| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2020 |
| Session | October |
| Marks | 5 |
| Topic | Exponential Functions |
| Type | Solve exponential equation by substitution |
| Difficulty | Moderate -0.8 Part (i) requires spotting two algebraic errors in worked solutions (misapplying index laws: 2^{2x+4} ≠ 2^{2x} + 2^4, and an arithmetic error in the quadratic constant term). Part (ii) is a standard exponential equation solved by substitution, leading to a straightforward quadratic. Both parts test routine A-level techniques with minimal problem-solving demand, making this easier than average. |
| Spec | 1.06g Equations with exponentials: solve a^x = b |
\begin{enumerate}[label=(\roman*)]
\item A student was asked to solve the equation $2^{2x+4} - 9(2^x) = 0$.
The student's attempt is written out below.
$$2^{2x+4} - 9(2^x) = 0$$
$$2^{2x} + 2^4 - 9(2^x) = 0$$
$$\text{Let } y = 2^x$$
$$y^2 - 9y + 8 = 0$$
$$(y - 8)(y - 1) = 0$$
$$y = 8 \text{ or } y = 1$$
$$\text{So } x = 3 \text{ or } x = 0$$
Identify the two mistakes that the student has made. [2]
\item Solve the equation $2^{2x+4} - 9(2^x) = 0$, giving your answer in exact form. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2020 Q6 [5]}}