3.1 Vectors. Each correct answer in this section is worth two marks.
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1 3.1 Vectors Paper1SectionA Each correct answer in this section is worth two marks. 3 1.Avectorvisgivenby 2. 6 Whatisthelength,inunits,of3v? A. 7 B. 15 C. 21 D. 49 Key Outcome Grade Facility Disc. Calculator Content Source C 3.1 C NC G16 HSN 135 hsn.uk.net Page 1
2 2.ThepointAhascoordinates (9,7,2)andB(5,5, 1). Whatisthevalueof AB? A. 3 B. 3 C. D Key Outcome Grade Facility Disc. Calculator Content Source C 3.1 C NC G16 HSN Whatisthedistancebetweenthepoints (3, 1, 1)and (2,7, 4)? A. B C. 2 D. 62 Key Outcome Grade Facility Disc. Calculator Content Source B 3.1 C NC G16, G1 HSN 040 hsn.uk.net Page 2
3 3 4.Ifu =k 1,wherek >0anduisaunitvector,determinethevalueofk. 0 A. 1 2 B. 1 8 C. 1 2 D Key Outcome Grade Facility Disc. Calculator Content Source D 3.1 C 0 0 NC G16, G P1 Q15 5.ThepointBhascoordinates ( 3,10, 5)andAB 3 = 9. 5 WhatarethecoordinatesofpointA? A. (0, 1, 10) B. (0, 1, 10) C. ( 6,1,0) D. (6, 1,0) Key Outcome Grade Facility Disc. Calculator Content Source C 3.1 C CN G17 HSN 12 hsn.uk.net Page 3
4 6.Vectorspandqaredefinedbyp =2i kandq=i +j+3k. Find2p qincomponentform. 3 A B C D. 3 3 Key Outcome Grade Facility Disc. Calculator Content Source A 3.1 C NC G17, G18 HSN 035 hsn.uk.net Page 4
5 3 7.Thevectoruhascomponents 0. 4 Whichofthefollowingisaunitvectorparalleltou? A. 3 5 i +4 5 k B. 3i +4k C. 3 7 i k D. 1 3 i +1 4 k Key Outcome Grade Facility Disc. Calculator Content Source A 3.1 C 0 0 NC G P1 Q17 k 8.Thevectoruisgivenby 2k wherek >0isaconstant. 2k Giventhatuisaunitvector,whatisthevalueofk? A. 1 9 B. 1 5 C. 1 5 D. 1 3 Key Outcome Grade Facility Disc. Calculator Content Source D 3.1 C NC G18 HSN 119 hsn.uk.net Page 5
6 Giventhatp = 5,q = 0 andr= 2,express2p q 1 2 rin component form. 1 A B C D Key Outcome Grade Facility Disc. Calculator Content Source C 3.1 C 0 0 NC G P1 Q1 hsn.uk.net Page 6
7 Giventhatu= 0 andv= 2,find3u 2vincomponentform A B C D. 4 5 Key Outcome Grade Facility Disc. Calculator Content Source D 3.1 C 0 0 CN G P1 Q3 hsn.uk.net Page 7
8 11.ABCDEisasquare-basedpyramid,andXisthecentreofthebase. E GiventhatAC 4 = 4 andce 2 = 2,findXE A B C D. 4 5 A B X D C Key Outcome Grade Facility Disc. Calculator Content Source B 3.1 C NC G20, G21 HSN 090 hsn.uk.net Page 8
9 12. Parallelogram OABC is shown below. a A D B O c C ThepointDdividesABintheratio3:1. FindCDintermsofaandc. A. a 1 4 c B. a 1 3 c C. 1 4 c a D. 1 3 c a Key Outcome Grade Facility Disc. Calculator Content Source A 3.1 C CN G21 HSN 116 hsn.uk.net Page 9
10 13.Thediagramshowsasquare-basedpyramidP,QRST. TS, TQand TPrepresent f, g and h respectively. P R h Q S f T g Express RPintermsof f,gandh. A. f +g h B. f g+h C. f g h D. f +g+h Key Outcome Grade Facility Disc. Calculator Content Source B 3.1 C 0 0 NC G22, G P1 Q10 hsn.uk.net Page 10
