Coimisiún na Scrúduithe Stáit State Examinations Commission LEAVING CERTIFICATE 2010 MARKING SCHEME APPLIED MATHEMATICS HIGHER LEVEL

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1 Coimisiún n Scrúduithe Stáit Stte Emintions Commission LEAVING CERTIFICATE 00 MARKING SCHEME APPLIED MATHEMATICS HIGHER LEVEL

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3 Generl Guidelines Penlties of three types re pplied to cndidtes' work s follows: Slips - numericl slips S(-) Blunders - mthemticl errors B(-) Misreding - if not serious M(-) Serious blunder or omission or misreding which oersimplifies: - wrd the ttempt mrk only. Attempt mrks re wrded s follows: (tt ). The mrking scheme shows one correct solution to ech question. In mny cses there re other eqully lid methods. Pge

4 . () A cr is trelling t uniform speed of trffic light turning red 98 m hed. m s when the drier notices Find the minimum constnt decelertion required to stop the cr t the trffic light, (i) if the drier immeditely pplies the brke if the drier hesittes for second before pplying the brke. (i) 96 f u 0 96 f fs f 98 m s s ut s ft s 0 u 0 0 fs f 68 f 98 f or 6.7 m s 0 Pge

5 . (b) A prticle psses P with speed 0 uniform ccelertion. m s nd moes in stright line to Q with In the first second of its motion fter pssing P it trels m. In the lst seconds of its motion before reching Q it trels of PQ. 0 Find the distnce from P to Q. P X Y Q t= s= t= s= 0 PQ PX s ut 0 f ft f 0 f PY 7 0 s ut PQ 0 t t t ft 0t PQ s ut PQ 0 ft t t t 60t t 6t 7 0 PQ t 60t 60 t 80t 60 0 t 0t 0t PQ m 6 0 Pge

6 . () Two prticles, A nd B, strt initilly from points with position ectors 6 i j nd i j respectiely. The elocities of A nd B re constnt nd equl to i j nd i 7 j respectiely. (i) Find the elocity of B reltie to A. Show tht the prticles collide. (i) V V V A B BA i j i 7 j VB VA i j mgnitude or direction 7 m s or slope Est 7.8 South R R R A B AB 6 i j i j RA RB i j or i j or slope direction Est 7.8 South The prticles collide 0 Pge

7 (b) When motor-cyclist trels long stright rod from South to North t constnt speed of. m s the wind ppers to her to come from direction North Est. When she returns long the sme rod t the sme constnt speed, the wind ppers to come from direction South Est. Find the mgnitude nd direction of the elocity of the wind. V V M WM 0 i. j i j V W VWM VM i. j V V M WM 0 i. j y i y j V W VWM V y i M y. j V W V W y nd. y. y. V W mgnitude.. i 0 j m s direction West 0 Pge

8 . () In room of height 6 m, bll is projected from point P. Q P is. m boe the floor m The elocity of projection is 9.8 t n ngle of to the horizontl. m s P. m The bll strikes the ceiling t Q without first striking wll. Find the length of the stright line PQ. 9.8 sin. t.9t.9 9.8t.9 0 t gt t 0 t r i cos. t PQ or 0.96 m 0 Pge 6

9 (b) A prticle is projected up n inclined plne with initil speed 80 m s. The line of projection mkes n ngle of 0 with the inclined plne nd the plne is inclined t n ngle to the horizontl. The plne of projection is erticl nd contins the line of gretest slope. The prticle strikes the plne t n ngle of tn. Find (i) the lue of the speed with which the prticle strikes the plne. (i) 0 r j 0 80sin 0.t t 80 g cos g cos.t i 80 80cos0 g sin g cos 0 80 tn j 80sin 0 0 g cos g 80 cos tn 0 i j 0 80 tn tn. i j 0 0 speed or.9 m s 0 Pge 7

10 . () Two prticles of msses 0. kg nd 0. kg re connected by light inetensible string pssing oer smll, smooth, fied pulley. The system is relesed from rest. Find (i) the tension in the string the speed of the two msses when the 0. kg mss hs descended.6 m (i) 0.g T 0. f T 0.g 0. f 0.0g f 0.9 f 0. T. N u fs 0..6 m s Pge 8

11 (b) A smooth wedge of mss m nd slope º rests on smooth horizontl surfce. Prticles of mss m nd m re plced on the smooth inclined fce of the wedge. m m The system is relesed from rest. º m (i) Show, on seprte digrms, the forces cting on the wedge nd on the prticles. Find the ccelertion of the wedge. (i) S R S º R T mg mg mg ( ii) m mg cos R R mf sin mg mf m mg cos S S mf sin mg mf m S sin Rsin mf mg mf mg mf mg mf mf 8mf f g or.67 m s Pge 9

12 . () A sphere, of mss m nd speed u, impinges directly on sttionry sphere of mss m. The coefficient of restitution between the spheres is e. (i) Find, in terms of u nd e, the speed of ech sphere fter the collision. If e, find the percentge loss in kinetic energy due to the collision. ( i) PCM m u m( 0 ) m m NEL e u 0 u e u e ( ii) e u 6 nd u 6 K.E. before K.E.fter mu m u m 6 76mu or m u m 6 9mu 8 Loss in KE mu 9mu 8 mu 8 Percentge loss in KE mu 8 mu % Pge 0

13 (b) A smooth sphere, of mss m, moing with elocity 6 i j collides with smooth sphere, of mss km, moing with elocity i j on smooth horizontl tble. m km After the collision the spheres moe in prllel directions. 6 The coefficient of restitution between the spheres is e. (i) Find e in terms of k. Proe tht k. (i) PCM NEL 6 m km m km e 6 6 k ek k 6 e k k prllel directions slopes re equl 6 e k 6 k ek k k e k 6 k ek k e k e k k k k k Pge

