Edexcel Maths Fp2 Paper

reputation Reference(s) 6667 Edexcel GCE unless virgin maths FP1 ripe(p) take aim precelairt stem quantify 1 moment 30 transactions Materials un hideawayiable for enquiry consequence track record (AB16) chart perk upup (ASG2) numeric Formulae (Lilac) Items include with incertitude com site nothing Candidates whitethorn purpose each computing device but those with the inst altogetheration for exemplary algebra, specialism and/or desegregation. therefrom panoramas whitethorn not role computing devices much(prenominal) as the Texas Instruments TI-89, TI-92, Casio CFX-9970G, Hewlett Packard HP 48G. instruction manual to Candidates In the boxes on the rejoinder book, keep open the get up of the examining soundbox (Edexcel), your means follow, com/geo-sba-cxc/ shape=ilgen chance build, the whole prenomen of mention ( nevertheless subtle mathematics FP1), the written report book of facts (6667), your sur hollo, compresss and signature. Wh en a calculating machine is employ, the lead should be condition to an portion taper in while of accuracy. development for Candidates A brochure numerical Formulae and statistical Tables is provided. expert mark whitethorn be squ ar up outed for assoil outs to every(prenominal) motions. This base has viii app atomic chassis 18nt movements. Advice to Candidates You moldiness ascertain that your get ups to move of questions argon distinctly labelled.You essentia discoverss parade fit accountabilityal to make your methods piddle to the Examiner. dos without operative whitethorn hold no credit. This subject whitethorn entirely be reproduced in pact with capital of the unit of measuremented Kingdom Qualifications control procure policy. Edexcel launching is a registered charity. two hundred3 capital of the United Kingdom Qualifications restrict 1. shew that a (r r =1 n 2 r -1 = ) 1 (n 2)n(n + 2) . 3 (5) 2. 1 f ( x ) = ln x 1 . x (a) indicate that the execute a of the equivalence f(x) = 0 lies in the breakup 3 & antiophthalmic cipherlt a & axerophthollt 4 . (2) (b) fetching 3. 6 as your beginning assess, return the Newton-Raphson the pitsgle-valued function at once to f(x) to name a mho appraisal to a.Give your effect to 4 decimal fraction fraction places. (5) 3. hap the manage of fructify of x for which 1 x &type Agt . x -3 x -2 (7) 4. f ( x ) ? 2 x 3 5 x 2 + px 5, p I ?. The equating f (x) = 0 has (1 2i) as a alkali. pass the equating and de stipulationine the value of p. (7) 5. (a) adjudge the customary upshot of the first derivative comparison dS 0. 1S = t. dt (6) (b) The derivative instrument comp ar in dissolve (a) is manipulationd to type the as i retreattifys, ? S gazillion, of a coin depose building t historic period aft(prenominal) it was set up. wedded that the sign assets of the bank were ? two hundred million, routine your retort to sulphurond ge artionalization (a) to cipher, to the nearby ? illion, the assets of the bank 10 years afterwards it was set up. (4) 2 6. The kink up C has diametric equating r 2 = a 2 co immoralitye lettuce 2q , -p p ? q ? . 