Mark scheme - June 2007 - 6675 - Further Pure Mathematics ...
GCE Mathematics (6675/01)
June 2007
6675 Further Pure Mathematics FP2
Mark Scheme
|Question Number |Scheme |Marks |
| | | |
|1. | [pic] |B1 |
| | [pic] |M1 A1ft |
| | ft their completing the square, requires arcosh | |
| | [pic] | |
| | [pic] |M1 A1 (5) |
| | | [5] |
| | | |
| | | |
| |Alternative | |
| | [pic] |B1 |
| |Let [pic], [pic] | |
| | [pic] |M1 |
| | [pic] |A1ft |
| | [pic] |M1 A1 (5) |
| | | |
| | | |
| | | |
|Question Number |Scheme |Marks |
| | | |
|2. |(a) y | |
| | 3 | |
| | | |
| | E D | |
| | | |
| | O 2 5 x | |
| | | |
| | | |
| | | |
| | One ellipse centred at O |B1 |
| | Another ellipse, centred at O, touching on y-axis |B1 |
| | Intersections: At least 2, 5, and 3 shown correctly |B1 (3) |
| | | |
| |(b) Using [pic], or equivalent, to find e or ae for D or E. |M1 |
| | | |
| |For S: [pic] and [pic], [pic], [pic] ignore sign with ae |A1 |
| | | |
| |For T: [pic] and [pic], [pic], [pic] ignore sign with [pic] |A1 |
| | | |
| | [pic] |M1 A1 (5) |
| | | [8] |
| | | |
| | | |
| | | |
|Question Number |Scheme |Marks |
| | | |
|3. | [pic] |B1 |
| | | |
| |[pic] |M1 |
| | [pic] |M1 A1 |
| | | |
| | [pic] |A1 |
| | | |
| | [pic] |M1 A1 (7) |
| | [pic] | [7] |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
|Question Number |Scheme |Marks |
| | | |
|4. |(a) | |
| |[pic] |M1 |
| | [pic] | |
| | [pic] ( cso |M1 A1 (3) |
| | | |
| |(b) [pic] |M1 |
| | [pic] |M1 |
| | [pic] ( cso |M1 A1 (4) |
| | | [7] |
| | | |
| | | |
| |Alternative for (b) | |
| | [pic] |M1 |
| | Leading to [pic] |M1 |
| | [pic] ( cso |M1 A1 (4) |
| | | |
| | | |
| | | |
| | | |
|Question Number |Scheme |Marks |
| | | |
|5. |(a) [pic] either |M1 |
| | [pic] both |A1 |
| | | |
| | Obtaining [pic] and [pic] in terms of t |M1 |
| | [pic] |A1 |
| | [pic] |A1 |
| | | |
| | [pic] |A1 (6) |
| | | |
| |(b) The least value of y [pic] is [pic] |B1 |
| | | |
| | [pic] accept equivalent fractions or 2.25 |B1 (2) |
| | | [8] |
| | | |
| | | |
| | | |
| | | |
| | | |
|Question Number |Scheme |Marks |
| | | |
|6. |(a) [pic] |M1 A1 |
| | [pic] ft numeric constants only |A1ft |
| | [pic] |M1 A1 |
| | [pic] ( cso |A1 (6) |
| | | |
| |(b) [pic] |M1 A1 |
| | [pic] |M1 |
| | [pic], [pic] |M1 A1 |
| | [pic] | |
| | [pic] |A1 (6) |
| | | [12] |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
|Question Number |Scheme |Marks |
| | | |
|7. |(a) [pic] |M1 A1 |
| | At [pic], [pic] accept equivalents |A1 (3) |
| | | |
| | | |
| | | |
| |(b) [pic] [pic] | |
| | [pic] |M1 A1 |
| | [pic] |M1 A1 + A1 |
| | [pic] |M1 |
| | [pic] … attempt at substitution |M1 |
| | [pic] |M1 A1 |
| | [pic] |A1 (10) |
| | | [13] |
| | | |
| | | |
| |Alternative for (a) | |
| | | |
| | [pic] |M1 |
| | [pic] |A1 |
| | At [pic], [pic] accept equivalents |A1 (3) |
| | | |
| |An alternative for (b) is given on the next page | |
| | | |
| | | |
|Question Number |Scheme |Marks |
| | | |
|7. |Alternative for (b) | |
| | | |
| | [pic] |M1 A1 + A1 |
| | [pic] | |
| | | |
| | Let [pic] [pic] | |
| | [pic] |M1 A1 |
| | [pic] |M1, M1 |
| | [pic] |M1 A1 |
| |[pic] |A1 (10) |
| | | |
| |The last 7 marks of the alternative solution can be gained as follows | |
| | | |
| | Let [pic] [pic] | |
| | [pic] dependent on first M1 |M1 A1 |
| | [pic] | |
| | [pic] |M1 |
| | [pic] | |
| |Hence [pic] | |
| | [pic] |M1 |
| | [pic] |M1 A1 |
| | [pic] |A1 |
| | | |
| | | |
|Question Number |Scheme |Marks |
| | | |
|8. | (a) [pic] Can be implied |B1 |
| |Use of any correct method or formula to obtain an equation of PQ in any |M1 |
| |form. | |
| | Leading to [pic] ( |A1 (3) |
| | | |
| |(b) Gradient of normal at P is [pic]. Given or implied at any stage |B1 |
| | Obtaining any correct form for normal at either point. |M1 A1 |
| |Allow if just written down. | |
| | [pic] | |
| | [pic] | |
| | Using both normal equations and eliminating x or y. |M1 |
| |Allow in any unsimplified form. | |
| | [pic] Any correct form for x or y |A1 |
| | Leading to [pic] ( cso |A1 |
| | [pic] ( cso |A1 (7) |
| | | |
| |(c) [pic] |B1 |
| | Using [pic] in both [pic] and [pic]. |M1 |
| | [pic] | |
| | Any complete method for relating x and y, independently of p and q. |M1 |
| | A correct equation in any form. |A1 |
| | [pic] | |
| | Leading to [pic] Accept equivalent forms of [pic] |A1 (5) |
| | | [15] |
| |The algebra above can be written in many alternative equivalent forms. | |
| | | |
| | | |
| | | |
| | | |
| | | |
-----------------------
Mark Scheme (Post-Standardisation)
Summer 2007
Edexcel Limited. Registered in England and Wales No. 4496750
Registered Office: One90 High Holborn, London WC1V 7BH
GCE
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