MATHEMATICS Higher Grade - Paper I

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1 Higher Mathematics - Practice Eaminatin D Please nte the frmat f this practice eaminatin is different frm the current frmat. The paper timings are different and calculatrs can be used thrughut. MATHEMATICS Higher Grade - Paper I Time allwed - hurs Read Carefull. Full credit will be given nl where the slutin cntains apprpriate wrking.. Calculatrs ma be used.. Answers btained b readings frm scale drawings will nt receive an credit. Pegass 005

2 FORMULAE LIST The equatin ( g f c) g + f + c 0 represents a circle centre ( g, f ) and radius The equatin ( a) + ( b) r represents a circle centre ( a, b ) and radius r. Scalar Prduct: a. b a b cs θ, where θ is the angle between a and b. r a. b ab+ ab + ab where a a a a and b b b b Trignmetric frmulae: ( ) ( ) sin A± B sin AcsB ± cs AsinB cs A± B cs AcsB m sin AsinB csa cs A sin A cs A sin A sina sin Acs A Table f standard derivatives: f ( ) f ( ) sin a a cs a cs a a sin a Table f standard integrals: f ( ) f ( ) d sin a cs a a cs a + a sin a + C C Pegass 005

3 All questins shuld be attempted. The parallelgram PQRS has three f its vertices as P(,5), Q(-,-) and R(,-). (a) Establish the crdinates f the furth verte S. () (b) Hence find the equatin f the diagnal QS. () d. Find, given that d ( ) cs (4). Part f the graph f f () is shwn in the diagram. Sketch the graph f all the relevant pints. f ( + ), shwing (,-) 4 () 4. The functins f and g are defined n suitable dmains and are given as f ( a) a b and g ( a) 5( a + b), where b is a cnstant. Find an epressin fr g ( f (a)) in its simplest frm. () 5. A recurrence relatinship is defined as U 0 5U 6 with U 8 n+ n + (a) Find the limit (L) f this sequence. () (b) Given that U L 6, find n. () n 0 Pegass 005

4 6. A small plastic blck has a rectangular whle punched thrugh it as shwn. All the lengths are in centimetres. + (a) Given that the vlume f plastic in the blck + 6 is 80 cubic centimetres, shw, that t find, the equatin must be slved. () (b) Slve the equatin and shw that it has nl ne real rt. Hence state the utside dimensins f the blck. (4) 7. Find the equatin f the tangent t the curve at the pint where the curve crsses the -ais. (5) 8. A lgarithmic equatin is given as lg ( ), where. Slve this equatin fr. (4) 6 9. The diagram shws a sketch f the graph f + 8. A B (a) Find algebraicall the crdinates f the tw statinar pints A and B. (5) (b) Find the equatin f the line AB and hence prve that this line passes thrugh ne f the pints where the curve crsses the -ais. (4) Pegass 005

5 0. A circle has as its equatin Find the equatin f the tangent at the pint P(-,5) n the circle. (4). The gradient f the tangent at an pint (, ) n a curve is d 8 given b +. d Given that the pint (, 4) lies n this curve, epress in terms f. (4). The diagram features tw right-angled triangles psitined as shwn. QS PT, ST, PQ 5 and SR 5. Q R 5 P S T 5 (a) B appling Pthagras' Therem t PQT, frm an equatin in and slve it t find. () (b) Hence calculate the length f QR. () (c) Shw that the eact value f Sin PQR is given as Sin PQR 7 6. (4) 8 [ END OF QUESTION PAPER ] Pegass 005

