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. | |

| | | |

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| | | |

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Mark Scheme (Post-Standardisation)

Summer 2007

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