IYGB. Special Extension Paper E. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas

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1 YGB Special Extesio Paper E Time: 3 hours 30 miutes Cadidates may NOT use ay calculator. formatio for Cadidates This practice paper follows the Advaced Level Mathematics Core ad the Advaced Level Further Pure Mathematics Syllabi of recet years. Booklets of Mathematical formulae ad statistical tables may NOT be used. Full marks may be obtaied for aswers to ALL questios. The marks for the parts of questios are show i roud brackets, e.g. (). There are 0 questios i this questio paper. The total mark for this paper is 00. Advice to Cadidates You must esure that your aswers to parts of questios are clearly labelled. You must show sufficiet workig to make your methods clear to the Examier. Aswers without workig may ot gai full credit. No exact aswers should be give to a appropriate degree of accuracy. The examier may refuse to mark ay parts of questios if deemed ot to be legible. Scorig Total Score = T, Number of o attempted questios = N, Percetage score = P. P = T + N (rouded up to the earest iteger) Distictio P 70, Merit 55 P 69, Pass 40 P 54

2 Questio A curve has equatio f ( x) 3 ax + b, x R, where a ad b are o zero costats. Fid the value of a ad the value of b, give further that f ( ) = 3 3 ad ( ) f 3 = 3. ( 5) Questio The positive iteger fuctios f ad g are defied as Evaluate 3 ( ) = r ad g ( ) = + ( r + ) f r=. r= 39 f ( ) g ( ). = ( 6) Questio 3 A curve with equatio y = f ( x) passes through the poit with coordiates ( 0, ) ad satisfies the differetial equatio y dy y 4e x dx + =. 3 Solve the differetial equatio, by fidig a suitable itegratig factor, to show that y 3 x 3x = 3e e. ( 8)

3 Questio π cot x = dx. cosec x π 3 Use appropriate itegratio techiques to show that = a + b 3 6, where a ad b are itegers to be foud. ( 7) Questio 5 Two joggers, A ad B ra a stadard route of 5 km, which cosists of a dowhill sectio to start with, a flat sectio i the middle of the ru ad a uphill sectio all the way to the fiish lie. A ra the three sectios with respective speeds.4 A took 3 miutes ad 40 secods to complete the ru. ms, 3. ms ad ms. B ra the three sectios with respective speeds 3.6 A took exactly 7 miutes to complete the ru. ms, 3 ms ad.5 ms. Assumig that both ruers started at the same time, determie the distace betwee A ad B, as B crosses the fiish lie. 7 ( )

4 Questio 6 The quadratic equatio x 4x = 0, has roots α ad β i the usual otatio, where α > β. t is further give that f α β. Determie the value of f f f ( 7) Questio 7 t is give that a +, Z, 0 0 = x a x dx where a is a positive costat. a) Use itegratio by parts to show a + = 0,. 4 + ( 8) b) Determie the value of x 0 4 x dx. 0 ( 4)

5 Questio 8 Solve the followig iequality. ( x)( x ) 5 5 > 9, x R ( 7) Questio 9 The variable poit P lies o the rectagular hyperbola, with Cartesia equatio xy = a, where a is a positive costat. The ormal to the hyperbola at P meets the hyperbola agai at the poit Q. The poit M is the midpoit of PQ. Determie, i the form f ( x, y ) = 0, a equatio of the locus of M, for all the possible positios of P. ( 0) Questio 0 ( ) ( ) 3 z + 4i z 3 3i z + 4 i = 0, z C. Fid the three solutios of the above equatio give that oe of these solutios is purely imagiary. ( ) Questio Determie, as exact simplified atural logarithms, the solutios of the followig simultaeous equatios cosh x + cosh y = 4 ad sih x + sih y =. ( )

6 Questio C A O Q C P iitial lie The figure above shows the curves C ad C with respective polar equatios ( ) r = secθ ta θ ad 3 = sec θ, 0 θ r < π. 4 The poits P ad Q are the respective poits where C ad C meet the iitial lie, ad the poit A is the itersectio of C ad C. a) Fid the exact area of the curviliear triagle OAQ, where O is the pole. ( ) The agle OAP is deoted by ψ. b) Show that taψ = 3 3. ( 4) You may assume without proof ( ) 6 4 sec x dx = 8 + 4sec x + 3sec x ta x + C 5 Questio 3 f e e x x e 4 x ( x) ( )( ) +, x R. a) Fid i exact simplified form the solutio of the equatio f ( x ) = 0. b) Determie, i terms of l, the two solutios of the equatio f ( x ) =. ( 6) ( 6)

7 Questio 4 The part of the graph of the expoetial curve y = e x, l ( 3 ) x l ( 4 ), 4 3 is rotated by π radias i the x axis, formig a surface of revolutio S. Show that area of S is 85 3 π l ( 0) Questio 5 Fid the sum to ifiity of the followig series You may fid the series expasio of arcta x useful i this questio. ( ) Questio 6 Sketch the graph of the curve with equatio y = x l x, x R. The sketch must iclude the coordiates of ay poits where the curve meets the coordiate axes.... ay statioary poits. ( 0)

8 Questio 7 The poit P i a Argad diagram represets the complex umber z, which satisfies z i arg z i, z i. t further give that P lies o the arc AB of a circle cetred at C ad of radius r. a) Sketch i a Argad diagram the circular arc AB, statig the coordiates of C ad the value of r. b) Give further that PA = PB, fid the complex umber represeted by P. ( 0) ( 6) Questio 8 Three circles, C, C ad C 3, have their cetres at A, B ad C, respectively, so that AB = 5, AC = 4 ad BC = 3. The positive x ad y axis are tagets to C. The positive x axis is a taget to C. C ad C touch each other exterally at the poit M. Give further that C 3 touches exterally both C ad C, fid, i exact simplified form, a equatio of the straight lie which passes through M ad C. ( 0)

9 Questio 9 The itegral is defied as = π si x dx. 0 + cos x ( 4) a) Show by a detailed method that + π = π 4 dx. o + cos x b) Hece, fid the value of i exact simplified form. ( 7) c) Verify the aswer obtaied i part (b) by a alterative method by first writig the itegrad of as a fuctio of cot x. ( 8) Questio 0 t is give that a + 3b is a multiple of 3 a, where a N, b N. t is the asserted that ( 3 a)( b) + + is also a multiple of 3 a. Prove the validity of this assertio. ( 5)

Edexcel GCE Further Pure Mathematics FP1 Advanced/Advanced Subsidiary

Cetre No. Cadidate No. Surame Sigature Paper Referece(s) 6667/0 Edexcel GCE Further Pure Mathematics FP Advaced/Advaced Subsidiary Moday 28 Jauary 203 Morig Time: hour 30 miutes Materials required for