6 Higher Mathematics Practice Eam D Marking Scheme - Paper. (a) Fr finding S(5,7) (b stepping-ut... r equiv.) [ mark ] (b) Fr gradient... 5 m 0 QS 6 Fr equ. 7 5 ( 5), (r equiv.), t 5 4 [ marks ]. Fr... 4 ( + ) ( + )... () 8 ( fr 4 and pwer f.. (), fr.. ()) Fr ( sin ) sin... () ( fr sin... (), fr ( ) +... () ) [ 4 marks ]. Fr knwing t reflect in -ais Fr knwing t mve units t the left Fr cmpleting the sketch and marking the tw pints ( ) [ marks ] 4. Fr g f (a) 5 ( a b) + 0b (r equivalent) Fr simplifing t answer... g ( f (a)) 5a [ marks ] 5. (a) Fr b 6 L (r equiv.) a 0 5 [ mark ] (b) Fr realising U n 8 (stated r implied) and starting with U.. Fr U 0. 5 (8) Fr successive lines f wrking until U 4 8 n 4 [ marks ] 6. (a) Fr ( + )( + 6) 6( + ) 80 (r eqiv.) Fr simplifing t ans. given [ marks ] (b) Fr deciding n methd e.g. snthetic divisin Fr finding that gives a remainder 0. Fr using the qutient and shwing that it has n real rts (i.e. b 4ac < 0 ), n mre rts Fr final ans.... blck 8 4 4cm [ 4 marks ] 7. Fr btaining -intercept (0,5) Fr knwing t differentiate fr gradient d Fr diff. crrectl t... d + m Fr sub. 0 int derivative t find that m Fr sub. m and (0,5) int equ. f line t ans [ 5 marks ] Pegass 005

7 8. Fr lifting t pwer. lg ( ) Fr remving lgs. ( ) Fr squaring bth sides. Slving t answer (( )( ) 0 [ 4 marks ] 9. (a) Fr knwing t differentiate and slve t zer d Fr differentiating crrectl t d Fr slving t 4 r Fr cmpleted pints ( mark each) A(-4,8) and B(,-8)... () [ 5 marks ] (b) Fr gradient f AB... m 6 Fr equ. f AB ( + 4) 6 6 r equiv. Fr finding where line crsses -ais i.e (-,0) Fr sub. - int equ. f curve t shw that 0, etc. (i.e. prving that (-,0) lies n curve and the line) [ 4 marks ] 0. Fr drawing ut the centre C(-,) Fr finding the gradient f the radius m CP Fr gradient f tangent m tan Fr ans. ( + ) + (r equiv.) 5 [ 4 marks ]. Fr knwing t integrate fr " " Fr integrating t C (r equiv.) 8 Fr knwing t sub. t find C.. i.e. 4 () + C Fr C (r equiv.) [ 4 marks ]. (a) Fr... ( + ) + ( 5) Fr answer [ marks ] (b) Fr... QR + ( 5) QR 4 r 6 [ mark ] (c) Fr knwing t use... sin PQR sin( a + b) sin acsb + csasin b (r equiv.) 5 5 Fr lifting crrect values t (r equiv.) () Fr simplifing t sin PQR 7 6 (r equiv.) Fr ratinalising denm. t final ans... Sin PQR [ 4 marks ] Ttal : 55 marks Pegass 005

8 Higher Mathematics - Practice Eaminatin D Please nte the frmat f this practice eaminatin is different frm the current frmat. The paper timings are different and calculatrs can be used thrughut. MATHEMATICS Higher Grade - Paper II Time allwed - hurs 40 mins Read Carefull. Full credit will be given nl where the slutin cntains apprpriate wrking.. Calculatrs ma be used.. Answers btained b readings frm scale drawings will nt receive an credit. Pegass 005

9 FORMULAE LIST The equatin ( g f c) g + f + c 0 represents a circle centre ( g, f ) and radius The equatin ( a) + ( b) r represents a circle centre ( a, b ) and radius r. Scalar Prduct: a. b a b cs θ, where θ is the angle between a and b. r a. b ab+ ab + ab where a a a a and b b b b Trignmetric frmulae: ( ) ( ) sin A± B sin AcsB ± cs AsinB cs A± B cs AcsB m sin AsinB csa cs A sin A cs A sin A sina sin Acs A Table f standard derivatives: f ( ) f ( ) sin a a cs a cs a a sin a Table f standard integrals: f ( ) f ( ) d sin a cs a a cs a + a sin a + C C Pegass 005

10 All questins shuld be attempted. Triangle PQR has as its vertices P(-6,-), Q(,-7) and R(5,9). R P S Q (a) Find the equatin f the median RS. () (b) Shw that this median is at right-angles t side PQ. () (c) What tpe f triangle is PQR? (). The functins f and g, defined n suitable dmains, are given as f () ( + ) and g ( ) k (6 ). (a) Given that f ( 4) g(5), find the value f k. () (b) Find an epressin fr g ( f ( )). () (c) Find when g ( f ( )) g( ). (). Slve algebraicall the equatin sin + 4 sin 0, 0 < < 60. (6) 4. Evaluate ( ) d 0 (4) Pegass 005

