the $x$-axis. Find, in exact form, the volume of the solid of revolution generated. [7]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core Q12 [13]}}