11 14.GiventhatthepointsS( 4,5,1),T( 16, 4,16)andU( 24, 10,26)are collinear, calculate the ratio in which T divides SU. A. 2:3 B. 3:2 C. 2:5 D. 3:5 Key Outcome Grade Facility Disc. Calculator Content Source B 3.1 C 0 0 CN G P1 Q15 hsn.uk.net Page 11
12 15.ThepointPhascoordinates (4, 3,7)andQ(7, 9,4).ThepointRdivides PQin theratio1:2. FindthecomponentsofPR. A B. C. D Key Outcome Grade Facility Disc. Calculator Content Source B 3.1 C NC G25 HSN 015 hsn.uk.net Page 12
13 3 16. The vectors 1 and 7 Whatisthevalueofk? A. 3 k 2 areperpendicular. 1 B. 3 C. D Key Outcome Grade Facility Disc. Calculator Content Source B 3.1 C NC G26, G27 HSN 019 hsn.uk.net Page 13
14 17. An equilateral triangle of side 3 units is shown.the vectors pandqareasrepresentedinthediagram.what isthevalueofp.q? p A. 9 q B. 9 2 C. 9 2 D. 0 Key Outcome Grade Facility Disc. Calculator Content Source B 3.1 A/B 0 0 CN G26, G P1 Q Ifu = 1 andv= t are perpendicular, what is the value of t? 2t 1 A. 3 B. 2 C. 2 3 D. 1 Key Outcome Grade Facility Disc. Calculator Content Source A 3.1 C 0 0 NC G P1 Q7 hsn.uk.net Page 14
15 19.Thevectorsxi +5j +7kand 3i +2j kareperpendicular. Whatisthevalueofx? A. 0 B. 1 C. 4 3 D Key Outcome Grade Facility Disc. Calculator Content Source B 3.1 C 0 0 CN G27, G P1 Q10 a The vectors 1 and 2a areperpendicular. b 3b Findanexpressionforaintermsofb. A. a =3b 2 B. a = 3 2 b2 C. a = 3 2 b2 1 2 D. a =3b 2 1 Key Outcome Grade Facility Disc. Calculator Content Source B 3.1 C NC G27, G26 HSN 118 hsn.uk.net Page 15
16 21.Fortwovectorsuandv, u =4, v =7andu.v =3. Whatisthevalueofu.(u +v)? A. 7 B. 12 C. 19 D. 44 Key Outcome Grade Facility Disc. Calculator Content Source C 3.1 C NC G29, G16 HSN Giventhata= 4 anda.(a +b) =7,whatisthevalueofa.b? 0 A B C. 6 D. 18 Key Outcome Grade Facility Disc. Calculator Content Source D 3.1 C 0 0 NC G29, G P1 Q17 [ENDOFPAPER1SECTIONA] hsn.uk.net Page 16
17 Paper1SectionB k 1 23.Vectorsuandvaregivenbyu= 1 andv= k 3 wherekisaconstant. 2 k (a)giventhatu.v =2,showthatk 3 3k 2 =0. 2 (b)show that (k +1) is a factor of k 3 3k 2 and hence fully factorise k 3 3k 2. 5 (c)giventhat u =3,findthevalueofk. 2 (d)giventhattheanglebetweenuandvis θ,findtheexactvalueofcos θ. 2 (a) 2 C CN G26 proof AT102 (b) 5 C CN A21 (k +1) 2 (k 2) (c) 2 C CN G16 k =2 (d) 2 C CN G beginmi mi[pd] find scalar product mi[ic] complete proof newline mi[ss]knowtousek = 1 mi[pd] complete evaluation and conclusion mi[ic] start to find quadratic factor mi[ic] complete quadratic factor mi[pd] factorise completely newline mi[ss] find magnitude mi[ic] interpret k newline mi[pd] find magnitude mi[ss] use formula endmi beginmi mi vectu vectv =k times( 1) + 1 timesk 3 + ( 2) timesk mik 3 3k =2andcomplete newline miknowtousek = 1 mi ( 1) 3 3( 1) 2 =0 mi (k +1)(k 2 cdots) mi (k +1)(k 2 k 2) mi (k +1)(k +1)(k 2) newline mi vectu 2 =k 2 +5 =9 mik =2 newline mi vectv = sqrt65 mi cos theta = f rac23sqrt65 endmi hsn.uk.net Page 17
