4755 Mark Scheme June Question Answer Marks Guidance M1* Attempt to find M or 108M -1 M 108 M1 A1 [6] M1 A1

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1 4755 Mark Scheme Jue 05 * Attempt to fid M or 08M - M * Divide by their determiat,, at some stage Correct determiat, (A0 for det M= 08 stated, all other OR x, y,oe 8 7 4xy 8xy dep* [6] marks are available) Attempt to pre-multiply by iverse or by M - Correct matrix multiplicatio (allow oe slip) For both, cao x ad y must be specified, may be i colum vectors SC aswers oly B Usig M to create two equatios Correct equatios Elimiatig x or y Ay valid method Fidig secod ukow Valid method 5 x, y Allow dp or better. 8 7 For each cao. SC Aswers oly B [6] j ad j B For both, accept + j Modulus Attempt at modulus of their complex roots Argumet arcta 0.98 Attempt at arcta ft their complex roots j has modulus ad argumet 0.98 ft Moduli specified, ft their roots. Accept oly j has modulus ad argumet ft ft their roots - must be i (, ] Accept ±0.98, ±56. o If sf give accuracy MUST be stated. [5] 7

2 4755 Mark Scheme Jue 05 p, use of α for p ad use of αβγ for r - allow oe 6 p sig error; sig errors is M0 r 0 r 0 for p, cao for r, cao OR 4 6, 4 = 0 Implies, satisfy x 4x 5 0 OR Valid method to create a quadratic equatio 4 Roots p p r Product of roots 0 r 0 THEN THEN Attempt to solve a -term quadratic for p, cao for r, cao EITHER x 4 is a root, so 64 6 p 4q r 0 OR q q [6] Substitutio ad attempt to solve for coefficiet of x,(or for the remaiig ukow.) Allow makig q the subject if p ad r ot foud. OR usig αβ OR use of remaider after divisio for q, cao 8

4755 Mark Scheme Jue 05 4 (i) Accept u-umbered evely spaced marks o axes to show scale B Lie at acute agle, all or part i Im z>0 B Half lie from -- j through 0 [do t pealise if poit -- j is icluded]

3 4755 Mark Scheme Jue 05 4 (i) Accept u-umbered evely spaced marks o axes to show scale B Lie at acute agle, all or part i Im z>0 B Half lie from -- j through 0 [do t pealise if poit -- j is icluded] Allow ear miss to 0 if π/4 marked [] SC correct diagram, o aotatios see B B0 4 (ii) B Circle cetre + j B Radius Must touch real axis [] SC correct diagram, o aotatios see B B0 4 (iii) B The shaded regio must be outside their circle ad have a border with the circumferece B Fully correct SC correct diagram, o aotatios see allow B B 5 (i) r r r r [] Attempt to split ito two sums (May be implied) = Use of stadard result for r cao (must be i terms of ) SC Iductio: B case = : E sum to k + terms correctly foud : E argumet completely correct 5 (ii) r r r r Use of result from (i) i umerator of a fractio Expressig deomiator as explicit, or other valid method. Correct sums eed ot be r r k [4] k 9

4 4755 Mark Scheme Jue B Must show workig o give result with = u ad, so true for = Assume true for = k u k k 5 E Assumig true for k Allow Let = k ad (result) If = k ad (result) Do ot allow = k or Let = k, without the result quoted, followed by workig k 5 u 5 k k 5 5 k 5 0 k 5 5 u with substitutio of result for u z ad some workig k to follow Correctly obtaied whe k Or target see Therefore if true for = k it is also true for = k. Both poits explicit Depedet o ad previous E E Sice it is true for =, it is true for all positive itegers,. E Depedet o B ad previous E [6] 7 (i) Asymptotes: y =, B x =, x = - B (both) Allow x =, - Crosses axes at 0, B Must see values for x ad y if ot writte as co-ordiates B (both) Must see values for x ad y if ot writte as coordiates., 0,, 0 [4] 0

5 4755 Mark Scheme Jue 05 7 (ii) y B Itercepts labelled (sigle figures o axes suffice) (0, ) y = B Asymptotes correct ad labelled. Allow y = show by itercept labelled at (0,) ad x = ad x = - likewise (,0) (, 0) x B Three correct braches (- each error) Ay poorly illustrated asymptotic approaches pealised oce oly. x= x= Whe x is large ad positive, graph approaches y = from below, B Approaches to y = justified e.g. for x = 00, There must be a result for y Whe x is large ad egative, graph approaches y = from above, 980 e.g. for x = -00, [5] 7 (iii) y 0 x or x B x BB 0x (B for 0 < x < or 0 x ) isw ay more show

6 4755 Mark Scheme Jue j 5 4j 5 4j 5 40j j Use of j at least oce 8 (i) 5 4j 5 6j 8 (ii) q r 0 5 6j 9q40qj j r 0 Substitute for 6 40q 44 j 0, 5 9q 55 r 0 Compare either real or imagiary parts q 7 ft q 7 ft their r r ft their ft [4] 8 (iii) f ( z) z 7z z Sum of roots = 7 (5 4j) (5 4j) w 7 w ad ad Valid method for the third root. (divisio, factor theorem, attempt at liear x quadratic with complex roots correctly used) Roots are 5 4j ad 5 4j B quoted ad cao real root idetified, A0 if extra roots foud 8 (iv) zf ( z) f ( z) ( z ) f ( z) 0 z or f ( z) 0 z ad f( z) 0 (may be implied) solvig 0, z, z, z 5 4j, z 5 4j ft For all four solutios [ft (iii)] NB icomplete method givig z = oly is M0 A0 []

7 4755 Mark Scheme Jue 05 9 (i) Ay valid method may be implied Correct positio vectors foud (eed ot be idetified) 0 0 A 0, 0, B 4, 0, C, ft co-ordiates, ft their positio vectors A, B, C idetifiable. Coordiates oly, A0 9 (ii) M represets a two-way stretch B Stretch. ( elargemet B0) factor 4 parallel to the x axis factor parallel to the y axis B B Directios idicated 9 (iii) 4 0 Attempt at MT i correct sequece cao 6 0 Represets the composite trasformatio T followed by M cao OR B for T - ad M - correct for attempt at T - M cao OR a a a a Fidig A, B ad C coordiates or positio vectors 0 For correct positio vectors a a 0 8 Iverse matrix correctly foud a a

8 4755 Mark Scheme Jue 05 9 (iv) Area scale factor = 48 B Area of triagle ABC = 4 square uits Area of triagle AB C = 48 area of triagle ABC = 9 (square uits) Usig their 48 ad their area of triagle ABC, correct triagle Or other valid method cao OR Fidig A'' B'' C'' (0,0) (-6, 0) (8, 4) ad usig them B A'' B'' C'' may be i (iii) Fidig the area of A'' B'' C'' Ay valid method attempted Area of triagle 9 (square uits) cao (possibly after roudig to sf) 4

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.

REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains. The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values of the variable it cotais The relatioships betwee