| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric integration |
| Type | Parametric surface area of revolution |
| Difficulty | Challenging +1.2 This is a standard Further Maths parametric question requiring differentiation of fractional powers, application of arc length and surface area formulas, and algebraic simplification. While it involves multiple steps and careful algebra with surds, the techniques are routine for FP1 students and the expressions simplify cleanly without requiring novel insight. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian4.08d Volumes of revolution: about x and y axes8.06b Arc length and surface area: of revolution, cartesian or parametric |
| Answer | Marks | Guidance |
|---|---|---|
| \(\dot{x}^2 + \dot{y}^2 = \left(t^{\frac{3}{2}} - t^{-\frac{1}{2}}\right)^2 + 4t = \ldots = \left(t^{\frac{3}{2}} + t^{-\frac{1}{2}}\right)^2\) | M1A1 | SOI |
| \(s = \int_1^4 \left(t^{\frac{3}{2}} + t^{-\frac{1}{2}}\right)dt\) | M1 | |
| \(= \left[\frac{2}{5}t^{\frac{5}{2}} + 2t^{\frac{1}{2}}\right]_1^4\) | M1 | |
| \(= \left[\frac{64}{5} + 4\right] - \left[\frac{2}{5} + 2\right] = \frac{72}{5}\) | A1 |
| Answer | Marks |
|---|---|
| \(S = 2\pi\int_1^4 \frac{4}{3}t^{\frac{3}{2}}\left(t^{\frac{3}{2}} + t^{-\frac{1}{2}}\right)dt = \frac{8}{3}\pi\int_1^4(t^3 + t)dt\) | *M1 |
| \(= \left(\frac{8}{3}\pi\right)\left[\frac{1}{4}t^4 + \frac{1}{2}t^2\right]_1^4\) | DM1 |
| \(= \frac{8}{3}\pi\left\{[64+8] - \left[\frac{1}{4}+\frac{1}{2}\right]\right\} = 190\pi\) | A1 |
## Question 5(i):
| $\dot{x}^2 + \dot{y}^2 = \left(t^{\frac{3}{2}} - t^{-\frac{1}{2}}\right)^2 + 4t = \ldots = \left(t^{\frac{3}{2}} + t^{-\frac{1}{2}}\right)^2$ | M1A1 | SOI |
|---|---|---|
| $s = \int_1^4 \left(t^{\frac{3}{2}} + t^{-\frac{1}{2}}\right)dt$ | M1 | |
| $= \left[\frac{2}{5}t^{\frac{5}{2}} + 2t^{\frac{1}{2}}\right]_1^4$ | M1 | |
| $= \left[\frac{64}{5} + 4\right] - \left[\frac{2}{5} + 2\right] = \frac{72}{5}$ | A1 | |
**Total: 5**
---
## Question 5(ii):
| $S = 2\pi\int_1^4 \frac{4}{3}t^{\frac{3}{2}}\left(t^{\frac{3}{2}} + t^{-\frac{1}{2}}\right)dt = \frac{8}{3}\pi\int_1^4(t^3 + t)dt$ | *M1 | |
|---|---|---|
| $= \left(\frac{8}{3}\pi\right)\left[\frac{1}{4}t^4 + \frac{1}{2}t^2\right]_1^4$ | DM1 | |
| $= \frac{8}{3}\pi\left\{[64+8] - \left[\frac{1}{4}+\frac{1}{2}\right]\right\} = 190\pi$ | A1 | |
**Total: 3**
---5 A curve $C$ has parametric equations
$$x = \frac { 2 } { 5 } t ^ { \frac { 5 } { 2 } } - 2 t ^ { \frac { 1 } { 2 } } , \quad y = \frac { 4 } { 3 } t ^ { \frac { 3 } { 2 } } , \quad \text { for } 1 \leqslant t \leqslant 4$$
(i) Find the exact value of the arc length of $C$.\\
(ii) Find also the exact value of the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q5 [8]}}