3y ago

29 Views

2 Downloads

454.94 KB

28 Pages

Transcription

Coimisiún na ScrúduitheScrúduithe StáitStáitState ExaminationsExaminations CommissionStateCommissionLeaving CertificateCertificate 20122013Marking SchemeSchemeAppliedMathematicsGraphicsDesign andCommunicationHigher Level

Note to teachers and students on the use of published marking schemesMarking schemes published by the State Examinations Commission are not intended to bestandalone documents. They are an essential resource for examiners who receive training inthe correct interpretation and application of the scheme. This training involves, among otherthings, marking samples of student work and discussing the marks awarded, so as to clarifythe correct application of the scheme. The work of examiners is subsequently monitored byAdvising Examiners to ensure consistent and accurate application of the marking scheme.This process is overseen by the Chief Examiner, usually assisted by a Chief AdvisingExaminer. The Chief Examiner is the final authority regarding whether or not the markingscheme has been correctly applied to any piece of candidate work.Marking schemes are working documents. While a draft marking scheme is prepared inadvance of the examination, the scheme is not finalised until examiners have applied it tocandidates’ work and the feedback from all examiners has been collated and considered inlight of the full range of responses of candidates, the overall level of difficulty of theexamination and the need to maintain consistency in standards from year to year. Thispublished document contains the finalised scheme, as it was applied to all candidates’ work.In the case of marking schemes that include model solutions or answers, it should be notedthat these are not intended to be exhaustive. Variations and alternatives may also beacceptable. Examiners must consider all answers on their merits, and will have consultedwith their Advising Examiners when in doubt.Future Marking SchemesAssumptions about future marking schemes on the basis of past schemes should be avoided.While the underlying assessment principles remain the same, the details of the marking of aparticular type of question may change in the context of the contribution of that question tothe overall examination in a given year. The Chief Examiner in any given year has theresponsibility to determine how best to ensure the fair and accurate assessment of candidates’work and to ensure consistency in the standard of the assessment from year to year.Accordingly, aspects of the structure, detail and application of the marking scheme for aparticular examination are subject to change from one year to the next without notice.

General Guidelines1Penalties of three types are applied to candidates' work as follows:Slips- numerical slipsS(-1)Blunders- mathematical errorsB(-3)Misreading- if not seriousM(-1)Serious blunder or omission or misreading which oversimplifies:- award the attempt mark only.Attempt marks are awarded as follows:2The marking scheme shows one correct solution to each question.In many cases there are other equally valid methods.Page 15 (att 2).

1.(a)A ball is thrown vertically upwards with a speed of 44·1 m s 1 .Calculate the time interval between the instants that the ball is 39·2 m above thepoint of projection.s ut 12 at 239·2 44·1t 12 9·8 t 25t 2 9t 8 0 t 1 t 8 0 t 1, t 8t1 8 1 7sPage 25,5520

1.(b)A lift ascends from rest with constant acceleration f until it reaches a speed v. Itcontinues at this speed for t 1 seconds and then decelerates uniformly to rest withdeceleration f.The total distance ascended is d, and the total time taken is t seconds.(i)Draw a speed-time graph for the motion of the lift.1(ii) Show that v f t t1 .24d.(iii) Show that t1 t 2 f(i)v512 t t1 (ii)f (iii)d 12t1 t t1 12v v 12 f t t1 t t1 14 t t1 v t1v 14 t t1 v55or d 12 t t1 v 5 12 t 12 t1 t1 vd 12 t t1 12 f t t1 4d t 2 t12f t1 t 2 4dfPage 35530

2.(a)Two cars, A and B, travel along two straight roadswhich intersect at an angle .BCar A is moving towards the intersection at auniform speed of 9 m s 1 .Car B is moving towards the intersection at auniform speed of 15 m s 1 . AAt a certain instant each car is 90 m fromthe intersection and approaching the intersection.Blank(i)Find the distance between the cars when B is at the intersection.(ii)If the shortest distance between the cars is 36 m, find the value of .(i)(ii)AB 90 90 9 15 36 m55 VA 9 i VB 15cos i 15 sin j5 V AB V A V B 9 15 cos i 15 sin j5 V AB AB 9 15 cos 0 9 cos 1 . 53·13 . 15 55Page 4Page25

2(b)An aircraft P, flying at 600 km h 1 , sets outto intercept a second aircraft Q, which is a distanceaway in a direction west 30 south, and flyingdue east at 600 km h 1 .Find the direction in which P should flyin order to intercept Q.P30 600Q600 VP 600 cos i 600 sin j5 VQ 600 i5 VPQ VP VQ 600 cos 600 i 600 sin j600 sin 600 cos 6003 sin cos 1tan 30 553sin 2 cos2 2 cos 1 3 1 cos2 cos2 2 cos 10 4 cos2 2 cos 21cos 2 60 W 60 S or S 30 WPage 5525