18 24.Twovectorsaandbaregivenbya = Athirdvectorcisdefinedbyc=2a 3b. andb= Findthecomponentsofcand c. 3 3 C CN G20,G16 c = 1 pd: startvectoraddition 2 pd: completevectoraddition 3 pd: computemagnitude , c = WCHSU3Q = hsn.uk.net Page 18
19 25. Triangle ABC has vertices A(4, 0), B(4,16)andC(18,20),asshownin the diagram opposite. MediansAPandCRintersectatthe point T(6, 12). B y R T P Q C O A x (a)findtheequationofmedianbq. 3 (b)verifythattliesonbq. 1 (c)findtheratioinwhichtdividesbq. 2 (a) 3 C CN G7 y 16 = 2 5 (x ( 4)) 2010P1Q21 (b) 1 C CN A6 proof (c) 2 C CN G24 2:1 1 ss: knowandfindmidpointofac 2 pd: calculategradientofbq 3 ic: stateequation 4 ic: substituteinfortandcomplete 5 ss: valid method for finding the ratio 6 ic: completetosimplifiedratio 1 (11,10) orequiv 3 y 16 = 2 5 (x ( 4)) ory 10 = 2 5 (x 11) 4 2(6) +5(12) = =72 5 e.g.vectorapproach ( ) 10 BT =, ( ) 5 TQ = :1 hsn.uk.net Page 19
20 26.Thecubicwithequationy =x 3 4x 2 14x +45 andtheline y = 3x +15 areshowninthe diagram. ThelineandcurveintersectatthepointsP,Q andr. (a)giventhatthex-coordinateofqis2,find P Q R thecoordinatesofp,qandr. O x 7 (b)determinetheratioinwhichqdividesthe line PR. 2 (a) 7 C CN A23 P( 3, 24), Q(2, 9), R(5, 0) AT103 (b) 2 A CN G24 5:3 y 1 ss: knowtoequate 2 ss: usex=2 3 ic: starttofindquadraticfactor 4 ic: completequadraticfactor 5 pd: factorisecompletely 6 ic: interpretpoints 7 pd: statecoordinates 8 ss: chooseapproach(e.g.vectors) 9 ic: interpretratio 1 x 2 4x 2 14x +45 = 3x =0 3 (x 2)(x 2 ) 4 (x 2)(x 2 2x 15) 5 (x 2)(x 5)(x +3) 6 x P = 3andx R =5 7 P( 3,24),Q(2,9),R(5,0) 8 PR ( ) 5 = and ( ) 3 QR = :3 hsn.uk.net Page 20
21 27.ThecirclescentredatAandBhaveequationsx 2 +y 2 +8x +12y +36 =0and x 2 +y 2 4x 4y 28 =0respectively. (a)writedownthecoordinatesofaandb. 2 (b) Show that the circles touch externally. 4 (c)thecirclestouchatpointc. (i)findtheratioinwhichcdividesab. B O (ii) Hence find the coordinates of C. 4 x C A y (a) 2 C CN G9 A( 4, 6), B(2, 2) AT085 (b) 4 B CN G14 proof (c) 4 C CN G24,G25,G20 (i)2:3,(ii)c( 8 5, 14 5 ) 1 ic: interpreta 2 ic: interpretb 3 pd: finddistancebetweencentres 4 ic: interpretradius 5 ic: interpretradius 6 ss: comparesumofradiitodistance 7 ic: stateratio 8 pd: findvectorcomponents 9 ss: useparallelvectors 10 pd: processvectors 1 A( 4, 6) 2 B(2,2) 3 AB =10 4 rad A =4 5 rad B =6 6 AB =rad A +rad B sotouch 7 2:3 8 AB ( ) 6 = 8 9 c =a + 2 5AB 10 C( 8 5, 14 5 ) 28. ThepointQdividesthelinejoiningP( 1, 1,0)toR(5,2, 3)intheratio2:1. Find the coordinates of Q. 3 3 C NC G25 (3,1, 2) 2002P1Q2 1 pd: findvectorcomponents 2 ss: useparallelvectors 3 pd: processvectors 1 PR 6 = PQ = 2 3PR 3 Q = (3,1, 2) hsn.uk.net Page 21
