| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Surface area of revolution: parametric curve |
| Difficulty | Challenging +1.3 This is a Further Maths question on surface of revolution with parametric equations. Part (i) requires applying the formula S = 2π∫y√((dx/dt)² + (dy/dt)²)dt and simplifying using trigonometric identities to reach the given form—this is moderately technical but follows a standard procedure. Part (ii) uses a provided identity to evaluate the integral, which is straightforward once the substitution is recognized. The question requires solid technique and careful algebra but no novel insight, making it moderately above average difficulty for A-level. |
| Spec | 1.08d Evaluate definite integrals: between limits4.08d Volumes of revolution: about x and y axes |
| Answer | Marks |
|---|---|
| 4(i) | dx |
| Answer | Marks | Guidance |
|---|---|---|
| dt | B1 | dx dy |
| Answer | Marks | Guidance |
|---|---|---|
| dt | M1 | Uses an appropriate identity. |
| Answer | Marks |
|---|---|
| dt dt | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 0 | M1 A1 | Uses the correct formula for S |
| Answer | Marks |
|---|---|
| 4(ii) | π π |
| Answer | Marks | Guidance |
|---|---|---|
| 0 0 | M1 | Uses the result 4sin3t=3sint−sin3t |
| Answer | Marks |
|---|---|
| 0 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 3 3 | A1 | Answer must be exact. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 4:
--- 4(i) ---
4(i) | dx
=1−cos2t
dt
dy
=2sintcost
dt | B1 | dx dy
B1 for and .
dt dt
dx
=1−cos2t =2sin2t
dt | M1 | Uses an appropriate identity.
2 2
dx dy
+ = 4sin4t+4sin2tcos2t =2sint
dt dt | A1
π( )
S =2π∫ sin2t ( 2sint ) dt ⇒ a=4
0 | M1 A1 | Uses the correct formula for S
5
--- 4(ii) ---
4(ii) | π π
S =4π∫sin3t dt =π∫3sint−sin3t dt
0 0 | M1 | Uses the result 4sin3t=3sint−sin3t
π
1
=π −3cost+ cos3t
3
0 | A1
1 1 16π
=π3− −−3+ =
3 3 3 | A1 | Answer must be exact.
3
Question | Answer | Marks | GuidanceA curve is defined parametrically by
$$x = t - \frac{1}{2}\sin 2t \quad \text{and} \quad y = \sin^2 t.$$
The arc of the curve joining the point where $t = 0$ to the point where $t = \pi$ is rotated through one complete revolution about the $x$-axis. The area of the surface generated is denoted by $S$.
\begin{enumerate}[label=(\roman*)]
\item Show that
$$S = a\pi \int_0^\pi \sin^3 t \, dt,$$
where the constant $a$ is to be found. [5]
\item Using the result $\sin 3t = 3\sin t - 4\sin^3 t$, find the exact value of $S$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2018 Q4 [8]}}