3.(a)A particle is projected from a point on horizontal ground.The speed of projection is u m s 1 at an angle to the horizontal.The range of the particle is R and the maximum height reached by the particle isR.4 3(i)(ii)(i)2u 2 sin cos .gFind the value of .Show that R u sin .t 12 gt 2 0t 2u sin g5R ucos .t ucos 2u sin g2u 2 sin cos g(ii)t1 R4 32u 2 sin cos 4g 3cos 2 35u sin g5 u sin .t1 12 gt12 u sin g u sin u sin g 2 g 25 sin 12 sin tan 13 30 55Page 625

3(b)1to the horizontal.2A particle is projected up the plane with initial speed u m s 1 at an angle to theinclined plane.A plane is inclined at an angle tan 1The plane of projection is vertical and contains the line of greatest slope.Find the value of that will give a maximum range up the inclined plane.rj 0u sin t 12 g cos t 2 t 502u sin u 5 sin g cos g5R u cos t 12 g sin t 2 u2 5cos sin 12 sin 2 g u2 5sin 2 sin 2 2g dR u 2 5 2 cos 2 2 sin cos d 2gdR 0d 5 2 cos 2 2 sin cos 02 cos 2 sin 2 tan 2 5 2 31·7 Page 7525

4.(a)Two particles of masses 6 kg and 7 kg are connectedby a light inextensible string passing over a smoothlight fixed pulley which is fixed to the ceiling of a lift.The particles are released from rest.Find the tension in the string(i) when the lift remains at rest(ii) when the lift is rising vertically with constantgacceleration .8(i)g13g 7 g T 7 f 8 g T 6 g 6 f 8 f 559g104T 6g 6 f T 75T 6g 6 f84 gT or 63·3213(ii)657g T 7 fT 6g 6 ff 66g8189 gor 71·2426Page 8525

4(b)A light inextensible string passes over a smoothfixed pulley, under a movable smooth pulley ofmass m3 , and then over a second smooth fixed pulley.A particle of mass m1 is attached to one end ofthe string and a particle of mass m 2 is attachedto the other end.m1m3m2The system is released from rest.Find the tension in the string in terms of m1 , m2 and m3 .T m1 g m1 p5T m2 g m 2 q5m3 g 2T m3 m3 g 2T m3212 p q T T g g m2 m1 552m1 m2 m3 g 4m1 m2T m2 m3T m1 m3T 2m1 m2 m3 g4m1 m2 m3 g T m1 m3 m2 m3 4m1 m2 T 4m1 m2 m3 gm1 m3 m2 m3 4m1 m2or4g114 m1 m2 m3Page 9525

5.(a)A smooth sphere A, of mass 3m, moving with speed u, collides directly with asmooth sphere B, of mass 5m, which is at rest.The coefficient of restitution for the collision is e. Find(i) the speed, in terms of u and e, of each sphere after the collision(ii)the value of e if the magnitude of the impulse imparted to each sphere as aresult of the collision is 2mu.PCMNEL3m u 5m(0) 3mv1 5mv 2v1 v 2 v1 v2 I 2mu e u 0 u 3 5e 83u 1 e 85555mv 2 015 mu 1 e 816 mu 15 mu 1 e e 1155205Page 10

5(b)A ball is dropped on to a table and it rises after impact to one-quarter of the heightof the fall.(i) Find the value of the coefficient of restitution between the ball and the table.If sheets of paper are placed on the table the coefficient of restitution decreases bya factor proportional to the thickness of the paper. When the thickness of thepaper is 2·5 cm it rises to only one-ninth of the height of the fall.(ii) Find the value of the coefficient of restitution between the ball and thisthickness of paper.(iii) What thickness of paper is required in order that the rebound will be onesixteenth of the height of the fall?(i)v 2 u 2 2asv 2 0 2 gh5v 2 ghv 2 u 2 2as 2 h 0 e 2 gh 2 g 4 e (ii)12v 2 u 2 2as 0 e1 2 gh1 e1 (iii)5 2 h 2g 1 9 135v 2 u 2 2as 0 e2 2 gh2 e2 2 h 2g 2 16 145e k 2·5 e1 11 k 2·5 23 k 0·6e k x e2 x 103511 0·6 x 24or 3·33 cm.Page 115305

6.(a)A rectangular block of wood of mass 20 kg and height 2 m floats in a liquid.The block experiences an upward force of 400d N, where d is the depth, in metres,of the bottom of the block below the surface. Find(i)value of d when the block is in equilibrium(ii)the period of the motion of the block if it is pushed down 0·3 m from theequilibrium position and then released.400 d400 (d x)2m2md20 g(i )20 g400 d 20 g d 0·49(ii)d xF 20 g 400 d x 400 xa 5F 20 xm 20 or 2 5T 52 552 2 5or 1·4 s.5205Page 12

6(b)AA vertical rod BA, of length 4l, has one end Bfixed to a horizontal surface with the other end Avertically above B. The ends of a light inextensiblestring, of length 4l, are fixed to A and to a point C,a distance 2l below A on the rod.4lCA small mass m kg is tied to the mid-point of thestring. It rotates, with both parts of thestring taut, in a horizontal circle with uniform angularBvelocity .(i) Find the tension in each part of the string in terms of m, l and .(ii) At a given instant both parts of the string are cut.Find the time (in terms of l ) which elapses before the mass strikes thehorizontal surface.AT1α4lT2CmgB(i) 60 5T1 cos 60 T2 cos 60 mg5T1 T2 2mgT1 sin 60 T2 sin 60 mr 25 m 3 2T1 T2 2m 2 m g T1 m 2 gT225s ut 12 at 2(ii)3 0 12 gt 256 g5 t 5Page 13305