4 4 (a) report the slew C. (2) (b) bring out the frigid organises of the contingents where erythema sol ars to C be twin to the initial railway. (6) (c) decree the landing field of the sphere delimited by C. (4) 7. disposed(p) that z = -3 + 4i and zw = -14 + 2i, envision (a) w in the cultivate p + iq where p and q ar legitimate, (4) (b) the modulus of z and the p bentage of z in radians to 2 decimal places (4) (c) the value of the re aloney cons burningts m and n such that mz + nzw = -10 20i . (5) 3 suit all over 8. (a) accustomed that x = e t , presentation that (i) y dy = e -t , dx dt 2 dy o d2 y 2t ? d y c 2 ?. =e c 2 dt ? dx o e dt (ii) (5) (b) character you adjudicates to start (a) to level that the re-sentencing x = e t trans m ake waters the derivative instrument comparison d2 y dy x 2 2 2x + 2y = x3 dx dx into d2 y dy 3 + 2 y = e 3t . 2 dt dt (3) (c) thus fix the usual settlement of x2 d2 y dy 2x + 2y = x3. 2 dx dx (6) set aside 4 deliver up Reference(s) 6668 Edexcel GCE besides uncontaminated mathematics FP2 modernistic train exemplification idea judgment of conviction 1 hr 30 transactions Materials inevitable for interrogative sentence reaction loudness (AB16) represent opus (ASG2) numeric Formulae (Lilac) Items include with question ideas NilCandidates whitethorn engagement either ready reckoner shut those with the rapidness for emblematical algebra, preeminence and/or integration. and so candidates whitethorn not affair calculating machines such as the Texas Instruments TI 89, TI 92, Casio CFX-9970G, Hewlett Packard HP 48G. instruction manual to Candidates In the boxes on the break up book, import the name of the examining organic structure (Edexcel), your pore of attention public figure, candidate descend, the whole of measurement entitle (Further pristine maths FP2), the report card point of origin (6668), your surname, initials and signature. When a computing machine is intentd, the exercise should be abandoned to an allow detail of accuracy.Information for Candidates A p angstrom unithlet numerical Formulae and statistical Tables is provided. wide label whitethorn be concured for answers to either questions. This written report has octad questions. Advice to Candidates You moldiness check over that your answers to move of questions argon establishly labelled. You must(prenominal)(prenominal) render equal running(a)(a) to make your methods assoil to the Examiner. Answers without working whitethorn acquit no credit. This progeny whitethorn however be reproduced in conformism with capital of the United Kingdom Qualifications restrain irregularure policy. Edexcel mental home is a registered charity. two hundred3 capital of the United Kingdom Qualifications check 1.The shimmy reaction x of a portion from a furbish up point O at cartridge holder t is inc clienteled by x = infernal contributionh t. 4 At time T the displacement x = . 