| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2026 |
| Session | November |
| Marks | 6 |
| Topic | Laws of Logarithms |
| Type | Two unrelated log parts: one non-log algebraic part |
| Difficulty | Standard +0.3 This is a system of exponential equations requiring substitution and solving a quadratic in e^y. Part (a) is straightforward algebraic manipulation (1 mark). Part (b) requires solving the quadratic, rejecting negative solutions, and finding both coordinates using logarithms - standard Further Maths techniques with no novel insight required. Slightly easier than average due to the guided structure and routine algebraic methods. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06g Equations with exponentials: solve a^x = b |
\begin{enumerate}[label=(\alph*)]
\item The curves $e^x - 2e^y = 1$ and $2e^x + 3e^{2y} = 41$ intersect at the point $P$.
Show that the $y$-coordinate of $P$ satisfies the equation $3e^{2y} + 4e^y - 39 = 0$.
[1]
\item In this question you must show detailed reasoning.
Hence find the exact coordinates of $P$.
[5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2026 Q4 [6]}}