| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Simultaneous equations with logarithms |
| Difficulty | Standard +0.3 Part (i) is a straightforward application of logarithm laws (log addition and converting log to exponential form). Part (ii) requires simplifying both equations using log laws, then solving the resulting simultaneous equations - a standard C2 exercise combining logarithms with algebraic manipulation, but with no novel insight required. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks |
|---|---|
| (i) \(\log_2(y - 1) - \log_2 x = 1\), \(\log_2 \frac{y-1}{x} = 1\) | M1 |
| \(\frac{y-1}{x} = 2^1 = 2\) | M1 |
| \(y - 1 = 2x\), \(y = 2x + 1\) | A1 |
| (ii) \(2\log_3 y = 2 + \log_3 x \Rightarrow \log_3 y^2 - \log_3 x = 2\) | M1 |
| \(\log_3 \frac{y^2}{x} = 3^2 = 9\) | M1 |
| \(y^2 = 9x\) | A1 |
| Substitute \(y = 2x + 1\): \((2x + 1)^2 = 9x\) | M1 |
| \(4x^2 - 5x + 1 = 0\) | M1 |
| \((4x - 1)(x - 1) = 0\) | M1 |
| \(x = \frac{1}{4}, 1\) | A1 |
| Therefore \(x = \frac{1}{4}, y = \frac{3}{2}\) or \(x = 1, y = 3\) | A1 (10) |
**(i)** $\log_2(y - 1) - \log_2 x = 1$, $\log_2 \frac{y-1}{x} = 1$ | M1 |
$\frac{y-1}{x} = 2^1 = 2$ | M1 |
$y - 1 = 2x$, $y = 2x + 1$ | A1 |
**(ii)** $2\log_3 y = 2 + \log_3 x \Rightarrow \log_3 y^2 - \log_3 x = 2$ | M1 |
$\log_3 \frac{y^2}{x} = 3^2 = 9$ | M1 |
$y^2 = 9x$ | A1 |
Substitute $y = 2x + 1$: $(2x + 1)^2 = 9x$ | M1 |
$4x^2 - 5x + 1 = 0$ | M1 |
$(4x - 1)(x - 1) = 0$ | M1 |
$x = \frac{1}{4}, 1$ | A1 |
Therefore $x = \frac{1}{4}, y = \frac{3}{2}$ or $x = 1, y = 3$ | A1 (10) |\begin{enumerate}
\item (i) Given that
\end{enumerate}
$$\log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x$$
show that
$$y = 2 x + 1$$
(ii) Solve the simultaneous equations
$$\begin{aligned}
& \log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x \\
& 2 \log _ { 3 } y = 2 + \log _ { 3 } x
\end{aligned}$$
\hfill \mbox{\textit{OCR C2 Q7 [10]}}