Unit1A/B. x ) Part Marks Level Calc. Content Answer U1 OC3 6 A/B CN C11 x =2 2000P2Q6. 1 A (x) =...

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1 Unit1A/B 1. A goldsmith has built up a solid which consists of a triangular prismoffixedvolumewitharegulartetrahedronateachend. Thesurfacearea,A,ofthesolidisgivenby A(x) = ( x ) x x wherexisthelengthofeachedgeofthetetrahedron. Find the value of x which the goldsmith should use to minimise the amount of gold plating required to cover the solid. 6 6 A/B CN C11 x =2 2000P2Q6 1 ss: knowtodifferentiate 2 pd: process 3 ss: knowtosetf (x) =0 4 pd: dealwithx 2 5 pd: process 6 ic: checkforminimum 1 A (x) = (2x 16x 2 )or3 3x 24 3x 2 3 A (x) = or 24 3 x 2 x 2 5 x =2 6 x A (x) ve 0 +ve sox =2ismin. hsn.uk.net Page 1 Questions marked c SQA

2 2. The shaded rectangle on this map represents the planned extension to the villagehall. Itishopedtoprovidethe largest possible area for the extension. 6m The Vennel Village hall 8m Manse Lane The coordinate diagram represents the right angled triangle of ground behind the hall. The extension has length l metres and breadth b metres, as shown. Onecorneroftheextensionisatthepoint (a,0). (a) (i)showthatl= 5 4 a. y O (0,6) l b (a,0) (8,0) (ii)expressbintermsofaandhencededucethatthearea, Am 2,ofthe extensionisgivenbya = 3 4a(8 a). 3 (b)findthevalueofawhichproducesthelargestareaoftheextension. 4 x (a) 3 A/B CN CGD proof 2002 P2 Q10 (b) 4 A/B CN C11 a =4 1 ss: select strategy and carry through 2 ss: select strategy and carry through 3 ic: completeproof 4 ss: knowtosetderivativetozero 5 pd: differentiate 6 pd: solveequation 7 ic: justifymaximum, e.g. nature table 1 proofofl = 5 4 a 2 b = 3 5 (8 a) 3 completeproofleadingtoa=... =... = a 6 a =4 7 e.g.naturetable,comp.thesquare 4 da da hsn.uk.net Page 2 Questions marked c SQA

3 3. A ball is thrown vertically upwards. Aftertsecondsitsheightishmetres,whereh = t 4 9t 2. (a)findthespeedoftheballafter1second. 3 (b) For how many seconds is the ball travelling upwards? 2 (a) 1 C CN C6, C P1 Q17 (a) 2 A/B CN C6,C6 (b) 2 A/B CN C6,C6 hsn.uk.net Page 3 Questions marked c SQA

4 4. Part Marks Level Calc. Content Answer U1 OC4 (a) 3 C CR A P2 Q8 (b) 3 C CR A10 (c) 1 C CR A14 (d) 3 C CR A12, A13 (d) 1 A/B CR A12, A13 hsn.uk.net Page 4 Questions marked c SQA

5 5.Onasuitabledomain,D,afunctiongisdefinedbyg(x) =sin 2 x 2. WhichofthefollowinggivestherealvaluesofxinDandthecorresponding values of g(x)? A. x 0 and 1 g(x) 1 B. x 0 and 0 g(x) 1 C. x 2 and 1 g(x) 1 D. x 2 and 0 g(x) 1 2 Key Outcome Grade Facility Disc. Calculator Content Source D 1.2 A/B 0 0 CN A1, T P1 Q20 6. f(x) =2x 1,g(x) =3 2xandh(x) = 1 4 (5 x). (a)findaformulafork(x)wherek(x) =f ( g(x) ). 2 (b)findaformulaforh ( k(x) ). 2 (c)whatistheconnectionbetweenthefunctionshandk? 1 Part Marks Level Calc. Content Answer U1 OC2 (a) 2 C NC A P1 Q13 (b) 2 C NC A4 (c) 1 A/B NC CGD hsn.uk.net Page 5 Questions marked c SQA

6 7. Thefunctions f andgaredefinedonasuitabledomainby f(x) =x 2 1and g(x) =x (a)findanexpressionforf ( g(x) ). 2 (b)factorisef ( g(x) ). 2 Part Marks Level Calc. Content Answer U1 OC2 (a) 2 C CN A P1 Q6 (b) 1 C CN A6 (b) 1 A/B CN A6 8. (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 6 Questions marked c SQA

7 9. 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 7 Questions marked c SQA

8 10. The Water Board of a local authority discovered it was able to represent the approximate amount of water W(t), in millions of gallons, stored in a reservoir tmonthsafterthe1stmay1988bytheformulaw(t) =1 1 sin πt 6. The board then predicted that under normal conditions this formula would apply for three years. (a)drawandlabelsketchesofthegraphsofy=sin πt πt 6 andy= sin 6,for 0 t 36,onthesamediagram. 4 (b)onaseparatediagramandusingthesamescaleonthet-axisasyouusedin part(a),drawasketchofthegraphofw(t) =1 1 sin πt 6. 3 (c)onthe1stapril1990aseriousfirerequiredanextra 1 4 milliongallonsofwater from the reservoir to bring the fire under control. Assuming that the previous trend continues from the new lower level, when will the reservoir run dry if water rationing is not imposed? 3 Part Marks Level Calc. Content Answer U1 OC2 (a) 4 A/B NC T1 1990P2Q10 (b) 3 A/B NC A3 (c) 3 A/B NC A6 hsn.uk.net Page 8 Questions marked c SQA

9 11. (a) 1 C CN CGD 1997 P2 Q10 (a) 3 A/B CN CGD (b) 3 C CN C11 (b) 3 A/B CN C11 hsn.uk.net Page 9 Questions marked c SQA

10 12. (a) 4 C CN C P2 Q6 (b) 1 C CN G3 (b) 1 A/B CN G3 (c) 4 A/B CN CGD hsn.uk.net Page 10 Questions marked c SQA

11 13. Aballisthrownverticallyupwards.Theheighthmetresoftheballtsecondsafter itisthrown,isgivenbytheformulah =20t 5t 2. (a)findthespeedoftheballwhenitisthrown(i.e.therateofchangeofheight withrespecttotimeoftheballwhenitisthrown). 3 (b)findthespeedoftheballafter2seconds. Explainyouranswerintermsofthemovementoftheball. 2 (a) 1 C NC C P1 Q21 (a) 2 A/B NC C6 (b) 2 A/B NC A6 14. Forwhatvaluesofxisthefunctionf(x) = 1 3 x3 2x 2 5x 4increasing? 5 2 C NC C7 1990P1Q16 3 A/B NC C7 hsn.uk.net Page 11 Questions marked c SQA

12 15. Acurvehasequationy =2x 3 +3x 2 +4x 5. Prove that this curve has no stationary points. 5 2 C NC C8,C7 1999P1Q16 3 A/B NC C8,C7 [END OF QUESTIONS] hsn.uk.net Page 12 Questions marked c SQA