22 29.Risthepoint (3, 1,7)andT(8,14, 3).ThepointSdivides RTintheratio3:2. Find the coordinates of S. 3 3 C CN G25 S(6,8,1) WCHSU3Q11 1 pd: findvectorcomponents 2 ss: useparallelvectors 3 pd: processvectors 1 RT 5 = RS = 3 5RT 3 OS = OR + RSleadingto S(6,8,1) 30.A triangle has vertices at the origin, A(8,6,2) A(8,6,2) andb(10,3, 1) asshownin the diagram. M ACisamedianoftriangleOAB,andthe pointmdividesacintheratio2:1. B(10,3, 1) C O Find the coordinates of M. 4 4 C CN G25,G20 M(6,3, 1 3 ) Ex ic: interpretratio 2 ss: use/findadirectedlinesegment 3 ss: usethefactthatcisamidpoint 4 pd: processscalarmultiplication 1 m =c + CM =c + 1 3CA 2 m =c (a c) 3 m = 1 3 (a +b) = som(6,3, 1 3 ) hsn.uk.net Page 22
23 31. VABCD is a pyramid with a rectangular base ABCD. Relative to some appropriate axes, VArepresents 7i 13j 11k ABrepresents6i +6j 6k ADrepresents8i 4j +4k. A KdividesBCintheratio1:3. B Find VK in component form. 3 D V 1 3 K C 1 3 C CN G25,G21,G P1Q ss: recognisecrucialaspect 2 ic: interpretratio 3 pd: processcomponents 1 VK = VA + AB + BK or VK = VB + BK 2 BK 2 = 1 4BC or 1 4AD or 1 or VK 1 = 8 16 hsn.uk.net Page 23
24 32. D,EandFhavecoordinates (10, 8, 15), (1, 2, 3)and ( 2,0,1)respectively. (a) (i)showthatd,eandfarecollinear. (ii)findtheratioinwhichedividesdf. 4 (b)ghascoordinates (k,1,0). GiventhatDEisperpendiculartoGE,findthevalueofk. 4 (a) 4 C CN G23,G24 3:1 2009P1Q22 (b) 4 C CN G27 k =7 1 ss: usevectorapproach 2 ic: comparetwovectors 3 ic: completeproof 4 ic: stateratio 5 ss: usevectorapproach 6 ss: knowscalarproduct =0for vectors 7 pd: starttosolve 8 pd: complete 1 DE 9 = 6 oref 3 = nd column vector and (DE) =3EF 3 DEand EFhavecommonpointand commondirection;henced,eandf are collinear 4 3:1 5 GE 1 k = DE. GE =0 7 9(1 k) +6 ( 3) +12 ( 3) 8 k =7 hsn.uk.net Page 24
25 33. (a) Roadmakers look along the tops of a set C of T-rods to ensure that straight sections of road are being created. Relative to suitableaxesthetopleftcornersofthe B T-rods are the points A( 8, 10, 2), B( 2, 1,1)andC(6,11,5). A Determine whether or not the section of roadabchasbeenbuiltinastraightline. 3 C (b)afurthert-rodisplacedsuchthatdhas coordinates (1, 4, 4). Show that DB is perpendicular to AB. B 3 A D (a) 3 C CN G23 the road ABC is straight 2001 P1 Q3 (b) 3 C CN G27, G17 proof 1 ic: interpretvector(e.g. AB) 2 ic: interpretmultipleofvector 3 ic: completeproof 4 ic: interpretvector(i.e. BD) 5 ss: staterequirementforperpend. 6 ic: completeproof hsn.uk.net Page 25 or 1 e.g. AB 6 = e.g. BC = 12 = 4 3AB or 4 2 AB =3 3 andbc 2 = a common direction exists and a common point exists, so A, B, C collinear 4 BD 3 = AB. BD =0 6 AB. BD = =0 5 AB. BD = AB. BD =0soABisatrightanglesto BD
