| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Topic | Laws of Logarithms |
| Type | Two unrelated log parts: one non-log algebraic part |
| Difficulty | Standard +0.8 Part (a) requires solving 3a·2^(2a) = 96√2 by recognizing powers of 2, which is moderately challenging. Part (b) involves equating two exponential expressions, applying logarithm laws strategically, and algebraic manipulation to reach a specific form—requiring multiple non-routine steps and careful handling of logarithms. This is above average difficulty but not exceptionally hard for A-level. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
\includegraphics{figure_6}
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
Figure 6 shows a sketch of part of the curve with equation
$$y = 3x \cdot 2^{2x}.$$
The point $P(a, 96\sqrt{2})$ lies on the curve.
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $a$. [3]
\end{enumerate}
The curve with equation $y = 3x \cdot 2^{2x}$ meets the curve with equation $y = 6^{3-x}$ at the point $Q$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the $x$ coordinate of $Q$ is $\frac{3 + 2\log_2 3}{3 + \log_2 3}$. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2022 Q16 [7]}}