| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2020 |
| Session | February |
| Marks | 7 |
| Topic | Laws of Logarithms |
| Type | Solve by showing reduces to polynomial |
| Difficulty | Moderate -0.3 This is a straightforward logarithm manipulation question requiring standard log laws (combining logs, exponentiating) to reach a given quadratic, then factorising and checking domain restrictions. All steps are routine with no novel insight required, making it slightly easier than average but still requiring multiple techniques. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06f Laws of logarithms: addition, subtraction, power rules |
1
\begin{enumerate}[label=(\alph*)]
\item Given that
$$2 \ln ( 3 - x ) - \ln ( 21 - 2 x ) = 0$$
show that
$$x ^ { 2 } - 4 x - 12 = 0$$
[4]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the roots of the equation $x ^ { 2 } - 4 x - 12 = 0$.
\item State which of these two roots is not a solution of
$$2 \ln ( 3 - x ) - \ln ( 21 - 2 x ) = 0$$
giving a reason for your answer.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2020 Q1 [7]}}