Prelim practice. Part Marks Level Calc. Content Answer U1 OC1 3 C CR G2 1992P1Q13

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1 Prelim practice 1. Part Marks Level Calc. Content Answer U1 OC1 3 C CR G2 1992P1Q13 2. Find the equation of the perpendicular bisector of the line joining A(2, 1) and B(8,3). 4 Part Marks Level Calc. Content Answer U1 OC1 4 C CN G2,G5 1996P1Q1 hsn.uk.net Page 1

2 3. Part Marks Level Calc. Content Answer U1 OC1 3 C CN G2,G5,G3 1997P1Q1 4. Part Marks Level Calc. Content Answer U1 OC1 (a) 2 C CN G P1 Q2 (b) 3 C CN G8 5. Find the equation of the straight line which is parallel to the line with equation 2x +3y =5andwhichpassesthroughthepoint (2, 1). 3 Part Marks Level Calc. Content Answer U1 OC1 3 C CN G3,G2 2x +3y =1 2001P1Q1 1 ss: expressinstandardform 2 ic: interpretgradient 3 ic: stateequationofstraightline 1 y = 2 3 x +5 3 statedorimpliedby 2 2 m line = 2 3 statedorimpliedby 3 3 y ( 1) = 2 3 (x 2) hsn.uk.net Page 2

3 6. A quadrilateral has vertices A( 1, 8), B(7, 12), C(8, 5) and D(2, 3) as shown in the diagram. y B A E C O x D (a) Find the equation of diagonal BD. 2 (b)theequationofdiagonalacisx +3y =23. Find the coordinates of E, the point of intersection of the diagonals. 3 (c) (i) Find the equation of the perpendicular bisector of AB. (ii) Show that this line passes through E. 5 Part Marks Level Calc. Content Answer U1 OC1 (a) 2 C CN G3,G2 y 12 =3(x 7) 2011P1Q21 (b) 3 C CN G8 E(5, 6) (ci) 4 C CN G7 y 10 = 2(x 3) (cii) 1 C CN A6 proof 1 pd: findgradientofbd 2 ic: stateequationofbd 3 ss: start solution of simultaneous eqs 4 pd: solveforonevariable 5 pd: solveforsecondvariable 6 ss: knowandfindmidpointofab 7 pd: findgradientofab 8 ic: interpretperpendiculargradient 9 ic: stateequationofperp.bisector 10 ic: justificationofpointonline hsn.uk.net Page orequiv. 2 y ( 3) =3(x 2) 3 3x y =9andx +3y =23 4 x =5ory =6 5 y =6orx=5 6 (3,10) orequiv orequiv 9 y 10 = 2(x 3) 10 whenx =5,y = =6

4 7. Part Marks Level Calc. Content Answer U1 OC1 (a) 6 C NC G3,G5,G8 1992P1Q2 (b) 2 C NC G8 hsn.uk.net Page 4

5 8. TrianglePQRhasvertexPonthex-axis, asshowninthediagram. QandRarethepoints (4,6)and (8, 2) respectively. Q(4, 6) 6 x 7 y + 18 = 0 T TheequationofPQis6x 7y +18 =0. (a) State the coordinates of P. O P x 1 (b)findtheequationofthealtitudeof the triangle from P. R(8, 2) 3 y (c)thealtitudefrompmeetstheline QRatT.FindthecoordinatesofT. 4 Part Marks Level Calc. Content Answer U1 OC1 (a) 1 C CN G4 P( 3, 0) 2009 P1 Q21 (b) 3 C CN G7 y = 1 2 (x +3) (c) 4 C CN G8 T(5, 4) 1 ic: interpretx-intercept 2 pd: findgradient(ofqr) 3 ss: knowandusem 1 m 2 = 1 4 ic: stateequ.ofaltitude 5 ic: stateequ.ofline(qr) 6 ss: preparetosolvesim.equ. 7 pd: solveforx 8 pd: solvefory 1 P = ( 3,0) 2 m QR = 2 3 m alt. = y 0 = 1 2 (x +3) 5 y +2 = 2(x 8) 6 x 2y = 3and2x +y =14 7 x =5 8 y =4 hsn.uk.net Page 5