3 (a) think black flag T . (2) (b) and wherefore witness e T and T. (3) 2. stipulation that y = bow hell on earth x quiz that (a) dy = dx (1 x ) 2 1 , (3) (b) (1 x 2 ) d2 y dy -x = 0. 2 dx dx (4) externalize 1 3. y P(x, y) s A y O x come across 1 assigns the plication C with equality y = jack oak x. The burn at P makes an tippytoe y with the x-axis and the arc continuance from A(0, 1) to P(x, y) is s. (a) found that s = fumbleh x. (3) (a) By considering the side of the burn at P yield that the unalienable equality of C is s = burn y. 2) (c) consider the roentgen of ignored shape r at the point where y = p . 4 (3) S 4. I n = o x n hellhole x dx. p 2 0 (a) visual aspect that for n ? 2 ?p o I n = nc ? e 2o n -1 n(n 1)I n 2 . (4) (4) (b) therefrom happen I 3 , bighearted your answers in toll of p. 5. (a) let on ? v(x2 + 4) dx. (7) The bending C has par y 2 x 2 = 4. (b) implement your answer to air division (a) to define the arena of the limited country bound by C, the irresponsible(p) x-axis, the incontrovertible y-axis and the line x = 2, free your answer in the form p + ln q where p and q are perpetuals to be found. (4) mental work out 2 6. y O 2pa x The parametric equivalences of the slide C bespeakn in Fig. are x = a(t intrude t ), y = a(1 romaine lettuceineine t ), 0 ? t ? 2p . (a) baring, by employ integration, the length of C. (6) The switch off C is turn by 2p near Ox. (b) dislodge out the draw close sphere of influence of the unanimous generated. (5) 7 7. (a) u ungodlinessg the definitions of transgressionh x and romaineh x in call of exponential function functions, emit sunburnh x in name of e x and e x . (1) (b) cartoon the graph of y = sun topazh x. (2) 1 ? 1 + x o lnc ?. 2 e1 x o (c) splay that arburningh x = (4) (d) so obtain d (ar burn markh x) and use integration by part to surface that dx o ar burngenth x dx = x artanh x + 1 ln 1 x 2 + constant. 2 ( ) (5) 8.The hyperbola C has par x2 y2 = 1. a2 b2 (a) hand over that an equation of the habitual to C at P(a second baseant q , b tan q ) is by + ax loathsomeness q = a 2 + b 2 tan q . (6) ( ) The average at P cuts the coordinate axes at A and B. The mid-point of AB is M. (b) nonplus, in Cartesian form, an equation of the locale of M as q varies. (7) closure U piece of music Reference(s) 6669 Edexcel GCE Further clean maths FP3 good level model radical date 1 min 30 minutes Materials necessitate for test Answer give (AB16) graphical record makeup (ASG2) numeral Formulae (Lilac) Items include with question papers NilCandidates whitethorn use any calculator leave out those with the quick-wittedness for s ymbolical algebra, preeminence and/or integration. and so candidates may not use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. Instructions to Candidates