Mark Scheme (Results) Summer 2008

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1 Mark (Results) Summer 8 GCE GCE Mathematics (7/) Mark Eecel Limite. Registere in Englan an Wales No. 97 Registere Office: One9 High Holborn, Lonon WCV 7BH

2 7 Further Pure FP Mark Question. cos. (.8 ) Ma be imlicit B. (..8.) M.. (..) Allow awrt A. cos... [or. cos(..cos.) ] Aft.88. (..88. ) Requires use of the ifferential equation to fin M (..) A ()

3 Degree moe in calculator: Gives answers:. (.999 ).7 (.78 ) This can score B M A Aft M A

4 . (a) q q is M A ( eqns imlie) ) ( q q is M A ( eqns, use of arameter imlie) q M: Two equations, one in, one in q M A q q (*) A () (b) or (or with q instea of ) M [ (8 ) ( 8) ( ) ] or q A < q M: Use q to fin value of an of q A: Positive values must be rejecte M A () (c) z, z, z An eqns, with value of M z (or searate equations) M E.vector is k (An non-zero value of k) A () () (a) Assuming a value for, e.g., gives M A A. (a) Assuming result an working backwars : ( ), gives M A A (b) Alternative: z z or z (or q instea of ) M z, z, z z z (i), z (ii), z (iii) From (i) an (iii) z From (ii) (or equiv. in terms of an/or q) A <,, q A: Positive values must be rejecte M A (b) Using the eigenvector scores no marks in this art.

5 . (a) ( ) ) ( M A ( ) ( ) ( ) (*) A () (b) B Follow through: Bft... M A () (c)., (..) [awrt.77] B () (8) (a) M: Use of rouct rule (at least once) an imlicit ifferentiation (at least once). (b) M: Use of series eansion with values for the erivatives (can be allowe without the first term, an can also be allowe if final term uses rather than!)

6 . (a) ( ) i i ( ) M A ( ) M Centre (, ), raius A, A () (b) Circle, centre on -ais B B B C (, ), r Bft O Follow through centre an raius, but eenent on first B. There must be inication of their, or on the -ais an no contraictor evience for their raius. Straight line B Straight line through (, ), or er. bisector of (, ) an (, ). B Straight line through oint of int. of circle & ve -ais, or through (, ) B B B B () (c) Shaing (onl) insie circle Insie correct circle an all of the correct sie of the correct line this mark is eenent on all the revious B marks in arts (b) an (c). (a) st M: Use z i, an attemt square of moulus of each sie. Not squaring the on the RHS woul be M A. n M: Attemting to eress in the form ( a) ( b) k, or attemting centre an raius from the form g f c B B () ()

7 . (a) k t t( k ) M k k t t( k) t( k) t(k ) M k 9 A () (b) et A k ( k) (Must be seen in art (b)) M (k ), which is alwas ositive M A is non-singular Acso () (c) A k k k k k M A () () k, A B A q A q M A () Alt., B M A for solving two sim. eqns. in an to give.,. (o.e.) () (b) n M: Alternative is to use quaratic formula on the quaratic equation, or to use the iscriminant, with a comment about no real roots, or can t equal zero, or a comment about the conition for singularit. ± 8 A Conclusion. (c) M: Nee, k's unchange an attemt to change sign for their et A either (leaving as to right) or k (leaving as bottom left). () M: Requires an attemt to multil the matrices.

8 . (a) ( cosθ isinθ ) cosθ isinθ (b) (c) true for n B Assume true for n k, ( cos θ isinθ ) cos kθ isin kθ k ( cosθ isinθ ) (cos kθ isin kθ )( cosθ isinθ ) k cos k θ cosθ sin kθ sinθ i(sin kθ cosθ cos kθ sinθ ) (Can be achieve either from the line above or the line below) cos( k ) θ isin( k ) θ Requires full justification of ( cosθ isinθ ) cos( k ) θ isin( k ) θ k M M A ( true for n k if true for n k) true for n Z b inuction Acso () cos θ Re[ (cosθ isinθ ) ] cos θ cos θ i sin θ cosθ i sin θ M A cos θ cos θ sin θ cosθ sin θ M cos θ cos θ ( cos θ ) cosθ ( cos θ ) M cos θ cos θ cos θ cosθ (*) Acso () cosθ cosθ cos θ M π π θ θ A cosθ, (a) Alternative: For the n i i i ( ) M mark: ( e )( e ) e π cos is a root (*) A () kθ θ θ k (b) Alternative: z z z z z z M z z z z z z cosθ cosθ cosθ A ( cosθ )... an attemt to ut cos θ in owers of cos θ M Correct metho (or formula) for utting cos θ in owers of cos θ M cosθ cos θ cos θ cosθ Acso (c) Alternatives: (i) Substitute given root into : π π π π cos cos cos cos M π Multil b cos θ an use result from art (b):... cos A an conclusion A π (ii) Use θ in result from art (b) M π π π cos cos cos an ivie b cos θ A an conclusion A ()

9 7. (a) PQ i j k, PR i j k B i j k PQ PR i j k M A () (b) r. (i j k) (i k). (i j k) [ma use OQ uuur uuur or OR ] M r. (i j k) o.e. ft from (a) Aft () (c) z (i), z (ii) (i) (ii) 7 7z, z (M: Eliminate one variable) M In (ii) z z, z (M: Substitute back) M z an z o.e. ( ) z () () () Writing own irection vector of PS from art (c). (Two correct -term equations) A o.e. (M: Form cartesian equations) M A () QR i j k PS PS // QR (or cross-rouct ) A () (e) PT i j (or QT i j k or RT i j k) M Volume uuur PQ PR. PT uuur uuur (i j k). (i j) ft from (a) (Instea of PQ PR, it coul be PQ QR or PR QR ) M Aft ( ) o.e. A () (a) If both vectors are reverse, B M A is ossible () (c) Alternative: Direction of line: 7 M A Through P (,, ) : z M A (e) Alternative: gives M A irectl. Here ft from st line of art (a). Secial case: or instea of, but metho otherwise correct: M A A

FP2 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002)

FP2 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002) FP Mark Schemes from old P, P5, P6 and FP, FP, FP papers (back to June 00) Please note that the following pages contain mark schemes for questions from past papers. The standard of the mark schemes is