| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Topic | Exponential Functions |
| Type | Solve exponential equation by substitution |
| Difficulty | Moderate -0.3 This is a standard exponential equation requiring substitution (explicitly given), simplification using index laws, solving a quadratic, then back-substitution. While it requires multiple steps and careful algebraic manipulation, the method is routine and the substitution is provided. Slightly easier than average due to the guided approach. |
| Spec | 1.06g Equations with exponentials: solve a^x = b |
3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
\begin{enumerate}[label=(\alph*)]
\item By substituting $p = 3 ^ { x }$, show that the equation
$$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$
can be rewritten in the form
$$9 p ^ { 2 } + 26 p - 3 = 0$$
\item Hence solve
$$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q3 [5]}}