| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | June |
| Marks | 13 |
| Topic | Parametric integration |
| Difficulty | Challenging +1.2 This is a multi-part parametric curves question requiring: finding a point coordinate (routine), finding a tangent equation using dy/dx = (dy/dt)/(dx/dt) (standard A-level technique), and computing a volume of revolution about the y-axis (standard Further Maths topic). The volume calculation involves subtracting cone volume from curve volume, requiring careful setup but following standard methods. The 13 total marks and 'show detailed reasoning' indicate substantial work, but all techniques are standard Further Maths content without requiring novel insight. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation4.08d Volumes of revolution: about x and y axes |
In this question you must show detailed reasoning.
\includegraphics{figure_12}
The curve $C$ has parametric equations
$$x = \frac{1}{\sqrt{2 + t}}, \quad y = \ln(1 + t), \quad 2 \leq t < \infty$$
The point $P$ on curve $C$ has $x$-coordinate $\frac{1}{2}$.
\begin{enumerate}[label=(\alph*)]
\item Find the exact $y$-coordinate of $P$.
[1]
\end{enumerate}
The tangent to $C$ at $P$ meets the $y$-axis at point $Y$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Determine the exact coordinates of $Y$.
[4]
\end{enumerate}
The curve $C$ and the line segment $PY$ are rotated $2\pi$ radians about the $y$-axis.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Determine the exact volume of the solid generated.
Give your answer in the form $\pi(\ln p + q)$, where $p$ and $q$ are rational numbers.
[8]
\end{enumerate}
[You are given that the volume of a cone with radius $r$ and height $h$ is $\frac{1}{3}\pi r^2 h$]
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q12 [13]}}