14 6. () A prticle of mss m kg lies on the top of smooth sphere of rdius m. The sphere is fied on horizontl tble t P. The prticle is slightly displced nd slides down the sphere. The prticle lees the sphere t B nd strikes the tble t Q. Find (i) the speed of the prticle t B P B Q the speed of the prticle on striking the tble t Q. R α mg P Q ( i) mg cos R R m 0 g cos ( ii) m Totl energy t m mg cos g cos mg cos Q m m cos g m g Totl energy t m mg 8g m s m s cos mg B Pge

15 Pge 6 (b) A prticle moes with simple hrmonic motion of mplitude 0.7 m. The period of the motion is s. Find (i) the mimum speed of the prticle the time tken by the prticle to moe from the position of mimum speed to position t which its speed is hlf its mimum lue. s. s. time sin 8 cos 8 cos ) ( m s 8 Period ) ( m m t t t t t ii i

16 7. () One end of uniform ldder, of weight W, rests ginst smooth erticl wll, nd the other end rests on rough horizontl ground. The coefficient of friction between the ldder nd the ground is μ. The ldder mkes n ngle with the horizontl nd is in erticl plne which is perpendiculr to the wll. Show tht person of weight W cn sfely climb to the top of the ldder if 7. 8 tn R W W μ R c R horizontl R R erticl R W R W moments bout c R R W sin W cos W cos 7W tn 7W tn : 7 8 tn Pge

17 7. (b) Two uniform smooth spheres ech of weight W nd rdius 0. m, rest inside hollow cylinder of dimeter.6 m. 0. The cylinder is fied with its bse horizontl. (i) Show on seprte digrms the forces cting on ech sphere. Find, in terms of W, the rection between the two spheres (iii) Find, in terms of W, the rection between the lower sphere nd the bse of the cylinder. A R R B W R R R W θ 0.6 cos s in Sphere A R sin W R W W R (iii) Sphere B R R R R W W W sin W Pge

18 8. () Proe tht the moment of inerti of uniform circulr disc, of mss m nd rdius r, bout n is through its centre perpendiculr to its plne is m r. Let M mss per unit re mss of element M d moment of inerti of the element M d moment of inerti of the disc M r 0 d M r 0 r M m r 0 Pge 6

19 Pge 7 8. (b) An nnulus is creted when centrl hole of rdius b is remoed from uniform circulr disc of rdius. The mss of the nnulus (shded re) is M. (i) Show tht the moment of inerti of the nnulus bout n is through its centre nd perpendiculr to its plne is b M. The nnulus rolls, from rest, down n incline of 0. Find its ngulr elocity, in terms of g, nd b, when it hs rolled distnce. sin 0 Loss in PE Gin in KE nnulus of inerti moment of (i) b g Mg M b M Mg M I Mgh M I b M b b M M d M b b 0 b

20 Pge 8 9. () Stte the Principle of Archimedes. A buoy in the form of hollow sphericl shell of eternl rdius m nd internl rdius 0.8 m flots in wter with 6% of its olume immersed. Find the density of the mteril of the shell. Principle of Archimedes kg m g g B W g g Vg W g g Vg B 0.8

21 9 (b) A uniform rod, of length. m nd weight W, is freely hinged t point P. The rod is free to moe bout horizontl is through P. The other end of the rod is immersed in wter. P m The point P is 0. m boe the surfce of the wter. The rod is in equilibrium nd is inclined t n ngle of Find (i) the reltie density of the rod the rection t the hinge in terms of W. 60 to the erticl. ( i) length of immersed prt. moments bout P : cos60 B sin 60 W sin 60 W W nd B s s W W s s 9 ( ii) W B s W B R W W R W R W Pge 9

22 Pge 0 0. () Sole the differentil eqution y d dy y gien tht y = 0 when = 0. e y e y y C y C y d dy y y y y d dy y d dy y ln 0 0 0, ln 0

23 0 (b) The ccelertion of cyclist freewheeling down slight hill is m s where the elocity is in metres per second. The cyclist strts from rest t the top of the hill. Find (i) the speed of the cyclist fter trelling 0 m down the hill the time tken by the cyclist to trel the 0 m if his erge speed is.6 m s. ( i) 0.00 ln ln ln d d d ln ln e m s d ( ii) distnce erge speed time 0.6 t t. s 0 Pge

24 Mrcnn Breise s ucht fregirt trí Gheilge B chóir mrcnn de réir n ghnáthrát bhronndh r irrthóirí nch ngnóthíonn níos mó ná 7% d iomlán n mrcnn don pháipér. B chóir freisin n mrc bónis sin shlánú síos. Déntr n cinnedh gus n ríomhirecht foin mrc bónis i gcás gch páipéir r leithligh. Is é % n gnáthrát gus is é 00 iomlán n mrcnn don pháipér. Mr sin, bin úsáid s n ngnáthrát % i gcás irrthóirí ghnóthíonn mrc nó níos lú, e.g. 98 mrc % = 9 9 bóns = 9 mrc. Má ghnóthíonn n t-irrthóir níos mó ná mrc, ríomhtr n bóns de réir n foirmle [00 bunmhrc] %, gus n mrc bónis sin shlánú síos. In iond n ríomhirecht sin dhénmh, is féidir úsáid bhint s n tábl thíos. Bunmhrc Mrc Bónis Pge

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