6 9. Triangle ABC has vertices A( 1, 6), B( 3, 2) and C(5, 2). Find A( 1, 6) (a)theequationoftheline p,the median from C of triangle ABC. C(5, 2) 3 (b)theequationoftheline q, the O perpendicular bisector of BC. x 4 (c)thecoordinatesofthepointof B( 3, 2) intersectionofthelinespandq. 1 Part Marks Level Calc. Content Answer U1 OC1 (a) 3 C CN G7 y =2 2002P2Q1 (b) 4 C CN G7 y = 2x +2 (c) 1 C CN G8 (0, 2) y 1 ss: determinemidpointcoordinates 2 pd: determinegradientthro 2pts 3 ic: stateequationofstraightline 4 ss: determinemidpointcoordinates 5 pd: determinegradientthro 2pts 6 ss: determinegradientperp.to 5 7 ic: stateequationofstraightline 8 pd: processintersection 1 F =mid AB = ( 2,2) 2 m FC =0 statedorimpliedby 3 3 equ.fcisy =2 4 M =mid BC = (1,0) 5 m BC = m = 2 7 y 0 = 2(x 1) 8 (0,2) hsn.uk.net Page 6

7 10. The diagram shows a sketch of the y functiony =f(x). ( 4, 8) (2, 8) (a)copythediagramandonitsketch thegraphofy =f(2x). y = f( x) 2 (b)on a separate diagram sketch the graphofy =1 f(2x). 3 O x Part Marks Level Calc. Content Answer U1 OC2 (a) 2 B CN A3 sketch 2009 P1 Q23 (b) 3 B CN A3 sketch 1 ic: scalingparalleltox-axis 2 ic: annotategraph 3 ss: correctorderforrefl(x)andtrans 4 ic: starttoannotatefinalsketch 5 ic: completeannotation 1 sketch and one of (0,0), (1,8), ( 2, 8) 2 remainingpoints 3 reflect in x-axis then vertical translation 4 sketch and one of (0,1), (1, 7), ( 2, 7) 5 remainingpoints 11. Part Marks Level Calc. Content Answer U1 OC2 (a) 2 C NC A P1 Q9 (b) 3 C NC A3, C11 hsn.uk.net Page 7

8 12. Part Marks Level Calc. Content Answer U1 OC2 (a) 2 C CN A P1 Q10 (b) 2 C CN A3 13. Part Marks Level Calc. Content Answer U1 OC2 3 C NC A3,A1 1993P1Q14 hsn.uk.net Page 8

9 14. Part Marks Level Calc. Content Answer U1 OC2 3 A/B NC A3,A2 1990P1Q f(x) =3 xandg(x) = 3,x =0. x (a)findp(x)wherep(x) =f(g(x)). 2 (b)ifq(x) = 3,x =3,findp(q(x))initssimplestform. 3 3 x Part Marks Level Calc. Content Answer U1 OC2 (a) 2 C CN A4 3 3 x 2000P2Q3 (b) 2 C CN A4 x (b) 1 A/B CN A4 1 ic: interpretcompositefunc. 2 pd: process 3 ic: interpretcompositefunc. 4 pd: process 5 pd: process 1 f ( ) 3 x statedorimpliedby x 3 p ( ) 3 3 x statedorimpliedby x 5 x hsn.uk.net Page 9

10 16. Onasuitablesetofrealnumbers,functionsfandgaredefinedbyf(x) = 1 x +2 andg(x) = 1 x 2. Findf ( g(x) ) initssimplestform. 3 Part Marks Level Calc. Content Answer U1 OC2 3 C NC A4 1992P1Q6 17. Thefunctions f andg,definedonsuitabledomains,aregivenby f(x) = 1 x 2 4 andg(x) =2x +1. (a)findanexpressionforh(x)whereh(x) =g ( f(x) ). Giveyouranswerasa single fraction. 3 (b)stateasuitabledomainforh. 1 Part Marks Level Calc. Content Answer U1 OC2 (a) 2 C NC A P1 Q11 (a) 1 A/B NC A4 (b) 1 A/B NC A1 hsn.uk.net Page 10