11 5. The height, H metres, after t secnds, f an arrw fired verticall upwards frm a t bw is given b the equatin H 40t 6t. (a) Find the velcit f the arrw after and after secnds. Eplain the significance f the change f sign between these tw velcities. (4) (b) Find the maimum height t which the arrw rises. () 6. In a cmmercial laundr a certain material is washed in a slutin f water and bleach. This washing slutin is kept in a large vat and is cntinuusl reccled thrugh the cleaning machines.the slutin is renewed each da. At the beginning f each da fresh bleach and clean water are mied in the vat. The initial strength f the bleach is 0 units/galln. It has been fund, that in ever hur f use, the bleach lses % f its strength. (a) Calculate the strength f the bleach after 4 hurs f use. () Give ur answer crrect t tw decimal places. (b) It is knwn that if the strength f the bleach drps belw 0 units/galln the slutin will n lnger be an efficient cleaning agent. The cmpan decide t add a quantit f fresh bleach t the slutin ever 4 hurs. This has the immediate effect f raising the verall strength f the bleach b 6 units/galln. Cmment n whether r nt this plic is effective. (4) Yur answer must be accmpanied with the apprpriate wrking. 7. A rectangular water tank has a vlume f 7 cubic metres. Its base has dimensins metres b + metres as shwn. h (a) (b) Frm this infrmatin, and frm the fact that V h, l b + write dwn a simple equatin, in,fr the height, h, f the tank. () Hence shw that this equatin can be written in the frm h + ( h ) + (7 h) 0 () (c) Shw that if h this equatin has equal rts. () (d) Using this value fr h, slve the equatin fr and hence state the dimensins and vlume f the tank. () Pegass 005

12 8. The pints A(-,0) and B(,) bth lie n the circumference f a circle, centre C, as shwn. (a) Find the equatin f the perpendicular bisectr f the chrd AB. A B () (b) Hence find the crdinates f the centre C f the circle, C, given that the line with equatin 5 passes thrugh C. () 5 (c) Find the equatin f the circle. () 9. The diagram shws the parablas + 4 and +. + Q P + 4 (a) Establish the crdinates f the tw intersectin pints P and Q. () (b) Find the area enclsed between the tw curves. (4) (c) Find the equatin f the line PQ () (d) Hence shw that the line PQ splits the shaded area in the rati :. (4) Pegass 005

13 0. One f the cncrete supprts fr a small rad bridge is clindrical in shape. Its ttal surface area, A s, is 4 square metres. (a) Given that the frmula fr the surface area, A s, f a slid clinder, is given as A s π r + π r h, shw that, fr this clinder, h r, where h is its height π r and r is the radius f its base. () (b) Hence shw that the vlume f this clinder can be epressed as V ( r) r π r. () (c) Find the area f the base when the vlume f the clinder is at a maimum. (5). A cubid with dimensins cm b 4cm b 4cm is placed relative t a set f crdinate aes as shwn in the diagram. F has crdinates (, 4, 4 ) M is the mid-pint f OA and N is the mid-pint f AB. z D G E F(, 4, 4 ) C O M A N B (a) Write dwn the crdinates f M and N. () (b) Calculate the size f angle MFN. (6) [ END OF QUESTION PAPER ] Pegass 005

14 Higher Mathematics Practice Eam D Marking Scheme - Paper. (a) Fr crdinates f S... S(-,-5) Fr gradient f median... m RS Fr equ. f median... ans... [ marks ] (b) Fr selecting a strateg i.e. m m Fr prving... i.e. right angled [ marks ] (c) Fr ans... issceles [ mark ]. (a) Fr f(4) 9 and g(5) 7k (r equiv.) Fr equating and slving t answer... k [ marks ] g ( ( + )) ( + ) (r equiv.) (b) Fr [ ] Fr simplifing t answer g ( f ( )) + [ marks ] (c) Fr... + ( ) (r equiv.) Fr slving t answer... i.e ( )( ) 0 r [ marks ]. Fr sin + 4(sin cs ) 0 Fr sin + 8sin cs sin ( + 8cs ) 0 Fr sin 0 r cs 8 Then fr... sin 0 then 80 Then fr... cs 8 then 0 r 48 0 ( each)... () [ 6 marks ] 4. Fr ( ) d 0 then 0 (r equivalent) then 7 8 ( 0) Fr answer 0 [ 4 marks ] Pegass 005