26 Findthevalueofcforwhichthevectors u = 4 and v = 5 are c 2 1 perpendicular. 3 3 C CN G27,G26 c = 23 OB ss: knowu.v =0 2 pd: computeu.v 3 pd: completeforc 1 u.v = (c 2) =0 3 c = Twovectorsuandvaresuchthat u =7, v =4andu.v =14. Thevectorwisdefinedbyw=2u v. Evaluatew.wandhencestate w. 4 4 B CN G29,G16 w.w =228, w = 228 Ex ss: usedistributivelaw 2 ic: usea.a = a 2 3 pd: substituteandprocess 4 ic: statelength 1 w.w =4u.u +2u.v v.v 2 4 u 2 +2u.v v = units C NC A6 1989P1Q9 4 A/B NC G29,G26 [ENDOFPAPER1SECTIONB] hsn.uk.net Page 26
27 Paper C CN G9,G10,G P2Q8 hsn.uk.net Page 27
28 2. 5 C CN G9,G P1Q12 hsn.uk.net Page 28
29 3. (a) 3 C CN G P2 Q6 (b) 3 C CN G9, G25 (c) 3 A/B CN CGD hsn.uk.net Page 29
30 4. (a) 1 C CN G P1 Q5 (b) 1 C CN G16 5. (a) 1 C CN G P1 Q1 (b) 2 C CN G16 hsn.uk.net Page 30
31 6. 3 A/B CN G P1Q18 7. (a) 2 C CN G P1 Q3 (b) 1 C CN G26 (c) 1 C CN G16 hsn.uk.net Page 31
32 8. (a) 2 C CN G P1 Q5 (b) 2 C CN G16 hsn.uk.net Page 32
33 9. (a) 3 C CR G P2 Q2 (b) 2 C CR G16 (c) 3 C CR G27 (d) 5 C CR G28 hsn.uk.net Page 33
34 10. (a) 3 C CR G P2 Q3 (b) 1 C CR G25 (c) 4 C CR G28 (d) 2 C CR CGD hsn.uk.net Page 34
35 11. (a) 2 C CR G P2 Q3 (b) 7 C CR G28 hsn.uk.net Page 35
36 12. (a) 2 C CR G P2 Q1 (b) 5 C CR G28 (c) 2 C CR CGD hsn.uk.net Page 36
37 13. The diagram shows a cuboid OPQR,STUV relative to the coordinate axes. Pisthepoint (4,0,0),Qis (4,2,0) anduis (4,2,3). MisthemidpointofOR. N is the point on UQ such that UN = 1 3 UQ. z S O M R V y U (4, 2, 3) N T Q (4, 2, 0) P (4, 0, 0) x (a)statethecoordinatesofmandn. 2 (b)expressthevectors VMand VN in component form. 2 (c) Calculate the size of angle MVN. 5 (a) 2 C CN G22,G25 M(0,1,0),N(4,2,2) 2010P2Q1 0 (b) 2 C CN G17 VM = 1, VN 4 = (c) 5 C CN G or1 339rad 1 ic: interpretmidpointform 2 ic: interpretratioforn 3 ic: interpretdiagram 4 pd: processvectors 5 ss: knowtousescalarproduct 6 pd: findscalarproduct 7 pd: findmagnitudeofavector 8 pd: findmagnitudeofavector 9 pd: evaluateangle 1 (0,1,0) 2 (4,2,2) 3 VM 0 = VN = cosm VN = VM. VN VM VN 6 VM. VN =3 7 VM = 10 8 VN = or1 339radsor85 2grads hsn.uk.net Page 37
38 14. Thevectorsp,qandraredefinedasfollows: p =3i 3j +2k,q =4i j+k,r=4i 2j +3k. (a)find2p q+rintermsofi,jandk. 1 (b)findthevalueof 2p q+r. 2 (a) 1 C CN G P1 Q3 (b) 2 C CN G C CN G P1Q6 hsn.uk.net Page 38
39 16. 3 C CN G P1 Q C CN G P1Q1 18. (a) 2 C CN G18, G P1 Q4 (b) 1 C CN G16 hsn.uk.net Page 39
40 19. (a) 3 C CN G18, G P1 Q12 (b) 2 A/B CN G C CN G18,G P1Q3 hsn.uk.net Page 40
41 21. (a) 2 C CN G P2 Q3 (b) 2 C CN G20 (c) 5 C CN G28 hsn.uk.net Page 41
42 22. 3 C CN G20,G P1Q7 23. Vectors p, q and r are represented A B on the diagram shown where angle ADC =30. q r Itisalsogiventhat p =4and q =3. 30 (a)evaluatep.(q +r)andr.(p q). D p C 6 (b)find q +r and p q. 4 (a) 6 B CN G29,G26 6 3, P2Q7 (b) 2 A CR G21,G30 q +r = (b) 2 B CR G21,G30 p q = ( )2 + ( 3 2 )2 1 ss: usedistributivelaw 2 ic: interpretscalarproduct 3 pd: processingscalarproduct 4 ic: interpretperpendicularity 5 ic: interpretscalarproduct 6 pd: completeprocessing 7 ic: interpret vectors on a 2-D diagram 8 pd: evaluate magnitude of vector sum 9 ic: interpret vectors on a 2-D diagram 10 pd: evaluate magnitude of vector difference 1 p.q +p.r 2 4 3cos ( 10 4) 4 p.r =0 5 r 3cos120 6 r = 3 2 and q +r fromdtotheproj.ofaonto DC 8 q +r = p q = AC 10 p q = ( )2 + ( 3 2 )2 ( 2 05) hsn.uk.net Page 42
