| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve using substitution or auxiliary variable |
| Difficulty | Moderate -0.3 Part (i) requires straightforward application of logarithm laws (sum and subtraction rules) with minimal algebraic manipulation. Part (ii) is a standard quadratic-in-disguise problem using substitution u = logâ‚…(y), which is a common textbook exercise. Both parts are routine applications of core techniques with no novel insight required, making this slightly easier than average for A-level. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_a x + \log_a 3 = \log_a 27 - 1\) so \(\log_a \frac{3x}{27} = -1\) | M1 A1 | M1: uses sum or difference of logs correctly; A1: uses two rules correctly |
| \(\frac{3x}{27} = a^{-1}\) | M1 | Removes logs correctly |
| \(x = 9a^{-1}\) or \(\frac{9}{a}\) | A1 | Correct simplified answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^2 - 7x + 12 = 0\) and attempt to solve | M1 | Recognise and attempt to solve quadratic |
| \(x\) (or \(\log_5 y\)) \(= 3\) and \(4\) | A1 | Both correct values |
| \(y = 5^3\) or \(5^4\) | dM1 | Uses powers correctly; dependent on first M1 |
| \(y = 125\) and \(625\) | A1 | Both values correct |
# Question 14:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_a x + \log_a 3 = \log_a 27 - 1$ so $\log_a \frac{3x}{27} = -1$ | M1 A1 | M1: uses sum or difference of logs correctly; A1: uses two rules correctly |
| $\frac{3x}{27} = a^{-1}$ | M1 | Removes logs correctly |
| $x = 9a^{-1}$ or $\frac{9}{a}$ | A1 | Correct simplified answer |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2 - 7x + 12 = 0$ and attempt to solve | M1 | Recognise and attempt to solve quadratic |
| $x$ (or $\log_5 y$) $= 3$ and $4$ | A1 | Both correct values |
| $y = 5^3$ or $5^4$ | dM1 | Uses powers correctly; dependent on first M1 |
| $y = 125$ and $625$ | A1 | Both values correct |
---\begin{enumerate}
\item (i) Given that
\end{enumerate}
$$\log _ { a } x + \log _ { a } 3 = \log _ { a } 27 - 1 , \text { where } a \text { is a positive constant }$$
find, in its simplest form, an expression for $x$ in terms of $a$.\\
(ii) Solve the equation
$$\left( \log _ { 5 } y \right) ^ { 2 } - 7 \left( \log _ { 5 } y \right) + 12 = 0$$
showing each step of your working.\\
\hfill \mbox{\textit{Edexcel C12 2016 Q14 [8]}}