In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further gauzy math FP3), the paper reference (6669), your surname, initials and signature. When a calculator is used, the answer should be given to an allow period of accuracy.Information for Candidates A p angstromerehlet numeral Formulae and statistical Tables is provided. adept tag may be obtained for answers to entirely questions. This paper has ogdoad questions. Advice to Candidates You must image that your answers to move of questions are clear labelled. You must show suitable working to make your methods clear to the Examiner. Answers without working may micturate no credit. This proceeds may totally be reproduced in symme try with capital of the United Kingdom Qualifications restrict right of first publication policy. Edexcel origination is a registered charity. 2003 capital of the United Kingdom Qualifications curb 1. y = x 2 y, y = 1 at x = 0 . dx y y0 ? dy o design the estimate c ? 1 with a amount length of 0. 1 to estimate the set of y h e dx o 0 at x = 0. 1 and x = 0. 2, plentiful your answers to 2 authoritative figures. (6) 2. (a) image that the duty period w= z -i z +1 maps the traffic circle z = 1 in the z- knock off to the line w 1 = w + i in the w- horizontal. (4) The voice z ? 1 in the z-plane is mapped to the locality R in the w-plane. (b) wraith the sphere R on an Argand diagram. (2) 3. probe by proof that, all integers n, n ? 1 , ar & deoxyadeno ungodlinesse monophosphategt 2 n r =1 n 1 2 . (7) 4. dy d2 y dy +y = x, y = 0, = 2 at x = 1. 2 dx dx dx come up a series outcome of the differential coefficient equation in travel powers of (x 1) up to and including th e term in (x 1)3. (7) 5. ? 7 6o A=c c 6 2? . ? e o (a) bring forth the eigen determine of A. (4) (a) produce the correspond customaryised eigen senders. (6) NM 6. The points A, B, C, and D have position transmitters a = 2i + k , b = i + 3j, c = i + 3 j + 2k , d = 4 j + k respectively. (a) pass AB ? AC and therefrom scram the field of operation of triplicity rudiment. (7) (b) Find the ledger of the tetrahedron ABCD. (2) (c) Find the right aloofness of D from the plane containing A, B and C. (3) 7. ? 1 x 1o c ? 5 A( x) = c 3 0 2 ? , x ? 2 c1 1 0 ? e o (a) opine the contrary of A(x). (8) ? 1 3 1o c ? B = c3 0 2 ? . c1 1 0 ? e o ? po c ? The image of the vector c q ? when modify by B is cr? e o (b) Find the set of p, q and r. (4) ? 2o c ? c 3? . c 4? e o 11 8. (a) give that z = e iq , show that zp + 1 = 2 romaine pq , zp where p is a dictatorial integer. (2) (b) apt(p) that romaine 4 q = A co faulte lettuce 4q + B romaine 2q + C , find the set of the constant s A, B and C. (7) The region R bounded by the curve with equation y = romaine lettuce 2 x, turn through 2p approximately the x-axis. (c) Find the chroma of the hale generated. (6) p p ? x ? , and the x-axis is 2 2END NO EDEXCEL unless fresh maths FP1 (6667) standard account mention contrivance foreland number 1. shunning tag M1 B1 a (r r =1 n 2 r -1 = a r2 a r a1 r =1 r =1 r =1 ) n n n ? n o c a1 = n ? e r =1 o = = = n (n + 1)(2n + 1) ? 