| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Logarithmic equation solving |
| Difficulty | Moderate -0.3 This is a straightforward logarithm question requiring standard techniques: combining logs using laws, exponentiating both sides to form a quadratic, then solving and rejecting the negative root (since ln requires x > 0). All steps are routine C3 material with clear signposting and no novel insight required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (i) \(\ln(3x^2)\) | B1 | \(2\ln x = \ln x^2\) |
| B1 | \(\ln x^2 + \ln 3 = \ln 3x^2\) | |
| (ii) \(\ln 3x^2 = \ln(5x+2) \Rightarrow 3x^2 = 5x+2\) | M1 | Anti-logging |
| \(\Rightarrow 3x^2 - 5x - 2 = 0\) * | E1 | |
| (iii) \((3x+1)(x-2) = 0 \Rightarrow x = -\frac{1}{3}\) or \(x = 2\) | M1, A1cao | Factorising or quadratic formula |
| \(x = -\frac{1}{3}\) is not valid as \(\ln(-\frac{1}{3})\) is not defined | B1ft | ft on one positive and one negative root |
## Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(i)** $\ln(3x^2)$ | B1 | $2\ln x = \ln x^2$ |
| | B1 | $\ln x^2 + \ln 3 = \ln 3x^2$ |
| **(ii)** $\ln 3x^2 = \ln(5x+2) \Rightarrow 3x^2 = 5x+2$ | M1 | Anti-logging |
| $\Rightarrow 3x^2 - 5x - 2 = 0$ * | E1 | |
| **(iii)** $(3x+1)(x-2) = 0 \Rightarrow x = -\frac{1}{3}$ or $x = 2$ | M1, A1cao | Factorising or quadratic formula |
| $x = -\frac{1}{3}$ is not valid as $\ln(-\frac{1}{3})$ is not defined | B1ft | ft on one positive and one negative root |
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# Section A (continued)3 (i) Express $2 \ln x + \ln 3$ as a single logarithm.\\
(ii) Hence, given that $x$ satisfies the equation
$$2 \ln x + \ln 3 = \ln ( 5 x + 2 )$$
show that $x$ is a root of the quadratic equation $3 x ^ { 2 } - 5 x - 2 = 0$.\\
(iii) Solve this quadratic equation, explaining why only one root is a valid solution of
$$2 \ln x + \ln 3 = \ln ( 5 x + 2 ) .$$
\hfill \mbox{\textit{OCR MEI C3 2006 Q3 [7]}}