| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Two unrelated log parts: one non-log algebraic part |
| Difficulty | Moderate -0.8 This question tests routine logarithm laws and trapezium rule application with no problem-solving required. Part (a)(i) is direct recall of log subtraction rule, (a)(ii) is a standard equation requiring one substitution, and part (b) is straightforward trapezium rule with given strip width and simple calculator work. All techniques are textbook exercises with clear methods. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) \(\log_3 \frac{4x+7}{9}\) | B1 (1 mark) | Correct single logarithm, as final answer, from correct working only |
| (a)(ii) \(\log_3 \frac{4x+7}{9} = 2\) \(\frac{4x+7}{9} = 9\) \(4x + 7 = 9x\) \(x = 1.4\) | B1, M1, A1 (3 marks) | State or imply \(2 = \log_3 9\); Attempt to solve equation of form \(f(x) = 8\) or \(9\); Obtain \(x = 1.4\), or exact equiv |
| (b) \(\int_1^9 \log_{10} x \, dx \approx \frac{1}{2} × 3 × (\log_{10} 3 + 2\log_{10} 6 + \log_{10} 9)\) \(\approx 4.48\) | B1, M1, A1, A1 (4 marks) | State, or imply, the 3 correct \(y\)-values only; Attempt to use correct trapezium rule; Obtain correct unsimplified expression; Obtain 4.48, or better |
**(a)(i)** $\log_3 \frac{4x+7}{9}$ | B1 (1 mark) | Correct single logarithm, as final answer, from correct working only
**(a)(ii)** $\log_3 \frac{4x+7}{9} = 2$ $\frac{4x+7}{9} = 9$ $4x + 7 = 9x$ $x = 1.4$ | B1, M1, A1 (3 marks) | State or imply $2 = \log_3 9$; Attempt to solve equation of form $f(x) = 8$ or $9$; Obtain $x = 1.4$, or exact equiv
**(b)** $\int_1^9 \log_{10} x \, dx \approx \frac{1}{2} × 3 × (\log_{10} 3 + 2\log_{10} 6 + \log_{10} 9)$ $\approx 4.48$ | B1, M1, A1, A1 (4 marks) | State, or imply, the 3 correct $y$-values only; Attempt to use correct trapezium rule; Obtain correct unsimplified expression; Obtain 4.48, or better
**Total: 8 marks**\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $\log_3(4x + 7) - \log_3 x$ as a single logarithm. [1]
\item Hence solve the equation $\log_3(4x + 7) - \log_3 x = 2$. [3]
\end{enumerate}
\item Use the trapezium rule, with two strips of width 3, to find an approximate value for
$$\int_3^9 \log_{10} x \, dx,$$
giving your answer correct to 3 significant figures. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2007 Q5 [8]}}