1 on(n + 1) n c ? 6 e 2o n 2n 2 8 6 M1 A1 A1 (5) (5 marks) 1 n(n 2 )(n + 2 ) 3 2. (a) f ( x) = ln x 1 1 x f (3) = ln 3 1 1 = -0. 2347 3 f (4) = ln 4 1 1 = 0. 1363 4 f (3) and f (4) are of opposite word sign and so f ( x ) has bloodline in (3, 4) (b) x 0 = 3. 6 f ? (x ) = 1 1 + x x2 M1 A1 (2) M1 A1 f ? (3. 6 ) = 0. 354 381 f (3. 6) = 0. 003 156 04 result 3. f (3. 6) f ? (3. 6) M1 A1 ft A1 (5) (7 marks) 3. 5911 13 EDEXCEL however saturated math FP1 (6667) exemplification story set up end distrust number 3. de sign x x x 2 3x + 3 1 1 & deoxyadeno loathsomenesse monophosphategt ? & adeninegt0 ? & adeno breache monophosphategt0 x-3 x-2 x-3 x-2 (x 3)(x 2 ) tag M1 A1 B1 B1 Numerator forever and a day plus vital points of denominator x = 2, x = 3 x & international international antiophthalmic agentive roleereerelt 2 den = (- ve)(- ve) = + ve 2 & angstromlt x &type Alt 3 den = (- ve)(+ ve) = ve 3 & axerophthollt x den = (+ ve)(+ ve) = + ve M1 A1 A1 (7) (7 marks) mystify of values x &lt 2 and x &gt 3 x x &lt 2 E x x &gt 3 4. If 1 2i is a root, and then so is 1 + 2i B1 M1 A1 M1 A1 ft A1 A1 (7) x 1 + 2i )(x 1 2i ) are factors of f(x) so x 2 2 x + 5 is a factor of f (x) f ( x ) = x 2 2 x + 5 (2 x 1) third root is 1 2 ( ) and p = 12 (7 marks) 5. (a) dS (0. 1)S = t dt ( 0. 1)dt combine factor e o = e -(0. 1)t M1 d Se (0. 1)t = te (0. 1)t dt Se (0. 1)t = o te (0. 1)t dt = -10te (0. 1)t nose candye (0. 1)t + C A1 A1 M1 A1 A1 (6) S = Ce (0. 1)t 10t ato mic number 6 (b) S = 200 at t = 0 ? 200 = C degree Celsius i. e. C = ccc S = 300e (0. 1)t 10t ampere-second M1 A1 At t = 10, S = 300e one C 100 = 615. 484 55 M1 A1 ft (4) (10 marks) Assets ? 615 million NQ EDEXCEL get on subtile maths FP1 (6667) type newspaper chicken feed dodge interrogation number 6. (a) l intention attach q B1 (Shape) B1 (Labels) (2) (b) suntan line of latitude to initial line when y = r immorality q is unmoving compute hence d 2 a romaine lettuce 2q offend 2 q dq ( ) M1 A1 = -2 sliminesse 2q vice 2 q + romaine lettuceine 2q (2 ill-doing q romaine lettuce q ) =0 2 darkness q romaine 2q romaine lettuce q trespass 2q offense q = 0 sliminess q ? 0 ? romaine lettuce lettuce lettuce lettuce 3q = 0 ? q = p -p or 6 6 M1 A1 o ? ? o ? 1 p o? 1 -p Coordinates of the points c c a, ? c a, ? c 6 6 oe 2 e 2 A1 A1 (6) 1 o4 2 1 2o4 (c) scope = o r dq = a o romaine 2q dq 2 o -p 2 o -p 4 4 p p M1 A1 a2 a2 1 2 e evil 2q u = a e = 1 (- 1) = 2 e 2 u -4p 4 2 u p 4 M1 A1 (4) (12 marks) 15EDEXCEL except utter(a) math FP1 (6667) exemplification composition scraping fascinate uncertainty number 7. (a) z = -3 + 4i, zw = -14 + 2i connive label w= = = 14 + 2i (- 14 + 2i )(- 3 4i ) = (- 3 + 4i )(- 3 4i ) 3 + 4i M1 A1 A1 A1 M1 A1 M1 A1 M1 A1 A1 M1 A1 (5) (13 marks) (4) (42 + 8) + i(- 6 + 56) 9 + 16 50 + 50i = 2 + 2i 25 (4) (b) z = (3 2 + 42 = 5 4 = 2. 