11 18. Functions f andg,definedonsuitabledomains,aregivenby f(x) = 2xand g(x) =sinx +cosx. Findf ( g(x) ) andg ( f(x) ). 4 Part Marks Level Calc. Content Answer U1 OC2 4 C NC A4 1997P1Q3 19. Functionsfandgaredefinedbyf(x) =2x+3andg(x) = x2 +25 x 2 25 wherex R, x = ±5. Thefunctionhisgivenbytheformulah(x) =g ( f(x) ). Forwhichrealvaluesofxisthefunctionhundefined? 4 Part Marks Level Calc. Content Answer U1 OC2 2 C CN A4,A1 1989P1Q19 2 A/B CN A4,A1 20. (a)express7 2x x 2 intheforma (x +b) 2 andwritedownthevaluesofa and b. 2 (b)statethemaximumvalueof7 2x x 2 andjustifyyouranswer. 2 Part Marks Level Calc. Content Answer U1 OC2 (a) 2 A/B NC A5 1991P1Q15 (b) 2 A/B NC A6 hsn.uk.net Page 11

12 21. Express (2x 1)(2x +5)intheforma(x +b) 2 +c. 3 Part Marks Level Calc. Content Answer U1 OC2 3 C NC A5 1996P1Q (a)showthatthefunction f(x) = 2x 2 +8x 3 canbewrittenintheform f(x) =a(x +b) 2 +cwherea,bandcareconstants. 3 (b) Hence, or otherwise, find the coordinates of the turning point of the function f. 1 Part Marks Level Calc. Content Answer U1 OC2 (a) 3 C NC A P1 Q9 (b) 1 C NC A6 hsn.uk.net Page 12

13 23. Part Marks Level Calc. Content Answer U1 OC2 3 C NC A6 1996P1Q3 24. Thediagramshowsasketchofpartofthegraph y y =log ofy =log 2 (x). 2 (x) (8,b) (a)statethevaluesofaandb. 1 (b)sketchthegraphofy =log 2 (x +1) 3. O (a,0) x 3 Part Marks Level Calc. Content Answer U1 OC2 (a) 1 A/B CN A7 a =1,b =3 2001P1Q10 (b) 3 A/B CN A3 sketch 1 pd: uselog p q = 0 q = 1and evaluatelog p p k 2 ss: useatranslation 3 ic: identifyonepoint 4 ic: identifyasecondpoint 1 a =1andb =3 2 a log-shaped graph of the same orientation 3 sketch passes through (0, 3) (labelled) 4 sketch passes through (7,0) (labelled) hsn.uk.net Page 13

14 25. Asketchofthegraphofy =f(x)wheref(x) =x 3 6x 2 +9xisshownbelow. ThegraphhasamaximumatAandaminimumatB(3,0). y A y =f(x) O B(3, 0) x (a)findthecoordinatesoftheturningpointata. 4 (b)hencesketchthegraphofy =g(x)whereg(x) =f(x +2) +4. Indicate the coordinates of the turning points. There is no need to calculate the coordinates of the points of intersection with the axes. 2 (c)writedowntherangeofvaluesofkforwhichg(x) =khas3realroots. 1 Part Marks Level Calc. Content Answer U1 OC3 (a) 4 C NC C8 A(1, 4) 2000 P1 Q2 (b) 2 C NC A3 sketch (translate 4 up, 2 left) (c) 1 A/B NC A2 4 <k<8 1 ss: knowtodifferentiate 2 pd: differentiatecorrectly 3 ss: knowgradient =0 4 pd: process 5 ic: interprettransformation 6 ic: interprettransformation 7 ic: interpretsketch 1 dy dx =... 2 dy dx =3x2 12x x 2 12x +9 =0 4 A = (1,4) translatef(x)4unitsup,2unitsleft 5 sketchwithcoord.ofa ( 1,8) 6 sketchwithcoord.ofb (1,4) 7 4 <k<8(accept4 k 8) hsn.uk.net Page 14

15 26. Part Marks Level Calc. Content Answer U1 OC3 (a) 5 C CN C P2 Q1 (b) 2 C CN A1 hsn.uk.net Page 15

16 27. Part Marks Level Calc. Content Answer U1 OC3 (a) 6 C CN C P2 Q2 (b) 2 C CN C8 28. Acurvehasequationy =2x 3 +3x 2 +4x 5. Prove that this curve has no stationary points. 5 Part Marks Level Calc. Content Answer U1 OC3 2 C NC C8,C7 1999P1Q16 3 A/B NC C8,C7 hsn.uk.net Page 16

