| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Geometric Sequences and Series |
| Type | Find year when threshold exceeded |
| Difficulty | Moderate -0.3 This is a straightforward application of geometric sequence formulas with minimal problem-solving required. Students identify the common ratio (0.95), apply the nth term formula, use the sum formula, and solve a simple inequality. The context is accessible and all steps are routine textbook exercises, making it slightly easier than average. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks |
|---|---|
| \(r = \frac{114}{120} = 0.95\) | M1 |
| \(u_5 = 120 \times (0.95)^4 = 97.74\) | M1 |
| \(\therefore\) 1 hour 38 minutes | A1 |
| Answer | Marks |
|---|---|
| \(S_8 = \frac{120[1-(0.95)^8]}{1-0.95}\) | M1 |
| \(= 807.79...\) minutes \(\approx\) 13 hours 28 minutes | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(120 \times (0.95)^{n-1} < 60\) | M1 | |
| \((n-1)\lg 0.95 < \lg 0.5\) | M1 | |
| \(n > \frac{\lg 0.5}{\lg 0.95} + 1\) | A1 | |
| \(n > 14.51 \therefore\) 15 papers | A1 | (9) |
# Question 7:
## Part (i):
| $r = \frac{114}{120} = 0.95$ | M1 | |
| $u_5 = 120 \times (0.95)^4 = 97.74$ | M1 | |
| $\therefore$ 1 hour 38 minutes | A1 | |
## Part (ii):
| $S_8 = \frac{120[1-(0.95)^8]}{1-0.95}$ | M1 | |
| $= 807.79...$ minutes $\approx$ 13 hours 28 minutes | A1 | |
## Part (iii):
| $120 \times (0.95)^{n-1} < 60$ | M1 | |
| $(n-1)\lg 0.95 < \lg 0.5$ | M1 | |
| $n > \frac{\lg 0.5}{\lg 0.95} + 1$ | A1 | |
| $n > 14.51 \therefore$ 15 papers | A1 | **(9)** |
---7. A student completes a mathematics course and begins to work through past exam papers. He completes the first paper in 2 hours and the second in 1 hour 54 minutes.
Assuming that the times he takes to complete successive papers form a geometric sequence,\\
(i) find, to the nearest minute, how long he will take to complete the fifth paper,\\
(ii) show that the total time he takes to complete the first eight papers is approximately 13 hours 28 minutes,\\
(iii) find the least number of papers he must work through if he is to complete a paper in less than one hour.\\
\hfill \mbox{\textit{OCR C2 Q7 [9]}}