Mark Scheme (Results) January 2011

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1 Mark (Resuls) January 0 GCE GCE Furher Pure Mahemaics FP (6667) Paper Edexcel Limied. Regisered in England and Wales No Regisered Office: One90 High Holborn, London WCV 7BH

2 Edexcel is one of he leading examining and awarding bodies in he UK and hroughou he world. We provide a wide range of qualificaions including academic, vocaional, occupaional and specific programmes for employers. Through a nework of UK and overseas offices, Edexcel s cenres receive he suppor hey need o help hem deliver heir educaion and raining programmes o learners. For furher informaion, please call our GCE line on , our GCSE eam on , or visi our websie a If you have any subjec specific quesions abou he conen of his Mark ha require he help of a subjec specialis, you may find our Ask The Exper service helpful. Ask The Exper can be accessed online a he following link: hp:// January 0 Publicaions Code UA06 All he maerial in his publicaion is copyrigh Edexcel Ld 0

3 General Insrucions for Marking. The oal number of marks for he paper is 75.. The Edexcel Mahemaics mark schemes use he following ypes of marks: M marks: mehod marks are awarded for knowing a mehod and aemping o apply i, unless oherwise indicaed. A marks: Accuracy marks can only be awarded if he relevan mehod (M) marks have been earned. B marks are uncondiional accuracy marks (independen of M marks) should no be subdivided.. Abbreviaions These are some of he radiional marking abbreviaions ha will appear in he mark schemes. bod benefi of doub f follow hrough he symbol will be used for correc f cao correc answer only cso - correc soluion only. There mus be no errors in his par of he quesion o obain his mark isw ignore subsequen working awr answers which round o SC: special case oe or equivalen (and appropriae) dep dependen indep independen dp decimal places sf significan figures The answer is prined on he paper The second mark is dependen on gaining he firs mark

4 January 0 Furher Pure Mahemaics FP 6667 Mark. z = 5 i, w = + i (a) z = ( 5 i)( 5 i) = 5 5i 5i + 9i = 5 5i 5i 9 An aemp o muliply ou he brackes o give four erms (or four erms implied). zw is M0 = 6 0i 6 0i A Answer only / () (b) z ( 5 i) = w ( + i) ( ) ( ) 5 i i = + i i Muliplies z w by ( i ) ( i) = 0 0i 6i Simplifies realising ha a real number is needed on he denominaor and applies i = on heir numeraor expression and denominaor expression. = 4 6i 8 = i i or a = and b = or equivalen Answer as a single fracion A0 A () [5] GCE Furher Pure Mahemaics FP (6667) January 0

5 . (a) A 0 =, = 5 B 5 0 AB = 5 5 ( ) + 0(5) ( ) + 0() = 5( ) + (5) 5( ) + () A correc mehod o muliply ou wo marices. Can be implied by wo ou of four correc elemens. 6 Any hree elemens correc A = 0 Correc answer A Correc answer only / () (b) Reflecion; abou he y-axis. Reflecion y-axis (or x = 0.) A () (c) 00 0 C = I = or I B () [6] GCE Furher Pure Mahemaics FP (6667) January 0

6 . (a) f( x) = 5x 4x 6, x 0 f (.6) = awr -.0 B f (.8) = awr 0.54 B α.6.8 α = " " " " " " α = " " + " " Correc linear inerpolaion mehod wih signs correc. Can be implied by working below. (b) (c) = awr.74 A Correc answer seen 4/4 (4) A leas one of ± ax or ± bx f( x) = 0x 6x correc. Correc differeniaion. A () f (.7) = f (.7) = awr 0.4 B f (.7) = f (.7) = awr 9.8 B α " " =.7 " " Correc applicaion of Newon- Raphson formula using heir values. = =.745 (dp).745 A cao Correc answer seen 4/4 (4) [0] GCE Furher Pure Mahemaics FP (6667) January 0

7 4. z + pz + q = 0, z = 4i (a) z = + 4i + 4i B () (b) ( z + 4i)( z 4i) = 0 z z z z + + z + = 4i 4 8i 4i 8i 6 0 z z + = An aemp o muliply ou brackes of wo complex facors and no i. Any one of p = 4, q = 0. A Boh p = 4, q = 0. A z 4z + 0 = 0 only / () [4] GCE Furher Pure Mahemaics FP (6667) January 0

8 5 (a) n r = rr ( + )( r+ 5) n r 6r 5r = + + r = = n n + + n n + n + + n n + Muliplying ou brackes and an aemp o use a leas one of he sandard formulae correcly. Correc expression. A 5 4 = n n + + n n + n + + n n + = nn nn 4( n ) Facorising ou a leas nn+ d = nn+ n + n+ n+ + = ( 9 4 ) 4 nn + n + n + Correc erm quadraic facor A = ( 7) 4 nn+ n+ n+ * Correc proof. No errors seen. A (5) (b) 50 S = r( r+ )( r+ 5) n r = 0 = S S 50 9 = (50)(5)(5)(57) (9)(0)()(6) 4 4 Use of S50 S9 = = A Correc answer only / () [7] GCE Furher Pure Mahemaics FP (6667) January 0

9 C: y = 6x a = = 9 (a) S (9, 0) (9, 0) B () (b) x + 9 = 0 or x = 9 x + 9 = 0 or x = 9 or f using heir a from par (a). B () (c) PS = 5 QP = 5 Eiher 5 by iself or PQ = 5. Do no award if jus PS = 5 is seen. B () (d) x-coordinae of P x = 5 9 = 6 x = 6 B y = 6(6) Subsiues heir x-coordinae ino equaion of C. y = 576 = 4 y = 4 Therefore P (6, 4) A () (e) Area OSPQ (9 5)4 heir + 5 heir or recangle and disinc riangles, correc for heir values. = 408 (unis) 408 A = + ( a )( y) () [8] GCE Furher Pure Mahemaics FP (6667) January 0