17 29. Acurvehasequationy =x 16,x >0. x Findtheequationofthetangentatthepointwherex =4. 6 Part Marks Level Calc. Content Answer U1 OC3 6 C CN C4,C5 y =2x P2Q2 1 ic: findcorrespondingy-coord. 2 ss: expressinstandardform 3 ss: starttodifferentiate 4 pd: diff.fractionalnegativepower 5 ss: findgradientoftangent 6 ic: writedownequ.oftangent 1 (4, 4) statedorimpliedby x dy dx = x m x=4 =2 6 y ( 4) =2(x 4) 30. Aballisthrownverticallyupwards.Theheighthmetresoftheballtsecondsafter itisthrown,isgivenbytheformulah =20t 5t 2. (a)findthespeedoftheballwhenitisthrown(i.e.therateofchangeofheight withrespecttotimeoftheballwhenitisthrown). 3 (b)findthespeedoftheballafter2seconds. Explainyouranswerintermsofthemovementoftheball. 2 Part Marks Level Calc. Content Answer U1 OC3 (a) 1 C NC C P1 Q21 (a) 2 A/B NC C6 (b) 2 A/B NC A6 hsn.uk.net Page 17

18 31. Findthex-coordinateofeachofthepointsonthecurvey =2x 3 3x 2 12x +20 atwhichthetangentisparalleltothex-axis. 4 Part Marks Level Calc. Content Answer U1 OC3 4 C NC C4 1993P1Q4 hsn.uk.net Page 18

19 32. Part Marks Level Calc. Content Answer U1 OC3 (a) 6 C CN C4, G P2 Q11 (b) 6 A/B CN G8,G1 hsn.uk.net Page 19

20 33.Theparabolaswithequationsy=10 x 2 andy= 2 5 (10 x2 )areshowninthe diagram below. y R S T Q P O x y = 10 x y = 5 (10 x ) ArectanglePQRSisplacedbetweenthetwoparabolasasshown,sothat: QandRlieontheupperparabola. RQandSPareparalleltothex-axis. T,theturningpointofthelowerparabola,liesonSP. (a) (i)iftp =xunits,findanexpressionforthelengthofpq. (ii)henceshowthatthearea,a,ofrectanglepqrsisgivenby A(x) =12x 2x 3 3 (b) Find the maximum area of this rectangle. 6 Part Marks Level Calc. Content Answer U1 OC3 (ai) 2 B CN C11 6 x P2Q5 (aii) 1 B CN C11 2x (6 x 2 ) =A(x) (b) 6 C CN C11 maxis8 2 1 ss: knowtoandfindot 2 ic: obtainanexpressionforpq 3 ic: completeareaevaluation 4 ss: knowtoandstarttodifferentiate 5 pd: completedifferentiation 6 ic: setderivativetozero 7 pd: obtain 8 ss: justifynatureofstationarypoint 9 ic: interpret result and evaluate area x x(6 x 2 ) =12x 2x 3 4 A (x) = x x 2 = x 2 A (x) Maxand8 2 hsn.uk.net Page 20

21 34. Part Marks Level Calc. Content Answer U1 OC3 (a) 2 C CN C P1 Q14 (b) 2 C CN A3 35. Differentiate2 x(x +2)withrespecttox. 4 Part Marks Level Calc. Content Answer U1 OC3 4 C NC C1 1998P1Q Iff(x) =kx 3 +5x 1andf (1) =14,findthevalueofk. 3 Part Marks Level Calc. Content Answer U1 OC3 3 C NC C1,A6 1994P1Q2 hsn.uk.net Page 21

22 37. Functionsfandgaregivenbyf(x) =3x +1andg(x) =x 2 2. (a) (i)findp(x)wherep(x) =f(g(x)). (ii)findq(x)whereq(x) =g(f(x)). 3 (b)solvep (x) =q (x). 3 Part Marks Level Calc. Content Answer U1 OC3 (a) 3 C CN A4 3(x 2 2) +1, (3x +1) P2Q2 (b) 3 C CN C1 x = ss: substituteforg(x)inf(x) 2 ic: complete 3 ic: sub.andcompleteforq(x) 4 ss: simplify 5 pd: differentiate 6 pd: solve 1 f(x 2 2) 2 3(x 2 2) +1 3 (3x +1) p(x) =3x 2 5,q(x) =9x 2 +6x 1 5 p (x) =6x,q (x) =18x +6 6 x = 1 2 hsn.uk.net Page 22