15 5. a) Fr knwing t differentiate fr velcit Fr substituting and int derivative... v ( t) 40 t Fr answers 8 m/s and -4 m/s Fr eplanatin i.e rising then falling, different directin, etc. [ 4 marks ] (b) Fr knwing t slve derivative t zer Fr t 5 maimum 4 Fr sub. in H... t answer H 5 metres r [ marks ] Fr slving 40t 6t 0 5 Fr t half-wa between rts 0 and, t 5 s 4 Fr sub. in H... t answer H 5 metres 6. (a) Fr crrect multiplier i.e Fr Strength (0 88) units (pupils ma use fur lines f wrking) [ marks ] 4 (b) Fr setting up recurr... U n+ (0 88). U n + 6 (r equiv.) Fr setting ut at least fur lines f calculatins Fr knwing t lk at lwer value befre adding 6 Fr discvering ans... that b end f the furth ccle (i.e. 6 hurs) strength is 9 64 ( immediatel befre 6 is added )... plic is nt acceptable (fr nl lking at upper value and... ans..k., fundamental errr, 4 ) [ 4 marks ] ** nte : limit cannt be applied as after the 6 ccles ( 4 hrs ) situatin is re-set. 7. (a) Fr ans. h V 7 l b ( )( + ) (r equiv.) [ mark ] (b) Fr knwing t change twards standard quad. frm Fr... h + h h Fr arranging t ans... h + ( h ) + (7 h) 0 [ marks ] (c) Fr knwing t sub. h int equatin Fr stating r shwing its a perfect square ( 5) 0 and eplaining it has nl ne rt (pupils ma use discriminant... fr equal rts b 4ac 0... () [ marks ]... fr shwing () (d) Fr slving... ( 5) 0... ans 5 metres Fr... dimensins 4 5 metres and V 8 m [ marks ] Pegass 005

16 8. (a) Fr m AB then m perp - Fr mid-pint... M(-,) Fr crrect gradient and pint t.. ( + ) () [ marks ] (b) Fr knwing perp. bisec. passes thrugh centre (stated r implied) Fr knwing t slve as a sstem + 5 Fr slving t answer... C(,-) [ marks ] (c) Fr finding r b Pth. r nticing that B is vert. abve C, r 5 Fr knwing t sub. r and C(,-) int ( a) + ( b) r... () Fr final answer... ( ) + ( + ) 5 [ marks ] 9. (a) Fr setting up the sstem (r equiv.) Fr simplifing t.. ( ) 0 Fr slving t and then t P(-,7) and Q(,5) [ marks ] (b) Fr setting-up crredt integral i.e. ( ) d Fr integrating t... A [ + + 9] Fr substituting in numbers Fr crrect answer Area square units. [ 4 marks ] (c) Fr m then... 7 ( + ) + 9 (r equiv.) [ mark ] (d) Fr crrect int. i.e ( + 9) ( + ) d (r equiv.) Fr simplifing and integrating t A Fr substituting t answer... A units squared Fr prving : b 0... (r equiv.) ( pupils ma use line and ther curve, giving A 0, etc. ) [ 4 marks ] Pegass 005

17 0. (a) Fr replacing A s with 4 t... 4 π r + π r h Fr making h the subject t answer [ marks ] (b) Fr writing V π r h π r π r r Fr simplifing t given answer [ marks ] (c) Fr knwing t differentiate Fr diff. crrectl and knwing t slve t zer... π r 0... () 4 4 Fr answer... fr ma vlume.. r r r π π Fr sub. in A base 4 π r... π. 4 sq. metres. π [ 5 marks ]. (a) Fr... M(6,0,0) and N(,,0) [ mark ] (b) Fr selecting crrect vectrs i.e. FM and FN 6 0 Fr FM 4 and FN 4 4 Fr scalar prduct FM FN 4 Fr bth magnitudes 68 and 0 (r equiv.) 4 Fr cs θ (r equiv.) 68 0 Fr ans. angle MFN 49 4 [ 6 marks ] Ttal : 8 marks Pegass 005

MATHEMATICS Higher Grade - Paper I

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