| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log/algebra parts - linked parts (hence) |
| Difficulty | Moderate -0.5 This question tests standard logarithm laws (quotient, product, power rules) in part (i), which is routine recall. Part (ii) requires algebraic manipulation to solve for y, but follows a predictable pattern once the laws are applied. The multi-step nature and algebraic solving elevates it slightly above pure recall, but it remains a standard textbook exercise with no novel insight required. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| \(\log_{10}x - \log_{10}y\) | B1 | For the correct answer |
| M1 | Sum of three log terms involving 10, \(x^2\), \(y\) | |
| A1 | ||
| A1 | For correct term \(2\log_{10}x\) For both correct terms 1 and \(\log_{10}y\) |
| Answer | Marks |
|---|---|
| \(1 + 2\log_{10} + \log_{10}y\) | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(2\log_{10}x - 2\log_{10}y = 2 + 2\log_{10}x + \log_{10}y\) | M1 | For relevant use of results from (i) |
| \(\text{Hence } 3\log_{10}y = -2\) | A1 | For a correct, unsimplified, equation in \(\log_{10}y\) only |
| \(y = 10^{-2/3} \approx 0.215\) | M1 | For correct use of \(a = \log_{10}c \Leftrightarrow c = 10^a\) |
| A1 | For the correct value 0.215 |
**(i)(a)**
$\log_{10}x - \log_{10}y$ | B1 | For the correct answer
| M1 | Sum of three log terms involving 10, $x^2$, $y$
| A1 |
| A1 | For correct term $2\log_{10}x$ For both correct terms 1 and $\log_{10}y$
**(i)(b)**
$1 + 2\log_{10} + \log_{10}y$ | M1 |
**(ii)**
$2\log_{10}x - 2\log_{10}y = 2 + 2\log_{10}x + \log_{10}y$ | M1 | For relevant use of results from (i)
$\text{Hence } 3\log_{10}y = -2$ | A1 | For a correct, unsimplified, equation in $\log_{10}y$ only
$y = 10^{-2/3} \approx 0.215$ | M1 | For correct use of $a = \log_{10}c \Leftrightarrow c = 10^a$
| A1 | For the correct value 0.215
---7 (i) Express each of the following in terms of $\log _ { 10 } x$ and $\log _ { 10 } y$.
\begin{enumerate}[label=(\alph*)]
\item $\log _ { 10 } \left( \frac { x } { y } \right)$
\item $\log _ { 10 } \left( 10 x ^ { 2 } y \right)$\\
(ii) Given that
$$2 \log _ { 10 } \left( \frac { x } { y } \right) = 1 + \log _ { 10 } \left( 10 x ^ { 2 } y \right)$$
find the value of $y$ correct to 3 decimal places.
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2006 Q7 [8]}}