23 38. Afunctionfisdefinedonthesetofrealnumbersbyf(x) = (x 2)(x 2 +1). (a)findwherethegraphofy =f(x)cuts: (i)thex-axis; (ii) the y-axis. 2 (b) Find the coordinates of the stationary points on the curve with equation y =f(x)anddeterminetheirnature. 8 (c) On separate diagrams sketch the graphs of: (i) y =f(x); (ii) y = f(x). 3 Part Marks Level Calc. Content Answer U1 OC3 (a) 2 CN A6 (2,0), (0, 2) 2011P1Q22 (b) 8 CN C8,C9 max: ( 1 3, ), min: (1, 2) (ci) 2 CN A8, A7 sketch (cii) 1 CN A3 reflect in x-axis 1 ic: interpretxintercept 2 ic: interpretyintercept 3 ic: writeindifferentiableform 4 ss: knowtoandstarttodifferentiate 5 pd: completederivativeandequate to0 6 pd: factorisederivative 7 pd: processforx 8 pd: evaluatey-coordinates 9 ic: justify nature of stationary points 10 ic: interpretandstateconclusions 11 ic: curveshowingpointsfrom(a) and(b) without annotation 12 ic: cubic curve showing all intercepts and stationary points annotated 13 ic: curvefrom(i)reflectedinx-axis 1 (2,0) 2 (0, 2) 3 x 3 2x 2 +x 2 4 3x x 2 4x +1 =0 6 (3x 1)(x 1) and and 2 9 x f (x) max.at ( ),min.at (1, 2) 11 sketch 12 sketch 13 reflectedsketch hsn.uk.net Page 23

24 39. Part Marks Level Calc. Content Answer U1 OC3 (a) 1 C CN CGD 1996 P2 Q11 (a) 3 A/B CN CGD (b) 2 C CN C11 (b) 3 A/B CN C11 hsn.uk.net Page 24

25 40. Part Marks Level Calc. Content Answer U1 OC3 (a) 3 A/B CR CGD 1998 P2 Q10 (b) 3 C CR C11 (b) 3 A/B CR C11 hsn.uk.net Page 25

26 41. Findthecoordinatesofthepointonthecurvey=2x 2 7x +10wherethetangent tothecurvemakesanangleof45 withthepositivedirectionofthex-axis. 4 Part Marks Level Calc. Content Answer U1 OC3 4 C NC G2,C4 (2,4) 2002P1Q4 1 sp: knowtodiff.,anddifferentiate 2 pd: processgradientfromangle 3 ss: equateequivalentexpressions 4 pd: solveandcomplete =4x 7 2 m tang =tan45 =1 3 4x 7 =1 4 (2,4) 1 dy dx hsn.uk.net Page 26

27 42.Functionsf,gandharedefinedonthesetofrealnumbersby f(x) =x 3 1 g(x) =3x +1 h(x) =4x 5. (a)findg(f(x)). 2 (b)showthatg(f(x)) +xh(x) =3x 3 +4x 2 5x 2. 1 (c) (i)showthat (x 1)isafactorof3x 3 +4x 2 5x 2. (ii)factorise3x 3 +4x 2 5x 2fully. 5 (d)hencesolveg(f(x)) +xh(x) =0. 1 Part Marks Level Calc. Content Answer U2 OC1 (a) 2 C CN A4 3(x 3 1) P2Q2 (b) 1 C CN A6 proof (c) 5 C CN A21 (x 1)(3x +1)(x +2) (d) 1 C CN A22 2, 1 3,1 1 ic: interpretnotation 2 ic: completeprocess 3 ic: substituteandcomplete 4 ss: knowtousex=1 5 pd: completeevaluation 6 ic: stateconclusion 7 ic: findquadraticfactor 8 pd: factorisecompletely 9 ic: interpretandsolveequationin (d) 1 g(x 3 1) 2 3(x 3 1) (x 3 1) +1+x(4x 5) =3x 3 +4x 2 5x 2 4 evaluatingatx= =0 6 (x 1)isafactor 7 (x 1)(2x 2 +7x +2) 8 (x 1)(3x +1)(x +2) 9 2, 1 3,1 hsn.uk.net Page 27