43 24. Acuboidmeasuring11cmby5cmby7cmisplacedcentrallyontopofanother cuboidmeasuring17cmby9cmby8cm. Coordinates axes are taken as shown. z A B C y x O (a)thepointahascoordinates (0,9,8)andChascoordinates (17,0,8). Write down the coordinates of B. 1 (b) Calculate the size of angle ABC. 6 (a) 1 C CN G22 B(3, 2, 15) 2000 P2 Q9 (b) 6 C CR G ic: interpret3-drepresentation 2 ss: knowtousescalarproduct 3 pd: processvectors 4 pd: processvectors 5 pd: processlengths 6 pd: processscalarproduct 7 pd: evaluatescalarproduct 3 1 B= (3,2,15) treat 2 asbadform 15 2 cosa BC = BA. BC BA BC 3 BA 3 = BC 14 = BA = 107, BC = BA. BC = 7 7 A BC =92 5 hsn.uk.net Page 43
44 25.D,OABCisasquarebasedpyramidasshowninthediagrambelow. z D(2, 2, 6) y C B O M A x Oistheorigin,Disthepoint (2,2,6)andOA =4units. Misthemid-pointofOA. (a) State the coordinates of B. 1 (b)express DBand DM in component form. 3 (c)findthesizeofanglebdm. 5 (a) 1 C CN G22 (4,4,0) 2011P2Q1 2 (b) 3 C CN G20, G22 DB = 2, DM 0 = (c) 5 C CN G ic: statecoordinatesofb 2 pd: statecomponentsofdb 3 ic: statecoordinatesofm 4 pd: statecomponentsofdm 5 ss: knowtousescalarproduct 6 pd: findscalarproduct 7 pd: findmagnitudeofavector 8 pd: findmagnitudeofavector 9 pd: evaluateanglebdm 1 (4,4,0) (2,0,0) cosbdm = 6 DB DM =32 7 DB = 44 8 DM = 40 DB DM DB DM or0 703rads hsn.uk.net Page 44
45 26. The square-based pyramid ABCDT is shown below. T B C A D AlloftheedgesofABCDThavelength4units. (a)findtheexactvalueofcostâc. 2 (b)hencefindtheexactvalueof AT.( AB + AC). 4 (a) 2 B CN G WCHSU3Q3 (b) 4 B CN G29, G ss: usepythagoras 2 ic: complete 3 ss: usedistributivelaw 4 pd: expandscalarproducts 5 ic: complete TAB value 6 ic: complete,usingcostâc 1 AC =4 2 2 letm =midpoint AC soam =2 2 costâc = AM 2 AT = 2 orequivalent 3 AT. AB + AT. AC 4 AT AB costâb+ AT AC costâc 5 4 4cos60 = ,so (a) 4 C CN G P1 Q7 (b) 1 C CN G25 hsn.uk.net Page 45
46 28. 3 C CN G P1Q C CN G P1Q C CN G23,G P1Q4 hsn.uk.net Page 46
47 31. 4 C CN G23,G P1Q6 32. ABCD is a quadrilateral with vertices A(4, 1,3), B(8,3, 1), C(0,4,4) and D( 4,0,8). (a)findthecoordinatesofm,themidpointofab. 1 (b)findthecoordinatesofthepointt,whichdividescmintheratio2:1. 3 (c)showthatb,tanddarecollinearandfindtheratioinwhichtdividesbd. 4 (a) 1 C CN G6, G P2 Q2 (b) 3 C CN G25 (c) 4 C CN G23, G25 hsn.uk.net Page 47
