| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | October |
| Marks | 6 |
| Topic | Proof |
| Type | Identifying errors in proofs |
| Difficulty | Moderate -0.3 This question requires identifying algebraic errors (incorrectly expanding a square and wrong final calculation) and solving a quadratic in log₃x using substitution or factorization. While it tests understanding of logarithms and quadratics, both parts are relatively straightforward applications of standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
6. A student was asked to solve the equation $2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0$. The student's attempt is written out below.
$$\begin{aligned}
& 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0 \\
& 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0 \\
& \log _ { 3 } x - 2 = 0 \\
& \log _ { 3 } x = 2 \\
& x = 8
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Identify the two mistakes that the student has made.
\item Solve the equation $2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0$, giving your answers in an exact form.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q6 [6]}}