28 43. (a)thefunctionfisdefinedbyf(x) =x 3 2x 2 5x +6. Thefunctiongisdefinedbyg(x) =x 1. Showthatf ( g(x) ) =x 3 5x 2 +2x (b)factorisefullyf ( g(x) ). 3 (c)thefunctionkissuchthatk(x) = 1 f ( g(x) ). Forwhatvaluesofxisthefunctionknotdefined? 3 Part Marks Level Calc. Content Answer U2 OC1 (a) 4 C NC A P2 Q6 (b) 3 C NC A21 (c) 2 C NC A1 hsn.uk.net Page 28

29 44. Part Marks Level Calc. Content Answer U2 OC1 (a) 1 C CN A P2 Q9 (b) 2 C CN C4, CGD (b) 4 A/B CN C4,CGD (c) 2 A/B CN A17 hsn.uk.net Page 29

30 45. (i)writedowntheconditionfortheequationax 2 +bx +c =0tohavenoreal roots. 1 (ii)henceorotherwiseshowthattheequationx(x +1) = 3x 2hasnoreal roots. 2 Part Marks Level Calc. Content Answer U2 OC1 3 C CN A P1Q8 46. Giventhatkisarealnumber,showthattherootsoftheequationkx 2 +3x +3 =k are always real numbers. 5 Part Marks Level Calc. Content Answer U2 OC1 1 C NC A P1 Q18 4 A/B NC A17 hsn.uk.net Page 30

31 47. Part Marks Level Calc. Content Answer U2 OC1 1 C NC A P1 Q17 3 A/B NC A (a) Giventhatx +2isafactorof2x 3 +x 2 +kx +2,findthevalueofk. 3 (b)hencesolvetheequation2x 3 +x 2 +kx +2 =0whenktakesthisvalue. 2 Part Marks Level Calc. Content Answer U2 OC1 (a) 3 C CN A21 k = P2Q1 (b) 2 C CN A22 x = 2, 1 2,1 1 ss: use synth division or f(evaluation) 2 pd: process 3 pd: process 4 ss: findaquadraticfactor 5 pd: process 1 f( 2) =2( 2) ( 2) 3 + ( 2) 2 2k +2 3 k = 5 4 2x 2 3x +1 or 2x 2 +3x 2 or x 2 +x 2 5 (2x 1)(x 1)or (2x 1)(x +2)or (x +2)(x 1) andx= 2, 1 2,1 hsn.uk.net Page 31

32 49. (a) (i)showthat (x 1)isafactoroff(x) =2x 3 +x 2 8x +5. (ii) Hence factorise f(x) fully. 5 (b)solve2x 3 +x 2 8x +5 =0. 1 (c)thelinewithequationy =2x 3isatangenttothecurvewithequation y =2x 3 +x 2 6x +2atthepointG. Find the coordinates of G. 5 (d)thistangentmeetsthecurveagainatthepointh. Write down the coordinates of H. 1 Part Marks Level Calc. Content Answer U2 OC1 (a) 5 C CN A21 (x 1)(x 1)(2x +5) 2010P1Q22 (b) 1 C CN A22 x =1, 5 2 (c) 5 C CN A23 (1, 1) (d) 1 C CN A23 ( 5 2, 8) 1 ss: knowtousex=1 2 ic: completeevaluation 3 ic: stateconclusion 4 pd: findquadraticfactor 5 pd: factorisecompletely 6 ic: statesolutions 7 ss: sety curve =y line 8 ic: expressinstandardform 9 ss: comparewith(a)orfactorise 10 ic: identifyx G 11 pd: evaluatey G 12 pd: statesolution 1 evaluatingatx= =0 3 (x 1)isafactor 4 (x 1)(2x 2 +3x 5) 5 (x 1)(x 1)(2x +5) 6 x =1andx= x 3 +x 2 6x +2 =2x 3 8 2x 3 +x 2 8x +5 =0 9 (x 1)(x 1)(2x +5) =0 10 x =1 11 y = 1 12 ( 5 2, 8) 50. Expressx 3 4x 2 7x +10initsfullyfactorisedform. 4 Part Marks Level Calc. Content Answer U2 OC1 4 C NC A P1Q2 hsn.uk.net Page 32

33 51. Part Marks Level Calc. Content Answer U2 OC4 (a) 4 C CN G5,G3 1991P2Q2 (b) 6 C CN G10, G1 hsn.uk.net Page 33

34 52. Part Marks Level Calc. Content Answer U2 OC4 (a) 3 C CN G5,G3 1993P2Q3 (b) 5 C CN G10 hsn.uk.net Page 34