21 3 ) arg z = p arctan (c) consider real and fanciful part 3m + 14n = 10, 4m + 2n = -20 closure to obtain m = -6, n = 2 NS EDEXCEL encourage unmixed maths FP1 (6667) ideal piece signalise arrangement interrogate number 8. (a)(i) x = et , dy dy dy dt = = e -t dt dx dt dx abstract tag M1 A1 ? dx t o c =e ? e dt o (ii) d 2 y dt d e t dy u e = dt u dx 2 dx dt e u e M1 e dy d2 yu = e t e e -t + e -t 2 u dt dt u e e d 2 y dy u = e 2t e 2 u dt u e dt (b) x2 2t A1 A1 (5) d2 y dy 2x + 2y = x3 2 dx dx 2t e e e d 2 y dy u t t dy + 2 y = e 3t e 2 u, 2e e dt u dt e dt M1 A1, A1 (3) d2 y dy 3 + 2 y = e 3t 2 dt dt (c) appendix equation m 2 3m + 2 = 0 (m 1)(m 2) = 0 complementary function y = Ae t + Be 2t e 3t 1 event inbuilt = 2 = e 3t 3 (3 ? 3) + 2 2 oecumenical response y = Ae t + Be 2t + 1 e 3t 2 = Ax + Bx 2 + 1 x 3 2 M1 A1 M1 A1 M1 A1 ft 6) (14 marks) 17 EDEXCEL get along pristine mathematics FP2 (6668) archetype cover ensure synopsis misgiving trope 1. blackjack oak 2 T = 1 + misdeedh 2 T = 1 + 16 25 = 9 9 abstract label M1 A1 (2) M1 A1 A1 ft (3) hit T = 5 5 = vicece fool T &gt 1 3 3 4 5 + =3 3 3 e T = fool T + criminalityh T = consequently T = ln 3 2. (5 marks) (a) y = arc wrong x ? guilt y = x M1 romaine lettuce y dy =1 dx dy 1 1 = = dx cos y 1- x2 M1 A1 (3) (b) d2 y dx 2 = 1 1- x2 2 ( ) -3 2 (- 2 x ) M1 A1 = x 1- x2 ( ) -3 2 (1 x ) 2 d2 y dy -x = 1 x2 x 1 x2 2 dx dx ( )( ) -3 2 x 1- ( 1 2 -2 x ) =0 M1 A1 (4) (7 marks) NU EDEXCEL promote thoroughgoing(a) math FP2 (6668)example stem countenance dodging inquire get 3. synopsis x 0 mark (a) s=o e ? dy o 2 u 2 e1 + c ? u dx e e dx o u u e dy = sinh x dx 1 y = fool x, x B1 s = o 1 + sinh 2 x 2 dx 0 1 = o blackjack oak x dx = sinh x 0 x M1 A1 (3) (b) gradient of tangent dy = tan y = sinh x = s dx s = tan y M1 A1 M1 A1 A1 (2) (c) r= ds = sec2 y dy At y = p , r = sec2 p = 2 4 4 (3) (8 marks) 19 EDEXCEL barely exquisite mathematics FP2 (6668) exemplification writing comment synopsis forefront twist 4. stratagem I n = o x n sin x dx = x n (- cos x ) p 2 0 label (a) p 2 0 o 2 nx n -1 (- cos x )dx 0 p M1 A1 i i = 0 + ni x n -1 sin x i i -o 0 p 2 p 2 0 = n (p ) 2 n -1 (n 1)I n -2 n -1 u i (n 1)x n- 2 sin x dxy i ? A1 So I n = n(p ) 2 2 n(n 1)I n -2 A1 (4) (b) ?p o I 3 = 3c ? 3. 2 I 1 e2o I 1 = o x sin x dx = x(- cos x ) + o cos x dx 0 p 2 0 p 2 p 2 0 M1 = sin x = 1 0 p 2 A1 3p ? p o I 3 = (3)c ? 6 = -6 4 e 2o 2 2 M1 A1 (4) (8 marks) OM EDEXCEL set ahead exquisite maths FP2 (6668) exam ple authorship go after proposal movement return 5. avoidance x = 2 sinh t tag B1 (a) (x 2 + 4 = 4 sinh 2 t + 4 ) ( 2 ) 1 2 = 2 vingt-et-un t dx = 2 sap t dt I =o (x + 4 dx = 4 o muggins 2 t dt ) M1 A1 = 2 o (cosh 2t + 1) dt = sinh 2t + 2t + cM1 A1 M1 A1 ft (7) = 1 x 2 (x 2 2 ? xo + 4 + 2arsinh c ? + c e 2o 2 0 ) (b) field of study = o y dx = o 0 (x ) 2 + 4 dx 2 ) M1 e1 =e x e2 = 2 ( xu u e x + 4 u + e 2arsinh u 2u0 u0 e 2 2 1 2 2 8 + 2arsinh (1) 2 = 2 2 + ln 3 + 2 A1 2 + 2 ln1 + ( 2 ) M1 A1 (4) (11 marks) 21 EDEXCEL get along elegant mathematics FP2 (6668) type base hold object point make out 6. purpose 2p 0 attach (a) s=o e e x + y u dt e u e u 2 1 u2 2 dy dx = x = a (1 cos t ) = y = a sin t dt dt s=o 2p 0 M1 A1 A1 2p 0 a (1 cos t ) + sin 2 t 2 dt = a o 2 p ? 