7.(a)Two forces 5 N and 12 N are inclined at anangle as shown in the diagram.5N They are balanced by a force of 15 N.12 NFind the acute angle .5 15 N121515 2 12 2 5 2 2 12 5 cos 5, 5225 144 25 120 cos cos 56 0·46671205 117 ·82 62·18 55Page 14205

7.(b)Two uniform rods AB and BC, of length 1 and weight W, are hinged at B and restin equilibrium on a smooth horizontal plane.A weight W is attached to AB at a distance b from A as shown in the diagram.A light inextensible string AC of length 2q prevents the rods from slipping.(i)Find the reaction at A and the reaction at C.(ii)Show that the tension in the string isq 1 b W2 1 q2.BR1AbWR2WαWC2qR 2 2q W 12 q W 32 q W b cos (i)cos q5R2 2q W 12 q W 32 q W bq R2 (ii)W 2 b 25R1 R2 3W5R1 5W 4 b 2R2 q W 12 q T sin W 2 b q Wq T 1 q222W 1 b q T 1 q22W 1 b qT 2 1 q2 5 5305Page 155

8.(a)Prove that the moment of inertia of a uniform circular disc, of mass m and1radius r, about an axis through its centre perpendicular to its plane is m r 2 .2Let M mass per unit areamass of element M 2 x dx moment of inertia of the element M 2 x dx x 2rmoment of inertia of the disc 2 M 0 x 3 dx55r x4 2 M 4 0 M 5r42 12 m r 25205Page 16

8.(b)A uniform circular lamina, of mass 8m and radius r,can turn freely about a horizontal axis through Pperpendicular to the plane of the lamina.Particles each of mass m are fixed at four points whichare on the circumference of the lamina and which arethe vertices of square PQRS.The compound body is set in motion.(i)(ii)FindPQSRthe period of small oscillations of the compound pendulumthe length of the equivalent simple pendulum.QmgPRmgmg8mgSmg(i)Mgh 8mg r mg r mg r mg 2r 12 mgrI 8m r 8m r m r 2 212225,5 mr 22 m 2r 25,5 20 mr 2T 2 IMgh5r20 mr 2 2 2 3g12 mgr(ii)2 55rL 2 3ggL 5r35Page 175305

9.(a)V1 cm3 of liquid A of relative density 0·8 is mixed with V 2 cm3 of liquid B ofrelative density 0·9 to form a mixture of relative density 0·88.The mass of the mixture is 0·44 kg.Find the value of V1 and the value of V2 .m A mB mM800 V1 900 V2 880 V1 V2 520V2 80V1V2 4V1880 V1 V2 0·4455880 V1 4V1 0·44V1 0·0001 m 3 or 100 cm 35V2 0·0004 m 3 or 400 cm 35255Page 18

9(b)Liquid C of relative density 0·8 rests on liquid D of relative density 1·2 withoutmixing. A solid object of density floats with part of its volume in liquid D andthe remainder in liquid C. 2aThe fraction of the volume of the object immersed in liquid D is.aFind the value of a.W BC BD5 V1 V2 g 800V1 g 1200V2 g5, 5 800 V1 1200 V2V21 V1 V2 V1 1V2 11200 1 8005 8001200 800 800400 a 40055Page 19255

10.(a)Ifx2dy 7 0dxand y 1 when x 7, find the value of y when x 14 .(b)(c)A particle starts from rest at O at time t 0. It travels along a straight line withacceleration 24t 16 m s 2 , where t is the time measured from the instant whenthe particle is at O. Find(i)its velocity and its distance from O at time t 3(ii)the value of t when the speed of the particle is 80 m s 1 .Water flows from a tank at a rate proportional to the volume of water remaining inthe tank. The tank is initially full and after one hour it is half full.After how many more minutes will it be one-fifth full?x2(a) dy 7dx dy 7 x 2 dxydy 1 y y17 14x 2 dx714 1 7 x 7 1 1 7 14 7 1 y 7 1 14 y 1·55y 1 51055Page 205

dv dt(b) (i)v 024t 163 dv 0(24t - 16) dt v 12t 2 16t 30v 60 m s 1 s03 ds 05(12t 2 - 16t) dt s 4t 3 8t 2 30s 36 m55v 12t 2 16t(b) (ii)80 12t 2 16t3t 2 4t 20 0 3t 10 t 2 0 t dVdt(c ) k V ln 2 ln V k V 10t 205dt k ln 2 or 0·693 V ln 5 ln V 55 kV1dV V1V210s35 ktln 5 2·322 hln 2t1 t 1 79·3 min555Page 215205

Blank PagePage 22

Blank PagePage 23

Blank PagePage 24

Blank Page

Marking Scheme Higher Level Design and Communication Graphics Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate 2013 Marking Scheme Applied Mathematics Higher Level. Note to teachers and students

Related Documents: