| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2003 |
| Session | November |
| Marks | 28 |
| Paper | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Rational function curve sketching |
| Difficulty | Challenging +1.2 This is a Further Maths question with two alternatives. The first involves partial fractions, differentiation, asymptotes, and range analysis—all standard FP1 techniques requiring multiple steps but no novel insight. The second involves arc length and surface of revolution formulas, which are more computationally intensive but still routine applications. Both alternatives are methodical multi-part questions typical of Further Maths, placing them moderately above average difficulty due to the extended working required, but they remain standard textbook exercises without requiring creative problem-solving. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07n Stationary points: find maxima, minima using derivatives4.05c Partial fractions: extended to quadratic denominators4.08d Volumes of revolution: about x and y axes |
Answer only one of the following two alternatives.
\textbf{EITHER}
The curve $C$ has equation $y = \frac{5(x-1)(x+2)}{(x-2)(x+3)}$.
\begin{enumerate}[label=(\roman*)]
\item Express $y$ in the form $P + \frac{Q}{x-2} + \frac{R}{x+3}$. [3]
\item Show that $\frac{dy}{dx} = 0$ for exactly one value of $x$ and find the corresponding value of $y$. [4]
\item Write down the equations of all the asymptotes of $C$. [3]
\item Find the set of values of $k$ for which the line $y = k$ does not intersect $C$. [4]
\end{enumerate}
\textbf{OR}
A curve has equation $y = \frac{5}{3}x^{\frac{3}{2}}$, for $x \geq 0$. The arc of the curve joining the origin to the point where $x = 3$ is denoted by $R$.
\begin{enumerate}[label=(\roman*)]
\item Find the length of $R$. [4]
\item Find the $y$-coordinate of the centroid of the region bounded by the $x$-axis, the line $x = 3$ and $R$. [5]
\item Show that the area of the surface generated when $R$ is rotated through one revolution about the $y$-axis is $\frac{232\pi}{15}$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2003 Q11 [28]}}