| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Logarithmic equation solving |
| Difficulty | Moderate -0.3 Part (a) is a straightforward logarithmic equation requiring rearrangement and application of log laws—routine C2 material. Part (b) involves simultaneous equations with logarithms requiring log laws and substitution, slightly more involved but still standard textbook fare. Overall slightly easier than average due to being mostly procedural with clear pathways. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\log 7^{w-3} = \log 184\) | M1* | Rearrange, introduce logs and use \(\log a^b = b \log a\). Must first rearrange to \(7^{w-3} = k\), with \(k\) from attempt at \(180 \pm 4\), before introducing logs. Can use logs to any base, as long as consistent on both sides. If taking \(\log_7\) then base must be explicit. |
| \((w-3)\log 7 = \log 184\) | A1 | Obtain \((w-3)\log 7 = \log 184\), or equiv e.g. \(w - 3 = \log_7 184\). Condone lack of brackets i.e. \(w - 3\log 7 = \log 184\), as long as clearly implied by later working. |
| \(w - 3 = 2.68\) | M1d* | Attempt to solve linear equation. Attempt at correct process i.e. \(w = \frac{\log k}{\log 7} \pm 3\), or equiv following expanding bracket first. |
| \(w = 5.68\) | A1 | Obtain \(5.68\), or better. More accurate final answer must round to \(5.680\). Answer only, or T&I, is 0/4. |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\log xy = \log 3\) hence \(xy = 3\) | M1 | Attempt correct use of log law to combine 2 (or more) logs. Must be used on at least two of \(\log x / \log y / \log 3\). Allow \(\log\!\left(\frac{xy}{3}\right)\) (condone no \(= 0\)). |
| \(3x + y = 10\) | A1 | Obtain \(xy = 3\). aef as long as no logs present, or equiv in one variable. |
| \(x(10 - 3x) = 3\) | B1 | Obtain \(3x + y = 10\). aef as long as no logs present. SR: if A0 B0 given above, then allow B1 for a correct combination of the 2 eqns e.g. \(9x + 3y = 10xy\) (others poss). |
| \(3x^2 - 10x + 3 = 0\) | ||
| \((3x-1)(x-3) = 0\) | M1 | Attempt to eliminate one variable, and solve the resulting three term quadratic. Elimination could happen prior to removal of logs — as long as logs are then removed completely to obtain three term quadratic. |
| or \(\frac{1}{3}(10-y)y = 3\), \(y^2 - 10y + 9 = 0\), \((y-1)(y-9)=0\) | ||
| \(x = \frac{1}{3},\ y = 9 \quad x = 3,\ y = 1\) | A1 | Obtain two correct values. Could be for two values of one variable, or for one pair of correct \((x, y)\) values. |
| A1 | Obtain \(x = \frac{1}{3},\ y = 9\) and \(x = 3,\ y = 1\). Pairings must be clear, but not necessarily as coordinates. SR: B1 for each pair of correct \((x, y)\) values but no method. M1A1B1B1 - 1 pair of \((x,y)\) values from 2 correct eqns but no other method shown (but 6/6 if both pairs found). | |
| [6] |
## Question 8(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\log 7^{w-3} = \log 184$ | M1* | Rearrange, introduce logs and use $\log a^b = b \log a$. Must first rearrange to $7^{w-3} = k$, with $k$ from attempt at $180 \pm 4$, before introducing logs. Can use logs to any base, as long as consistent on both sides. If taking $\log_7$ then base must be explicit. |
| $(w-3)\log 7 = \log 184$ | A1 | Obtain $(w-3)\log 7 = \log 184$, or equiv e.g. $w - 3 = \log_7 184$. Condone lack of brackets i.e. $w - 3\log 7 = \log 184$, as long as clearly implied by later working. |
| $w - 3 = 2.68$ | M1d* | Attempt to solve linear equation. Attempt at correct process i.e. $w = \frac{\log k}{\log 7} \pm 3$, or equiv following expanding bracket first. |
| $w = 5.68$ | A1 | Obtain $5.68$, or better. More accurate final answer must round to $5.680$. Answer only, or T&I, is 0/4. |
| **[4]** | | |
---
## Question 8(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\log xy = \log 3$ hence $xy = 3$ | M1 | Attempt correct use of log law to combine 2 (or more) logs. Must be used on at least two of $\log x / \log y / \log 3$. Allow $\log\!\left(\frac{xy}{3}\right)$ (condone no $= 0$). |
| $3x + y = 10$ | A1 | Obtain $xy = 3$. aef as long as no logs present, or equiv in one variable. |
| $x(10 - 3x) = 3$ | B1 | Obtain $3x + y = 10$. aef as long as no logs present. **SR:** if A0 B0 given above, then allow B1 for a correct combination of the 2 eqns e.g. $9x + 3y = 10xy$ (others poss). |
| $3x^2 - 10x + 3 = 0$ | | |
| $(3x-1)(x-3) = 0$ | M1 | Attempt to eliminate one variable, and solve the resulting three term quadratic. Elimination could happen prior to removal of logs — as long as logs are then removed completely to obtain three term quadratic. |
| **or** $\frac{1}{3}(10-y)y = 3$, $y^2 - 10y + 9 = 0$, $(y-1)(y-9)=0$ | | |
| $x = \frac{1}{3},\ y = 9 \quad x = 3,\ y = 1$ | A1 | Obtain two correct values. Could be for two values of one variable, or for one pair of correct $(x, y)$ values. |
| | A1 | Obtain $x = \frac{1}{3},\ y = 9$ and $x = 3,\ y = 1$. Pairings must be clear, but not necessarily as coordinates. **SR: B1** for each pair of correct $(x, y)$ values but no method. **M1A1B1B1** - 1 pair of $(x,y)$ values from 2 correct eqns but no other method shown (but 6/6 if both pairs found). |
| **[6]** | | |
---8
\begin{enumerate}[label=(\alph*)]
\item Use logarithms to solve the equation $7 ^ { w - 3 } - 4 = 180$, giving your answer correct to 3 significant figures.
\item Solve the simultaneous equations
$$\log _ { 10 } x + \log _ { 10 } y = \log _ { 10 } 3 , \quad \log _ { 10 } ( 3 x + y ) = 1$$
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2012 Q8 [10]}}