48 33.In the diagram, the circle centred at A has equationx 2 +y 2 10x 6y+9 =0andthecircle centredatbhasequation (x +1) 2 + (y +5) 2 =4. (a)findthecoordinatesofaandb. 2 A (b) Find the shortest distance between the two O circles. x 4 C (c)pointsa,bandcarecollinear,andthecircle centredatctouchestheothertwocircles B externally. (i)giventhatbc =kba,findthevalueof k. (ii)hencefindthecoordinatesofc. (iii) Write down the equation of the circle centred at C. 5 (a) 2 C CN G9 A(5, 3), B( 1, 5) Ex (b) 4 B CN G9, G1 3 (ci) 1 B CN G24 k = 7 20 (cii) 3 B CN G25 C( 11 10, 11 5 ) (ciii) 1 B CN G10 (x )2 +(y )2 =2 25 y 1 ic: interpretcircleequation 2 ic: interpretcircleequation 3 ss: usedistancebetweencentres 4 ic: useradiusformula 5 ic: interpretcircleequation 6 pd: conclusion 7 ic: interpretratio 8 ss: useavectorpathway 9 pd: process 10 pd: complete 11 ic: statecircleequation 1 A(5,3) 2 B( 1, 5) 3 d AB =10 4 radius A = ( 5) 2 + ( 3) 2 9 =5 5 radius B = =3 7 k = c =b + BC 9 BC = 7 20BA = ( 21 ) C( 11 10, 11 5 ) 11 (x )2 + (y )2 =2 25 hsn.uk.net Page 48
49 34. 3 C CN G P1 Q15 hsn.uk.net Page 49
50 35. (a) 3 C CR G P2 Q4 (b) 5 C CR G28 hsn.uk.net Page 50
51 36. (a) 3 C CR G P2 Q5 (b) 4 C CR G16, CGD (c) 5 C CR CGD hsn.uk.net Page 51
52 37. (a) 3 C CR G P2 Q2 (b) 7 C CR G28 hsn.uk.net Page 52
53 38. (a) 2 C CN G P1 Q17 (b) 4 A/B CN G29, G30 hsn.uk.net Page 53
54 39. (a) 3 C CN CGD 1992 P2 Q6 (b) 4 C CN G26, G27 hsn.uk.net Page 54
55 k Vectors u, v and w aregivenby u = 1, v = 2 and w = 4, 3k 5 1 wherekisaconstant. (a)giventhatu +visperpendiculartow,findthevalueofk. 3 (b)hencecalculatethesmallestanglebetweenuandv. 5 (a) 3 C CN G26,G27 k =2 New3.1G6 (b) 5 C CR G28,G26,G ss: knowtosetscalar productto zero 2 pd: expandscalarproduct 3 pd: process 4 pd: findlengthofvector 5 pd: findlengthofvector 6 pd: findscalarproduct 7 ss: usescalarproduct 8 pd: evaluateangle 1 sinceperp., (u +v).w =0 k =k k+5 3k k =2 4 u = ( 1) = 41 5 v = ( 1) 2 + ( 2) = 30 6 u.v =2.( 1) + ( 1).( 2) +6.5 =30 7 cos θ = u.v u v = θ = 31 2 (1d.p.) or0 544rads(3 d.p.) hsn.uk.net Page 55
56 41. A box in the shape of a cuboid is designed with circles of different sizes on each face. The diagram shows three of the circles, where the origin represents oneofthecornersofthecuboid.the z centresofthecirclesarea(6,0,7), B(0,5,6)andC(4,5,0). B FindthesizeofangleABC. 7 A O y x C 5 C CR G17,G16,G P2Q4 2 A/B CR G26,G ss: use BA. BC BA BC 2 ic: statevectore.g. BA 3 ic: stateaconsistentvectore.g. BC 4 pd: process BA 5 pd: process BC 6 pd: processscalarproduct 7 pd: findangle 1 use BA. BC BA BC statedorimpliedby 7 2 BA 6 = BC 4 = BA = 62 5 BC = 52 6 BA. BC =18 7 A BC =71 5 hsn.uk.net Page 56