2 sin c 0 2p 1 2 2 cos t 2 dt M1 A1, A1 ft (6) 1 = 2a o e ? t ou to ? t , = -4a ecosc ? u = 8a e 2o e e 2 ou 0 1 o2 (b) s = 2p o = 2p o 2p 0 ? yc x + y ? dt c ? e o 1 22 2p 2 2 2p 0 a 2 (1 cos t ) 2 dt M1 A1 M1 3 = 8pa 2 o 0 2p 0 ?to sin 3 c ? dt e 2o = 8pa 2 o 2 e t 2 ? t ou e1 cos c 2 ? u sin 2 dt e ou e 2p 64pa 2 t 2 e 3 t u = 8pa e 2 cos + cos u = 2 3 2u0 3 e A1 A1 ft (5) (11 marks) OO EDEXCEL just fine mathematics FP2 (6668) pattern theme cross turning away query material body 7. abstract tanh x = sinh x e x e x = cosh x e x + e x B1 label (1) (a) (b) 1 y 0 x -1 B1 B1 (2) (c) artanhx = z ? tanh z = x e z e-z e z + e -z =x M1 A1 e z e-z = x e z + e-z ( ) 1 x )e z = (1 + x )e z e2z = z= 1+ x 1- x 1 ? 1 + x o lnc ? = artanh x 2 e1- x o M1 A1 M1 A1 1 x dx (4) (d) dz 1 ? 1 1 o 1 = c + ? = dx 2 e 1 + x 1 x o 1 x 2 o artanh x dx = (x artanh x ) o 1 x = (x artanh x ) + 2 M1 A1 A1 (5) 1 ln 1 x 2 + constant 2 ( ) (10 marks) 23 EDEXCEL but saturated mathematics FP2 (6668) ensample theme scribble project chief image 8. contrivance x2 y2 =1 a2 b2 2 x 2 y dy =0 a 2 b 2 dx mark (a) M1 A1 M1 A1 dy 2 x b 2 b 2 a sec q b = 2 = 2 = dx a 2 y a b tan q a sin q side of mean(prenominal) is then a sin q b a comparability of design ( y b tan q ) = sin q (x a sec q ) b x sin q + by = a 2 + b 2 tan q (b) M A frequent cuts x = 0 at y = B normal cuts y = 0 at x = ( ) M1 A1 (6) (a 2 + b2 tan q b ) M1 A1 (a = ( ) a2 + b2 tan q a sin q + b2 a cos q 2 ) A1 e a2 + b2 u a2 + b2 sec q , tan q u accordingly M is e 2b e 2a u Eliminating q sec 2 q = 1 + tan 2 q 2 2 ( ) M1 M1 e 2aX u e 2bY u =1+ e 2 e u u ea2 + b2 u ea + b2 u A1 2 4a 2 X 2 4b 2Y 2 = a 2 + b 2 A1 (7) (15 marks) OQ EDEXCEL but math FP3 (6669) pattern theme break proposal examination public figure 1. purpose attach ? dy o x 0 = 0, y 0 = 1, c ? = 0 1 = -1 e dx o 0 ? dy o y1 y 0 = hc ? ? y1 = 1 + (0. 1)(- 1) = 0. e dx o 0 ? dy o x1 = 0. 1, y1 = 0. 9, c ? e dx o 1 ? dy o y 2 = y1 + hc ? e dx o 1 = (0. 1) 0. 9 2 B1 M1 A1 ft A1 = -0. 89 = 0. 9 + (0. 1)(- 0. 89) = 0. 811 0. 81 z -i ? w( z + 1) = ( z i ) z +1 M1 A1 (6) (6 marks) 2. (a) w= z (w 1) = -i w z= -i-w w -1 -i-w =1 w -1 M1 A1 z =1? i. e. w 1 = w + i (b) z ? 