10 7. (a) Im -4 Re Correc quadran wih ( 4, 7) indicaed. B -7 () 7 7 (b) arg z = π + an an or an ( 4 ) = =.86 ( dp) awr -.86 or awr.4 A () 5π 5 (c) w = 4,argw = r = 4, θ = w = rcosθ + irsinθ π 6 6 4cos ( 5 ) 4i sin( 5 ) π π w = ( ) = 4 + 4i Aemp o apply rcosθ + i rsin θ. Correc expression for w. A = + i eiher + i or awr.5 + i A a =, b = () (d) z = ( 4) + ( 7) = 5 z = 5 or zw = ( ) + (4 48) i or awr i B zw = z w = (5)(4) Applies z w or zw = A () [9] GCE Furher Pure Mahemaics FP (6667) January 0 4

11 8. A = (a) de A = () ( )( ) = 6 = 4 4 B () (b) A = 4 de A 4 A () (c) 7 Area( R ) = = 8 (unis) 4 7 heir de A or 7( heir de A ) 8 or f answer. A () (d) AR = S A AR = A S R = A S R 0 8 = = = 0 5 A leas one aemp o apply A by any of he hree verices in S. A leas one correc column o.e. A A leas wo correc columns o.e. A Verices are (, ), (4, 0) and (, 5). All hree coordinaes correc. A (4) [9] GCE Furher Pure Mahemaics FP (6667) January 0 5

12 9. n n u = 4u + +, u = and n u n = (4 ) n= ; u = (4 ) = () = So un is rue when n =. n Check ha u n = (4 ) yields when n =. B Assume ha for n= k ha, u = (4 k ) is rue + for k Z. k Then uk+ = 4uk + k ( ) = 4 (4 ) + Subsiuing u (4 k k = ) ino u = n 4u + + n. 8 k 8 = ( 4) + An aemp o muliply ou he brackes by 4 or 8 = 4 4 k = 4 k + = k + (4 ) k + (4 ) A Therefore, he general saemen, u (4 n n = ) is rue when n = k+. (As u n is rue for n =, ) hen u n is rue for all posiive inegers by mahemaical inducion Require True when n=, Assume rue when n=k and True when n = k+ hen rue for all n o.e. A (5) [5] GCE Furher Pure Mahemaics FP (6667) January 0 6

13 6 0. xy = ( ) (a) 6 a 6,. An aemp a d y. 6 dy 6 dx y = = 6x = 6x = x dx x or d y and d x d d 6 dy 6 A ( 6, ), = An aemp a d y. d (6 ) x dx dy dy So, m T dx = * dx Mus see working o award here A T: 6 y ( x 6) 6 y = heir m ( 6 ) T x T: y 6 = x + 6 T: y = x T: y x Correc soluion. A cso (5) (b) Boh T mee a ( 9,) gives = ( 9) + Subsiuing (-9,) ino T. 9 = + ( ) = 9 + An aemp o form a erm 9 = 0 quadraic 4 4 = 0 ( )( + ) = 0 An aemp o facorise. =, =, A 6 = x = 6 = 9, y = = 4 9, 4 = x = 6 =, ( ) 6 y = =, An aemp o subsiue eiher heir = or heir = ino x and y. A leas one of,. ( 9, 4) or Boh ( 9, 4) and A,. A (7) [] GCE Furher Pure Mahemaics FP (6667) January 0 7

14 Oher Possible Soluions = 0, = 4i z pz q z (a) (i) z = + 4i + 4i B Alier (ii) Produc of roos = ( 4i)( + 4i) No i. Aemp Sum and Produc Way of roos or Sum and discriminan =4+6=0 or b 4ac = ( 8 i) Sum of roos = ( 4i) + ( + 4i) = 4 z 4z 0 0 = + = Any one of p = 4, q = 0. A Boh p = 4, q = 0. A (4) = 0, = 4i z pz q z (a) (i) z = + 4i + 4i B Alier (ii) An aemp o subsiue eiher Way ( 4i) + p( 4i) + q = 0 z or z ino z + pz + q = 0 and no i. 6i + p( 4i) + q = 0 Imaginary par: 6 4 p = 0 Real par: + p + q = 0 4p = 6 p = 4 Any one of p = 4, q = 0. A q = p q = ( 4) = 0 Boh p = 4, q = 0. A (4) GCE Furher Pure Mahemaics FP (6667) January 0 8

15 Alier 5π 7. (c) w = 4,argw = and w = a + ib 6 Way 4 6 w = a + b = arg = arcan b = b = Eiher 5π 5π w 6 a 6 a Aemps o wrie down an equaion in erms of a and b for eiher he modulus or he argumen of w. a b + b = 6 or = a A a = b a = b So, b b b + = 6 = 4 b = ± and a = As w is in he second quadran w = + i eiher + i or awr.5 + i A a =, b = () GCE Furher Pure Mahemaics FP (6667) January 0 9

16 Furher copies of his publicaion are available from Edexcel Publicaions, Adamsway, Mansfield, Nos, NG8 4FN Telephone Fax Order Code UA06 January 0 For more informaion on Edexcel qualificaions, please visi Edexcel Limied. Regisered in England and Wales no Regisered Office: One90 High Holborn, London, WCV 7BH

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