C1 June 2013 Q7
7. A company, which is making 200 mobile phones each week, plans to increase its production.
The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week \(N\).
The company then plans to continue to make 600 mobile phones each week.
| Scheme | Marks |
|---|---|
| \(600 = 200 + (N - 1)20 \Rightarrow N = \ldots\) | M1 |
| \(N = 21\) | A1 |
| Accept correct answer only. | |
| (2) |
Notes
M1: Use of 600 with a correct formula in an attempt to find \(N\). A correct formula could be implied by a correct answer.
A1: cso
\(600 = 200 + 20N \Rightarrow N = 20\) is M0A0 (wrong formula)
\(\dfrac{600 - 200}{20} = 20 \therefore N = 21\) is M1A1 (correct formula implied)
Listing: All terms must be listed up to 600 and 21 correctly identified. A solution that scores 2 if fully correct and 0 otherwise.
| Scheme | Marks |
|---|---|
| Look for an AP first: | |
| \(S = \tfrac{21}{2}(2\times 200 + 20\times 20)\) or \(\tfrac{21}{2}(200 + 600)\) or \(S = \tfrac{20}{2}(2\times 200 + 19\times 20)\) or \(\tfrac{20}{2}(200 + 580)\) (= 8400 or 7800) | M1A1 |
| Then for the constant terms: | |
| \(600\times(52 - \text{"}N\text{"})\ (= 18600)\) | M1A1ft |
| So total is 27000 | A1 |
| Note that for the constant terms, they may correctly use an AP sum with \(d = 0\). | |
| There are no marks in (b) for just finding \(\boldsymbol{S_{52}}\) | |
| (5) | |
| (7 marks) |
Notes
M1: Use of correct sum formula with their integer \(n = N\) or \(N - 1\) from part (a) where \(3 < N < 52\) and \(a = 200\) and \(d = 20\).
A1: Any correct un-simplified numerical expression with \(n = 20\) or \(n = 21\) (No follow through here)
M1: \(600\times k\) where \(k\) is an integer and \(3 < k < 52\)
A1: A correct un-simplified follow through expression with their \(k\) consistent with \(n\) so that \(n + k = 52\)
A1: Cao
If they obtain \(N = 20\) in (a) (0/2) and then in (b) proceed with,
\(S = \tfrac{20}{2}(2\times 200 + 19\times 20) + 32\times 600 = 7800 + 19\ 200 = 27\ 000\)
allow them to ‘recover’ and score full marks in (b)
Similarly
If they obtain \(N = 22\) in (a) (0/2) and then in (b) proceed with,
\(S = \tfrac{21}{2}(2\times 200 + 20\times 20) + 31\times 600 = 8400 + 18\ 600 = 27\ 000\)
allow them to ‘recover’ and score full marks in (b)