35 53. (a)showthatthepointp(5,10)liesoncirclec 1 withequation (x +1) 2 + (y 2) 2 = (b)pqisadiameterofthiscircleas y showninthediagram. Findthe equationofthetangentatq. P(5, 10) 5 O x Q (c)twocircles,c 2 andc 3,touchcircleC 1 atq. TheradiusofeachofthesecirclesistwicetheradiusofcircleC 1. FindtheequationsofcirclesC 2 andc 3. 4 Part Marks Level Calc. Content Answer U2 OC4 (a) 1 C CN A6 proof 2009 P2 Q4 (b) 5 C CN G11 3x +4y +45 =0 (c) 4 A NC G15 (x 5) 2 + (y 10) 2 =400, (x+19) 2 +(y+22) 2 =400 1 pd: substitute 2 ic: findcentre 3 ss: usemid-pointresultforq 4 ss: knowto,andfindgradientof radius 5 ic: findgradientoftangent 6 ic: stateequationoftangent 7 ic: stateradius 8 ss: knowhowtofindcentre 9 ic: stateequationofonecircle 10 ic: stateequationoftheothercircle 1 (5 +1) 2 + (10 2) 2 =100 2 centre = ( 1,2) 3 Q = ( 7, 6) 4 m rad = m tgt = y ( 6) = 3 4 (x ( 7)) 7 radius =20 8 centre = (5,10) 9 (x 5) 2 + (y 10) 2 = (x +19) 2 + (y +22) 2 =400 hsn.uk.net Page 35

36 54. FindtheequationofthecirclewhichhasP( 2, 1)andQ(4,5)astheendpoints of a diameter. 3 Part Marks Level Calc. Content Answer U2 OC4 3 C CN G P1Q9 55. Part Marks Level Calc. Content Answer U2 OC4 (a) 3 C CN G P1 Q8 (b) 3 C CN G10 hsn.uk.net Page 36

37 56. (a) (i)showthatthelinewithequationy=3 xisatangenttothecirclewith equationx 2 +y 2 +14x +4y 19 =0. (ii) Find the coordinates of the points of contact, P. 5 (b)relativetoasuitablesetofcoordinateaxes,thediagrambelowshowsthe circlefrom(a)andasecondsmallercirclewithcentrec. P C Theliney =3 xisacommontangentatthepointp. Theradiusofthelargercircleisthreetimestheradiusofthesmallercircle. Find the equation of the smaller circle. 6 Part Marks Level Calc. Content Answer U2 OC4 (ai) 4 C CN G13 proof 2010 P2 Q3 (aii) 1 C CN G12 P( 1, 4) (b) 6 B CN G9,G15 (x 1) 2 + (y 6) 2 =8 1 ss: substitute 2 pd: expressinstandardform 3 ic: startproof 4 ic: completeproof 5 pd: coordinatesofp 6 ic: statecentreoflargercircle 7 ss: findradiusoflargercircle 8 pd: findradiusofsmallercircle 9 ss: strategyforfindingcentre 10 ic: interpretcentreofsmallercircle 11 ic: stateequation 1 x 2 +(3 x) 2 +14x+4(3 x) 19 =0 2 2x 2 +4x +2 =0 3 2(x +1)(x +1) 4 equalrootssolineisatangent 5 x = 1,y =4 6 ( 7, 2) e.g. Steppingout 10 (1,6) 11 (x 1) 2 + (y 6) 2 =8 hsn.uk.net Page 37

38 57. Explainwhytheequationx 2 +y 2 +2x +3y +5 =0doesnotrepresentacircle. 2 Part Marks Level Calc. Content Answer U2 OC4 2 C CN G9 1993P1Q18 hsn.uk.net Page 38

39 58. Part Marks Level Calc. Content Answer U2 OC4 (a) 4 C CN G9,G5 1999P2Q2 (b) 1 C CN A6 (c) 1 C CN CGD (d) 2 C CN G10 hsn.uk.net Page 39

40 59. 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 Part Marks Level Calc. Content Answer U3 OC1 (a) 1 C CN G6, G P2 Q2 (b) 3 C CN G25 (c) 4 C CN G23, G Part Marks Level Calc. Content Answer U3 OC1 5 C CN G9,G P1Q12 hsn.uk.net Page 40

