| Abbreviations used in the mark scheme | Meaning |
| dep* | Mark dependent on a previous mark, indicated by *. The * may be omitted if only one previous M mark |
| cao | Correct answer only |
| ое | Or equivalent |
| rot | Rounded or truncated |
| soi | Seen or implied |
| www | Without wrong working |
| AG | Answer given |
| awrt | Anything which rounds to |
| BC | By Calculator |
| DR | This question included the instruction: In this question you must show detailed reasoning. |
| 1 | (a) |
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| (25.92) | ||||||||||||
| 1 | (b) |
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| 1 | (c) |
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| \(93.31 \ldots - ( 33.59 \ldots +\) 22.39 ...) |
| 2 | (a) | \begin{tabular}{l} \(\frac { 60000 } { 10 } - R = 1500 \times 3.3\) |
| \(R = 1050\) \(\frac { 60000 } { v } = 1050\) | ||
| The greatest speed is \(57.1 \mathrm {~ms} ^ { - 1 }\) |
| M1 |
| A1 M1 |
| A1 [4] |
| 3.3 |
| 1.1 3.4 |
| 1.1 |
| = 4950 |
| May be -1050 |
| \(\frac { 60000 } { 10 } - k \times 10 = 1500 \times 3.3 k = 105\) |
| \(\frac { 60000 } { v } = 105 v\) \(v ^ { 2 } = 571.4 \ldots\) |
| \(v = 23.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) |
| M1 |
| A1 |
| M1 |
| A1 |
| A1 |
| [5] |
| 3.3 |
| 1.1 |
| 3.4 |
| 1.1 |
| 1.1 |
| Or \(1050 = 10 k\) |
| Must be positive |
| The constant resistance model does not seem to be very accurate |
| The refined (linear) model (is not perfect but) gives a much more accurate answer than the constant resistance model |
| B1ft |
| B1ft |
| B1 for each of two correct statements about the models. |
| If commenting on the accuracy of (a), must emphasise that (a) is very inaccurate or at least quite inaccurate |
| Do not allow e.g. |
| - model (a) is not very effective |
| - Neither model is accurate |
| - (a) and (b) are not very accurate |
| Clear comparison between the accuracy of the two models (must emphasise that (b) is fairly accurate or considerably more accurate than (a)), or other suitable distinct second comment |
| Do not allow e.g. |
| - model (b) is more accurate than model (a) |
| - (b) is not accurate |
| Do not allow statement claiming that resistance is proportional to speed, or to speed \({ } ^ { 2 }\) |
| Suitable comments for (a): |
| - is very inaccurate |
| - predicted speed is nearly three times the actual value |
| - constant resistance is not a suitable model |
| - both models underestimate the resistance (as top speed is lower than expected) |
| For the linear model (b) |
| - is fairly accurate (but probably underestimates the resistance at higher speeds) |
| - resistance is not proportional to speed but is a much better model than constant resistance |
| 3 | (a) | \(T _ { 2 } \cos \theta = m _ { 2 } g\) \(T _ { 2 } = \frac { m _ { 2 } \times 9.8 } { 0.8 } = 12.25 m _ { 2 }\) |
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| 3 | (b) | (i) | \(\begin{aligned} | T _ { 2 } \cos \theta + m _ { 1 } g = T _ { 1 } \cos \theta | |||||||||||||
| T _ { 1 } = T _ { 2 } + \frac { 9.8 m _ { 1 } } { 0.8 } = | |||||||||||||||||
| \qquad 12.25 m _ { 2 } + 12.25 m _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right) \end{aligned}\) |
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| 3 | (b) | (ii) | \(\begin{aligned} | T _ { 1 } \sin \theta + T _ { 2 } \sin \theta = m _ { 1 } a | |||||||||||||
| 12.25 \left( m _ { 1 } + m _ { 2 } \right) \times 0.6 + 12.25 m _ { 2 } \times 0.6 = m _ { 1 } \times 0.6 \omega ^ { 2 } | |||||||||||||||||
| \omega ^ { 2 } = \frac { 7.35 m _ { 1 } + 14.7 m _ { 2 } } { 0.6 m _ { 1 } } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } } \end{aligned}\) |
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| 3 | (c) | \(\begin{aligned} | \text { E.g } m _ { 1 } \gg m _ { 2 } \Rightarrow \frac { 2 m _ { 2 } } { m _ { 1 } } \approx 0 \text { or } \frac { 49 m _ { 2 } } { 4 m _ { 1 } } \approx 0 | |||||||||||
| \omega \approx \sqrt { \frac { 49 m } { 4 m } } = 3.5 \end{aligned}\) |
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| 3 | \multirow{3}{*}{(d)} |
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| [5] |