57 42. The parallelogram PQRS is shown in the diagram below. R v Q S u T P Thevectors u and v representlinesegments QPand QR respectively, and are suchthat u =5, v =2andu.v =5.ThepointTdivides QPintheratio2:3. (a)express RPand RTintermsofuandv. 3 (b)henceevaluate RP. RT. 3 (a) 3 C CN G21, G25 RP =u v, RT = 2 5u v OB (b) 3 B CN G26, G29 RP. RT =7 1 ss: usepathways 2 ss: usepathways 3 ic: interpretratio 4 ss: substituteandexpand 5 pd: usemagnitudes,andu.v 6 pd: complete 1 RP =u v 2 RT = RQ + QT 3 RT = 2 5 u v 4 RP. RT = (u v).( 2 5 u v) 5 u 2 7 5u v + v 2 = Forwhatvalueoftarethevectorsu = t 2 2 andv= 10 perpendicular? 2 3 t 2 C CN G27 t =4 2000P2Q7 1 ss: knowtousescalarproduct 2 ic: interpretscalarproduct 1 u.v =2t 20 +3t 2 u.v =0 t=4 hsn.uk.net Page 57
58 44. A(4,4,10), B( 2, 4,12) and C( 8,0,10) are the vertices of a right-angled triangle. Determine which angle of the triangle is the right angle. 3 3 C CN G P1Q C CN G P1Q4 k k 46.Thevectorsuandvhavecomponents 2 and k respectively. 4 1 Showthatthereisnovalueofkforwhichuandvareperpendicular. 4 4 C CN G27, G26, A17 proof AT042 1 ss: knowtousescalarproduct 2 pd: computescalarproduct 3 ss: knowtousediscriminant 4 ic: conclusion 1 u viffu.v =0 2 u.v =k 2 +2k +4 3 = 12 4 since <0,u.v =0 hsn.uk.net Page 58
59 47. The diagram shows a square-based pyramid of height 8 units. SquareOABChasasidelengthof6units. ThecoordinatesofAandDare (6,0,0) and (3,3,8). Cliesonthey-axis. (a) Write down the coordinates of B. O 1 (b)determine the components of DA A(6,0,0) x and DB. 2 z C D(3,3,8) y B (c) Calculate the size of angle ADB. 4 (a) 1 C CN G22 (6,6,0) 2002P2Q2 3 (b) 2 C CN G17 DA = 3, 8 DB = (c) 4 C CR G ic: interpretdiagram 2 ic: write down components of a vector 3 ic: write down components of a vector 4 ss: usee.g.scalarproductformula 5 pd: processlengths 6 pd: processscalarproduct 7 pd: processangle 1 B = (6,6,0) 2 DA 3 = DB 3 = cosa DB = DA. DB DA DB 5 DA = 82, DB = 82 6 DA. DB =64 7 A DB =38 7 hsn.uk.net Page 59
60 48. (a) 4 C CN G P2 Q5 (b) 6 C CN G25, G16 (c) 1 C CN G16, G1 (c) 1 A/B CN G16, CGD hsn.uk.net Page 60
61 6 49.Calculatetheacuteanglebetweenthevectorsu = 3 andv= B CR G28,G16 θ =88 15 WCHSU3Q8 1 ss: usescalarproductformula 2 pd: processlengths 3 pd: processscalarproduct 4 pd: processangle 1 cos θ = u.v u v 2 u = 48, v = u.v = cos θ ,so θ = (to2 d.p.) hsn.uk.net Page 61
62 50. (a) 7 C CR G28, G P2 Q5 (b) 3 C CR CGD hsn.uk.net Page 62
63 51.Calculatetheacuteanglebetweenthetwovectorsp =2i +4j k andq=i 3j +2k. 5 5 C CR G28,G26,G Ex pd: findlengthofvector 2 pd: findlengthofvector 3 pd: findscalarproduct 4 ss: usescalarproduct 5 pd: evaluateangle 1 p = ( 1) 2 = 21 2 q = ( 3) = 14 3 p.q = ( 3) + ( 1).2 = 12 4 cos θ = p.q p q = θ =134 4 (1d.p.) or2 346rads(3 d.p.) C CN G29,G P1Q18 hsn.uk.net Page 63
64 53. 1 C CN G P1 Q16 3 A/B CN G29,G C CN G29,G P1Q17 hsn.uk.net Page 64
65 55. 1 C CN G P1 Q13 3 A/B CN G29,G C CN G P1Q3 hsn.uk.net Page 65
66 57. PQRSisaparallelogramwithverticesP(1,3,3),Q(4, 2, 2)andR(3,1,1). Find the coordinates of S. 3 3 C CN G P1Q4 [ENDOFPAPER2] hsn.uk.net Page 66
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