June 2023 Paper 2 Q5
5 Ziad is training to become a long-distance swimmer.
He trains every day by swimming lengths at his local pool.
The length of the pool is 25 metres.
Each day he increases the number of lengths that he swims by four.
On his first day of training, Ziad swims 10 lengths of the pool.
(a) Write down an expression for the number of lengths Ziad will swim on his \(n\)th day of training. [1 mark]
(b)
(i) Ziad’s target is to be able to swim at least 3000 metres in one day.
Determine the minimum number of days he will need to train to reach his target. [3 marks]
(ii) Ziad’s coach claims that when he reaches his target he will have covered a total distance of over 50 000 metres.
Determine if Ziad’s coach is correct. [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Forms a correct expression for the number of lengths swum on the \(n\)th day. ACF Can be unsimplified. For example: \(10 + 4(n - 1)\) | B1 | 1.1b |
| (1) |
Typical solution
\[4n + 6\]| Scheme | Marks | AO |
|---|---|---|
| (i) Forms the linear equation \(25 \times \text{their part(a) expression} = 3000\) OE Condone incorrect inequalities | M1 | 3.1b |
| Solves their linear equation and rounds or truncates to the nearest positive integer. Condone incorrect inequalities | M1 | 3.2a |
| Obtains 29 CAO | A1 | 1.1b |
| (3) | ||
| (ii) Uses a correct formula for the sum to \(n\) terms of an arithmetic progression substituting \(a\) = 10 and \(d\) = 4 or \(l\) = 122 | M1 | 3.4 |
| Obtains either 47850 metres or 1914 lengths or AWRT 29.7 days. OE Condone missing units | A1 | 3.2a |
| Makes an appropriate comparison and concludes that the coach is wrong. The comparison must be explicit and can be one of the following: 47850 < 50 000 1914 < 2000 29.7 > 29 or 30 > 29 FT \(n\) = 28 only with one of the comparisons 44800 < 50 000 1792 < 2000 29.7 > 28 or 30 > 28 This latter case would be a maximum of M1 A0 R1F | R1F | 2.4 |
| (3) | ||
| (7 marks) |
Typical solution
(i)
\[4n + 6 = \frac{3000}{25}\]\[n = 28.5\]Ziad will need to train for 29 days
(ii)
\[\frac{29}{2}\left(2 \times 10 + (29 - 1) \times 4\right) \times 25 = 47850\]Swims 47850 metres
47850 < 50 000
Therefore the coach is not correct