41 61. Part Marks Level Calc. Content Answer U3 OC1 (a) 2 C CN G P1 Q3 (b) 1 C CN G26 (c) 1 C CN G Part Marks Level Calc. Content Answer U3 OC1 (a) 2 C CN G P1 Q5 (b) 2 C CN G16 hsn.uk.net Page 41

42 63. Part Marks Level Calc. Content Answer U3 OC1 (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 42

43 64. Thevector ai +bj +k isperpendiculartoboth the vectors i j+k and 2i +j+k. Findthevaluesofaandb. 3 Part Marks Level Calc. Content Answer U3 OC1 3 C CN G18 a =2,b =3 1990P1Q12 hsn.uk.net Page 43

44 65. Part Marks Level Calc. Content Answer U3 OC1 (a) 2 C CN G P2 Q3 (b) 2 C CN G20 (c) 5 C CN G28 hsn.uk.net Page 44

45 66. 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 Part Marks Level Calc. Content Answer U3 OC1 (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 45

46 67. Part Marks Level Calc. Content Answer U3 OC1 (a) 2 C CN G P1 Q17 (b) 4 A/B CN G29, G30 hsn.uk.net Page 46

47 68. 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 Part Marks Level Calc. Content Answer U3 OC1 (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) 69. Differentiatesin2x + 2 withrespecttox. 4 x Part Marks Level Calc. Content Answer U3 OC2 2 C NC C3 1989P1Q10 2 A/B NC C20 hsn.uk.net Page 47

48 70. Giventhatf(x) = (5x 4) 1 2,evaluatef (4). 3 Part Marks Level Calc. Content Answer U3 OC2 5 1 C CN C P2Q8 2 A/B CN C21 1 pd: differentiatepower 2 pd: differentiate2ndfunction 3 pd: evaluatef (x) (5x 4) f (4) = Givenf(x) =cos 2 x sin 2 x,findf (x). 3 Part Marks Level Calc. Content Answer U3 OC2 1 C NC C P1 Q19 2 A/B NC C21,C Differentiate2x2 3 +sin 2 xwithrespecttox. 4 Part Marks Level Calc. Content Answer U3 OC2 1 C NC C P1 Q11 3 A/B NC C21 hsn.uk.net Page 48

49 73. Iff(x) =cos 2 x 2 3x 2,findf (x). 4 Part Marks Level Calc. Content Answer U3 OC2 2 C NC C21,C1 1990P1Q19 2 A/B NC C21,C1 74. Givenf(x) = (sinx+1) 2,findtheexactvalueoff ( π 6 ). 3 Part Marks Level Calc. Content Answer U3 OC2 3 A/B NC C21,C20,T2 1998P1Q16 75.Functionsfandgaredefinedonsuitabledomainsbyf(x) =cosxand g(x) =x + π 6. Whatisthevalueoff ( g ( π 6 ))? A π 6 B. C π D Key Outcome Grade Facility Disc. Calculator Content Source D 1.2 C 0 0 NC A4, T P1 Q11 hsn.uk.net Page 49

50 76.Whatisthegradientofthetangenttothecurvewithequationy =cos2xatthe pointwherex = π 4? A. 2 B. 1 C. 0 D. 2 2 Key Outcome Grade Facility Disc. Calculator Content Source A 3.2 C NC C4, C20, T3 HSN 127 hsn.uk.net Page 50

51 77.Thediagramshowsthegraphwithequationoftheformy =acosbxfor 0 x 2π. y 2 O π 2π x 2 Whatistheequationofthisgraph? A. y =2cos3x B. y =2cos2x C. y =3cos2x D. y =4cos3x 2 Key Outcome Grade Facility Disc. Calculator Content Source A 1.2 C 0 0 CN T P1 Q4 hsn.uk.net Page 51

52 78.Thediagramshowsthecurvewithequationoftheformy =cos(x +a) +bfor 0 x 2π. y O π 6 7π 6 2π x 2 Whatistheequationofthiscurve? A. y =cos ( x π ) 6 1 B. y =cos ( x π ) 6 +1 C. y =cos ( x + π ) 6 1 D. y =cos ( x + π ) Key Outcome Grade Facility Disc. Calculator Content Source A 1.2 C 0 0 NC T4, T P1 Q9 [END OF QUESTIONS] hsn.uk.net Page 52