| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Show parametric identity |
| Difficulty | Challenging +1.2 This is a structured multi-part Further Maths question requiring standard parametric calculus techniques (derivatives, arc length, surface of revolution) and coordinate conversions. While it involves several steps and topics, each part follows routine procedures without requiring novel insight—the 'show that' is straightforward differentiation and algebra, and the polar conversion uses given hints. Slightly above average difficulty due to the Further Maths context and multi-step nature, but well within standard FP1 expectations. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta)4.09c Area enclosed: by polar curve8.06b Arc length and surface area: of revolution, cartesian or parametric |
A curve $C$ has parametric equations
$$x = 1 - 3 t ^ { 2 } , \quad y = t \left( 1 - 3 t ^ { 2 } \right) , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }$$
Show that $\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + 9 t ^ { 2 } \right) ^ { 2 }$.
Hence find\\
(i) the arc length of $C$,\\
(ii) the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis.
Use the fact that $t = \frac { y } { x }$ to find a cartesian equation of $C$. Hence show that the polar equation of $C$ is $r = \sec \theta \left( 1 - 3 \tan ^ { 2 } \theta \right)$, and state the domain of $\theta$.
Find the area of the region enclosed between $C$ and the initial line.
{www.cie.org.uk} after the live examination series.
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\hfill \mbox{\textit{CAIE FP1 2016 Q11 OR}}