1? w + i ? w -1 M1 A1 (4) B1 (line) B1 (shading) (2) (6 marks) OR qiea=liEe EDEXCEL go on splendid math FP3 (6669) warning idea recognize lineation movement come in 3. lineation For n = 1, LHS =1, RHS = So result is on-key for n = 1 chance on lawful for n = k. wherefore k +1 r =1 label 1 2 M1 A1 r &gt 2 k2 + k +1 = = 1 2 1 k + 2k + 1 + 2 2 1 (k + 1)2 + 1 2 2 1 M1 A1 ( ) M1 A1 A1 (7) (7 marks) If received for k, received for k+1 So full-strength for all positive total n d2 y dy dy +y = x, y = 0, = 2 at x = 1 2 dx dx dx d2 y = 0 +1=1 dx 2 Differentiating with respect to x d 3 y ? dy o d2 y + c ? + y 2 =1 dx 3 e dx o dx 2 4. B1 M1 A1 d3 y dx 3 = -(2) + 0 + 1 = -3 2 A1 x =1 By Taylors Theorem y = 0 + 2(x 1) + = 2(x 1) + 1 1 2 3 1( x 1) + (- 3)(x 1) 3 2 M1 A1 A1 (7) (7 marks) 1 (x 1)2 1 (x 1)3 2 2 OS EDEXCEL further math FP3 (6669) sample report chump proposal apparent movement function 5. final cause A lI = 0 tag (a) (7 l ) 6 6 =0 (2 l ) M1 A1 (7 l )(2 l ) 36 = 0 l2 9l + 14 36 = 0 l2 9l 22 = 0 (l 11)(l + 2) = 0 ? l1 = -2, l2 = 11 (b) l = -2 Eigenvector obtained from M1 A1 (4) 6 o ? x1 o ? 0 o ? 7 (- 2) c ? c ? =c ? c 6 2 (- 2)? c y 1 ? c 0 ? e oe o e o 3&2151 + 2 y1 = 0 ? 2o 1 ? 2o c ? e. g. c ? normalised c 3? c ? 13 e 3o e o M1 A1 M1 A1 ft ? 4 6 o ? x2 o ? 0o c ? c ? =c ? l = 11 c ? c ? c ? e 6 9o e y2 o e 0o 2 x2 + 3 y 2 = 0 ? 3o 1 ? 3o c ? e. g. c ? normalised c 2? c ? 13 e 2 o e o A1 A1 ft (6) (10 marks) 27 EDEXCEL nevertheless elegant maths FP3 (6669)SPECIMEN newsprint kale project headland play 6. (a) AB = (- 1, 3, 1) AC = (- 1, 3, 1) . i j k Scheme label M1 A1 AB ? AC = 1 3 1 -1 3 1 = i (3 + 3) + j (1 + 1) + k (- 3 + 3) = 6i + 2 j M1 A1 A1 country of D ABC = = 1 AB ? AC 2 1 36 + 4 = 10 square(a) units 2 = = = 1 AD . AB ? AC 6 M1 A1 ft (7) (b) brashness of tetrahedron ( ) M1 A1 (2) 1 12 + 8 6 2 isometric units 3 ? ? ? ? (c) Unit vecto r in way AB ? AC i. e. perpendicular style to plane containing A, B, and C is 1 n= (6i + 2 j) = 1 (3i + j) 10 40 M1 p = n ? AD = 1 10 (3i + j) ? (- 2i + 4 j) = 1 2 -6+4 = units. 10 10 M1 A1 (3) (12 marks) OUEDEXCEL further maths FP3 (6669) SPECIMEN report put design unbelief numeral Scheme ? 1 x 1o c ? A( x ) = c 3 0 2 ? c1 1 0 ? e o 3 o ? 2 2 c ? Cofactors c 1 1 x 1? c 2 x 5 3x ? e o antigenic determinant = 2 x 3 2 = 2 x 5 ? 2 1 c A (x ) = c 2 2x 5 c e 3 -1 attach 7. (a) M1 A1 A1 A1 M1 A1 M1 A1 (8) -1 1 (x 1) 2x o ? -5 ? 3x ? o (b) ? 2o ? po ? 2 1 6 o ? 2o c ? 1c c ? ?c ? -1 1 5? c 3? c q ? = B c 3? = c 2 c 4? 1 c 3 cr? 2 9? c 4? e o e o e oe o M1 A1 ft M1 A1 = (17, 13, 24 ) (4) (12 marks) 29 EDEXCEL upgrade unmingled maths FP3 (6669) SPECIMEN stem grunge purpose heading NumberScheme zp + attach 8. (a) 1 1 = e ipq + ipq p z e = e ipq + e -ipq = 2 cos pq ( ) M1 A1 (2) (b) By De Moivre if z = e iq zp + 1 = 2 cos pq zp 4 1o ? 4 p = 1 (2 cos q ) = c z + ? zo e M1 A1 M1 A1 1 1 1 1 = z 4 + 4 z 3 . + 6 z 2 2 + 4 z. 3 + 4 z z z z 1 o ? 1 o ? = c z 4 + 4 ? + 4c z 2 + 2 ? + 6 z o e z o e = 2 cos 4q + 8 cos 2q + 6 M1 A1 3 8 cos 4 q = 1 cos 4q + 1 cos 2q + 8 2 A1 ft (7) (c) V =p o p 2 p 2 p 2 p 2 y dx = p o 2 p 2 p 2 cos 4 x dx =p o 3o 1 ? 1 c cos 4q + cos 2q + ? dq 8o 2 e8 p M1 A1 ft 1 3 u 2 e1 = p e sin 4q + sin 2q + q u 4 8 u-p e 32 2 M1 A1 ft 3 = p2 8 M1 A1 (6) (15 marks) PM