| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | October |
| Marks | 8 |
| Topic | Indices and Surds |
| Type | Solve exponential equations |
| Difficulty | Moderate -0.8 This question tests routine algebraic manipulation skills. Part (a)(i) involves standard rationalization of surds with common denominators, part (a)(ii) is a straightforward linear equation after simplification, and part (b) is a standard exponential equation solved by substitution (let y = 2^x). All techniques are textbook exercises requiring recall and careful algebra but no problem-solving insight or novel approaches. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.06g Equations with exponentials: solve a^x = b |
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}}$ can be written in the form $\frac{a}{b+cx}$, where $a$, $b$ and $c$ are constants to be determined. [2]
\item Hence solve the equation $\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}} = 2$. [2]
\end{enumerate}
\item In this question you must show detailed reasoning.
Solve the equation $2^{2x} - 7 \times 2^x - 8 = 0$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q1 [8]}}