2012 GCE A Level H2 Maths Solution Paper Let x,

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GCE A Level H Maths Solutio Pape. Let, y ad z be the cost of a ticet fo ude yeas, betwee ad 5 yeas, ad ove 5 yeas categoies espectively. 9 + y + 4z =. 7 + 5y + z = 8.

8 4 coect to d.p. (i) 5 u ad u. 4 (ii) As, u l ad u l. l Thus l l l l. (iii) 4 Let P be the popositio: u,. Whe : LHS u 4 RHS LHS. P is tue. Assume that P is tue fo some.

1 GCE A Level H Maths Solutio Pape. Let, y ad z be the cost of a ticet fo ude yeas, betwee ad 5 yeas, ad ove 5 yeas categoies espectively. 9 + y + 4z = y + z = y + 5z = 58.5 Fo ude, ticet costs $7.5. Fo betwee ad 5 yeas, ticet costs $9.85. Fo ove 5 yeas, ticet costs $8.5. (i) 4 d 4 4 l C 4 4 d 4 (ii) Fom u, we have d u d d du 4 ta u C u ta C (iii) d.8 4 coect to d.p. (i) 5 u ad u. 4 (ii) As, u l ad u l. l Thus l l l l. (iii) 4 Let P be the popositio: u,. Whe : LHS u 4 RHS LHS. P is tue. Assume that P is tue fo some. Note: f' f d l f C, but 4 i this case, f, thus the modulus is omitted. Note: It is etemely cucial to get the value of l i (ii) coect i ode to popely aswe this questio.

2 4(i) i.e. u 4. P is tue too, i.e. To pove that 4 u. LHS u u = RHS P is tue. Theefoe, Sice P is tue ad tue too, by mathematical iductio, fo all. By Sie Rule, AC si si 4 4 si 4 AC si 4 P is tue P P is tue AC si cos cos si 4 4 AC cos si AC (show) cos si is

3 4(ii) Fo small, AC (show) 5(i) OC OA OB 5(ii) Aea of OAC OA OC ( ) ( ) 4 9 sice 4 4 u Sice the legth OC 4 8 ( 4) (8 ) , Thus the coodiates ae 5, 7, o,,. Note: We appoimate the tigoometic tems usig the stadad seies i the MF5 (selectig tems up to ad icludig sice the fial epessio has tems up to ). This questio also tests o how to maage a -tem biomial epasio. I this case, it is cucial to goup the tems ad goup. as oe Note: To fid the aea of OAC, we ca also mae use of the vectos OA ad AC, o OC ad AC istead of OA ad OC. Howeve, usig the combiatio OA ad OC will be moe efficiet sice we ca avoid calculatig a ew vecto AC.

4 (i) (ii) z ic ic ic ic ic c ic c i c c Give that z is eal c c c ( c ) c sice c is o-zeo c z i o i (iii) Let z i. 7(i) 7(ii) Sice z z i, theefoe, z. By tial ad eo (o fom the GC), the smallest positive is. Thus, modulus is z 4, ad the agumet is z z ag ag. Let y. The y y = + (y ) = y + y y Theefoe g : Hece g is self ivese. y =, O, y = Note: z is eal meas that the imagiay pat of z is zeo. Note: Give the picipal value of the agumet by addig/subtactig multiples of so that ag z. Note: The domai of the fuctio also plays a impotat ole i detemiig if a fuctio has this selfivese popety. Fo eample, if we defie g :,,, the ca we coclude g is self-ivese? Note: To see the shape of the cuve o the gaphig calculato, studets ca ty puttig i a positive value of. 4

5 7(iii) Lie of symmety is y =. y taslate by uit i the diectio of +ve ais y scale by facto paallel to the y ais y taslate by uit i the diectio of +ve y ais y 8(i) dy dy y d y y y d d d d y y y d d y y y d dy y d dy (Show) d y 8(ii) d y dy d y d 8(iii) dy y d dy (Show) d Whe d y d, d y d Hece the tuig poit is a maimum poit. 9(i) AB OB OA Equatio of lie AB is 7 8, 9 Note: Sice the fuctio is self ivese, the gaph is symmetical about the lie y. What is the equatio of the othe lie of symmety, if ay? Note: Diffeetiate the equatio implicitly with espect to, ad the eaage the tems. Note: This questio gives the possibility that though we do ot ow the coodiates of the tuig poit, we ca still detemie the atue of this poit. Note: It is advisable to use the simplest fom of the diectio vecto which will aid i simplifyig woig late o. Thus the commo facto 8 was tae out fom the diectio vecto i this case. 5

6 9(ii) 7 ON 8 fo some 9 7 CN CN the lie AB 4 5 Thus ON 4 7 AN ON OA AB AN : AB : 9(iii) Let C be the poit of eflectio of C i lie AB. AC AC CN Equatio of the lie of eflectio is y 8 7 z 9 4 (i) Volume V h Thus h () Suface aea A h (substitute ()) 4 Note: It is easie to epess h i tems of, istead of epessig i tems of h because thee is oly oe h tem i the epessio fo V.

7 (ii) (i) 5 da da ad let d d Theefoe 5 5 d A 4 d So A is miimum whe. 5 Substitute this value of ito (): Suface aea A The GC shows that this is a cubic equatio with positive oots ad egative oot. Sice is positive, thee ae possible values of. Fom GC, =.75,.75 h = 4.88,. Sice h, thus.4, h 4.88 coect to s.f. d Compute cos d dy ad si d si cos dy si cot d cos si (Show) Whe, gadiet of C. As o, d y d. The tagets become paallel to the y ais. Note: The aswe must be epessed i its simplest fom. This ivolves applyig the laws of idices to simplify the idices. 7

8 (ii) y Note: Whe dawig the cuve, studets must show that the gadiet of the cuve is paallel to the vetical ais at ad, ad that the cuve attais its statioay value at. O (iii) Whe, ad whe,. Thus the equied aea ( cos ) d cos cos d cos cos d cos cos d si si 4 (iv) Equatio of omal at P is y( cos p) ta p p si p Substitute y : p cos p ta p si p p si p si p si p p cos Note: Though a calculato caot be used it his questio, studets ca still use the calculato to chec that the fial aswe is coect. p p si cos p si p si p p si p p Hece the omal cosses the ais at p,. 8

Advanced Higher Formula List

Advanced Higher Formula List Advaced Highe Fomula List Note: o fomulae give i eam emembe eveythig! Uit Biomial Theoem Factoial! ( ) ( ) Biomial Coefficiet C!! ( )! Symmety Idetity Khayyam-Pascal Idetity Biomial Theoem ( y) C y 0 0