| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Exponential inequality |
| Difficulty | Standard +0.3 This is a straightforward geometric progression question with standard applications. Part (i) is simple GP term calculation, part (ii) requires setting up a GP sum inequality (though the algebra is guided), and part (iii) applies logarithms to solve an exponential inequality. All techniques are routine for C2 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(u_5 = 1.5 × 1.02^4 = 1.624 \text{ tonnes A.G.}\) | M1, A1 (2 marks) | Use \(1.5r^4\), or find \(u_2, u_3, u_4\); Obtain 1.624 or better |
| (ii) \(\frac{1.5[(1.02)^N - 1]}{1.02 - 1} ≤ 39\) | M1 (1 mark) | Use correct formula for \(S_N\) |
| \((1.02^N - 1) ≤ 39 × 0.02 ÷ 1.5\) \((1.02^N - 1) ≤ 0.52\) Hence \(1.02^N ≤ 1.52\) | M1, M1 (4 marks) | Correct unsimplified expressions for \(S_N\); Link \(S_N\) to 39 and attempt to rearrange |
| \(N ≤ 21.144...\) \(N = 21 \text{ trips}\) | A1 (4 marks) | Obtain given inequality convincingly, with no sign errors |
| (iii) \(\log 1.02^N ≤ \log 1.52\) \(N \log 1.02 ≤ \log 1.52\) \(N ≤ 21.144...\) \(N = 21 \text{ trips}\) | M1, A1, M1, A1 (4 marks) | Introduce logarithms on both sides and use \(\log a^b = b \log a\); Obtain \(N \log 1.02 ≤ \log 1.52\) (ignore linking sign); Attempt to solve for \(N\); Obtain \(N = 21\) only |
**(i)** $u_5 = 1.5 × 1.02^4 = 1.624 \text{ tonnes A.G.}$ | M1, A1 (2 marks) | Use $1.5r^4$, or find $u_2, u_3, u_4$; Obtain 1.624 or better
**(ii)** $\frac{1.5[(1.02)^N - 1]}{1.02 - 1} ≤ 39$ | M1 (1 mark) | Use correct formula for $S_N$
$(1.02^N - 1) ≤ 39 × 0.02 ÷ 1.5$ $(1.02^N - 1) ≤ 0.52$ Hence $1.02^N ≤ 1.52$ | M1, M1 (4 marks) | Correct unsimplified expressions for $S_N$; Link $S_N$ to 39 and attempt to rearrange
$N ≤ 21.144...$ $N = 21 \text{ trips}$ | A1 (4 marks) | Obtain given inequality convincingly, with no sign errors
**(iii)** $\log 1.02^N ≤ \log 1.52$ $N \log 1.02 ≤ \log 1.52$ $N ≤ 21.144...$ $N = 21 \text{ trips}$ | M1, A1, M1, A1 (4 marks) | Introduce logarithms on both sides and use $\log a^b = b \log a$; Obtain $N \log 1.02 ≤ \log 1.52$ (ignore linking sign); Attempt to solve for $N$; Obtain $N = 21$ only
**Total: 10 marks**On its first trip between Maltby and Grenlish, a steam train uses 1.5 tonnes of coal. As the train does more trips, it becomes less efficient so that each subsequent trip uses 2% more coal than the previous trip.
\begin{enumerate}[label=(\roman*)]
\item Show that the amount of coal used on the fifth trip is 1.624 tonnes, correct to 4 significant figures. [2]
\item There are 39 tonnes of coal available. An engineer wishes to calculate $N$, the total number of trips possible. Show that $N$ satisfies the inequality
$$1.02^N < 1.52.$$ [4]
\item Hence, by using logarithms, find the greatest number of trips possible. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2007 Q9 [10]}}