| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2011 |
| Session | June |
| Marks | 6 |
| Topic | Laws of Logarithms |
| Type | Two unrelated log/algebra parts - linked parts (hence) |
| Difficulty | Moderate -0.8 This is a straightforward logarithm manipulation question requiring only standard log laws (subtraction, multiplication, and power rules) followed by solving a simple equation by exponentiating. Part (i) is pure algebraic manipulation with no problem-solving, and part (ii) requires only one additional step (setting the argument equal to 1). Easier than average A-level content. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\ln x^2 - \ln(3x - 2) - \ln x^2\) | M1, M1 | Use power law at least once. Use division or multiplication law at least once |
| \(\ln \frac{x^2}{3x - 2}\) | A1 | AG so NIS |
| (ii) \(\frac{x^2}{3x - 2} = 1\) | B1 | Use \(e^0 = 1\) or state \(x^2 = 3x - 2\) from \(\ln x^2 = \ln(3x - 2)\) |
| \(x^2 - 3x + 2 = 0\) | M1 | |
| \(x = 2\) or \(x = 1\) | A1 | [6] |
(i) $\ln x^2 - \ln(3x - 2) - \ln x^2$ | M1, M1 | Use power law at least once. Use division or multiplication law at least once
$\ln \frac{x^2}{3x - 2}$ | A1 | AG so NIS
(ii) $\frac{x^2}{3x - 2} = 1$ | B1 | Use $e^0 = 1$ or state $x^2 = 3x - 2$ from $\ln x^2 = \ln(3x - 2)$
$x^2 - 3x + 2 = 0$ | M1 |
$x = 2$ or $x = 1$ | A1 | [6] | Attempt soln of 3 term quadratic. Obtain 2 and 1\begin{enumerate}[label=(\roman*)]
\item Show that $4 \ln x - \ln(3x - 2) - \ln x^2 = \ln\left(\frac{x^2}{3x - 2}\right)$, where $x > \frac{2}{3}$. [3]
\item Hence solve the equation $4 \ln x - \ln(3x - 2) - \ln x^2 = 0$. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2011 Q4 [6]}}