| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Topic | Laws of Logarithms |
| Type | Two unrelated log parts: one non-log algebraic part |
| Difficulty | Standard +0.3 Part (a) requires straightforward substitution and solving an exponential equation using index laws. Part (b) involves equating two exponential expressions, taking logarithms, and algebraic manipulation to reach the given form. While part (b) requires multiple steps and careful logarithm work, these are standard A-level techniques without requiring novel insight. The 'show that' format provides a target to work towards, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06g 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
$$y = 3 \times 2^{2x}$$
The point $P\left(a, 96\sqrt{2}\right)$ lies on the curve.
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $a$. [3]
\end{enumerate}
The curve with equation $y = 3 \times 2^{2x}$ 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
$$\frac{3 + 2\log_2 3}{3 + \log_2 3}$$ [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2020 Q12 [8]}}