| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find second derivative d²y/dx² |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring parametric differentiation for d²y/dx², mean value integration, and centroid calculation. While it involves multiple techniques and careful algebraic manipulation, these are standard FP1 procedures without requiring novel insight. The parametric functions are straightforward, making this moderately above average difficulty but well within typical Further Maths scope. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation4.08e Mean value of function: using integral6.04d Integration: for centre of mass of laminas/solids |
The curve $C$ has parametric equations
$$x = t ^ { 2 } , \quad y = ( 2 - t ) ^ { \frac { 1 } { 2 } } , \quad \text { for } 0 \leqslant t \leqslant 2 .$$
Find\\
(i) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ in terms of $t$,\\
(ii) the mean value of $y$ with respect to $x$ over the interval $0 \leqslant x \leqslant 4$,\\
(iii) the $y$-coordinate of the centroid of the region enclosed by $C$, the $x$-axis and the $y$-axis.
\hfill \mbox{\textit{CAIE FP1 2014 Q12 EITHER}}