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8/14/2019 'OvcrPascrorJJ

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'ovcrPascrorJJ L? ffffiffii##ff.o.l$fl " Promorronal ExamrnationMATHEMATICSHigher 2 9740

22 September 20083 hours

AdditionallMaterialsr AnswerPaperCover Pag

READ THESE INSTRUCTIONS FIRSTWrile your name and civics class on all lhe wo.k you hand inWrile in dad( blue or black pen on both sides orthe paperYou may use a sofr pencilfor any diagrams orgraphsDo nor use 5qhllgnic s glue or cole r,on flJrdAnswe. all the questionsGive non exact numerical answers conect to 3 significant figures, o. 1 decimal place in thecase ofangles in degrees, unless a different levelofaccuracy is specified in the queslion.You are expected to use a graphic calcubtor.Unsupporled answers from a graphic calculato. are allowed unless a question specificallyWhere unsuppo.ted answers from a gmphic lculator are not allowed in a questioo, you arerequked to present the mathematical steps using mathematacal notalions and not calculatorYou are reminded of the need for clear presentation in your answers.The number ofmarks is given in b.ackets []at lhe end ofeach quesiion o. part quesiion_Al lhe end of the examination, fasten all your work together, with the cover page in

Can do the $ hole quesfion. t/Can do part ofques tioo only. './


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f-:-;::Expand ]Ija in ascending powers of r up to arld including thc tcrm ir, 12 and statethe range ofvalues of x for which this xpansion is valid- t5l

2 (a)(b)

Differentiate tar-r(hr) with respecl to J..Show that the tangent to the curve I = e 'sin2x ai

a

tzlr = I is pe.pendicular to the

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correct to 3

Prove by inducron rhar Y1 r'' ' '' 1 - l- n'ln t ti for n t'a'Hence evaluate ! (:'' + ")

(a) With the aid ofa calcutator(i) find thc values of r. , 13 and i1 , giving your answerssignificant figures.(ii) State the behaviour oathe seque.ce.

A sequence ot negative numbers ls defhed by -r",, = ' l'l , *n-. r, = ] f"r ,tf.r, - 4 7

(b) Giventhat -I,, ) 1as lI + oo, find, without the use of a calculator, thc value oft4l

t3l(a) An ariihmclic series I has first term a and a geomet.ic series G has first tel-ln 6.

The common difference of I is four times the first term of G and the commonralio ofG is lwlce rhe lirsl lerm of lEach tem of I is added to the corresponding term of C to folm the terms of athird series s. civen that the first iwo terms of ,tare -1 and0 resoectrvelv- findthe valu ofa, t5l


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o)3

/, ,*r \Urven that ,, = tn | _ l.wherer>-1.\ r+.r /" /,show rh,r tL = t"l '*" Ifr' (t+x/Hence frnd iz" in icrms ofi. when -t < jr< l.

The tunctions f and g are detured as follows:f -. x - 3 2d'l

A curve C is given by x1 +3yt +2ry=3 /i wher /r is a constant. Find the rangeof values of i for which C does not have any tangent parallel ro the r,axis. t5lA graph G is given by the parametdc equations r=2-sin2d, y=2+cos20,

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g: jr- (r+1)r+l , 'R. jr


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4(a) Given 1ha1jr is reai, prove thai x'z 5jr+9 is always positive

Hence hnd the rangc of values of r for which 'a 9x-l x+4

1hc diagrarn shows thc graph of l, = f(n) which interscts they axistufting poiol at ( 3, 0) and a horizoniat asyrnpiotel = 4 as r -) +.o.On separale diagrams. sketch the graphs of(a) r=l+f(2r),(b) r,.rI=r(.{),(c) rr = f(-t),

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(lt April, May, June and July entered a hot dog earing contest. 't'he table below showsthe anount offood and d.inks and total number ofcalorics consumed.By finding thc number ofcalorie.s ofcach item. determine thc numbcr of slasscsofju,ce corLurred hy lul! lsl

at (0, 2) and has a

(d) I = f'(r), where f'(-y) is ihe dcrivative tuncrion of f().).In each case, show clearly ihc axial iitercepts, the aslmpto(es and thc coordinatestuming points whe.e applicable.

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Number ofhot dogs

Number ofburu

Number ofglasses ofjuice Total numberofcalorics

ADril 21 l1 6 3343Ntay 33 2r) () f72)l3 31 i0 ,1668

JuLv 26 24 2948


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5

11 Relative to an origin O, points I and B have posiiion veclors 2i+j1-2kand 3i-2j +pkt2lt3t

(c) In the case where p : 5, hnd the position vector of the point M on the Lnet2lt4t

respectively, wherep is a constant-(a) Find the value of p for which Ol and OB are perpendicular.(b) In the case where p = 6, find angle lOB, correct to the nearest degree,

sgment .48 such that AM. MB = 1.2.(d) Find the values of p fo. which ihe length of,{B is JiT units.

An inverted right circular mne ofradius 2 cm and height 6 cm isinscribed ir a larger right ci.cularcore of radius 5 cm and height l0cm. The two bases are parallel andthe vertex of the smaller cone liesat the centre of the base of thelarger conc.Water is flowiog from a hole at thevedex of the larger cone to thesmaller cone at a constant rate oft6

(a)2

Al tirne r se{onds. the volume and lhe deplh of llle water in rhe smaller cone are/ cmr and I cm respectively-{i} Show tlnt t' = !rn'.)7(ii) Calculate th exact rat ofchange offt when ft = 2.

121

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(b)2

The diagrarn above shows a rectangle P0RS inscribd in a semi-circle r,{ith radiusl0 cm-If PO = x cm, fllld, in te(es of )., a[ expression for the perimeter p cm of therectangle POrRtShow that the area I cm2 ofthe rectangle POR,S is given by I =Find the exact value of p when,4 has its stationary value.

- End of Paper -

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1 tsinomial Series Inequalities I I I, I)4I"22 Applications of Differentiation:fangents and Normals I(a)

3 Vathematical lnduction I (3.' +,') = 826325 ri004 Sequences and Series ) (j) r, = {.586,rr = 0.820,r, = {) 925

(;i.) The sequence decreases aod coverges to) 1=. I5 AP,GP & Sigma Notation,Method of Difference ) a=-l){ri )r, ln{l+v)6 Functions (i) llrax value olk = -lg'i,1- r .[-,,e n.-.::rii) fg exists

fg(i) = 3-2c"'rr''r,-1 lR,r < IR," .'( .-.r -2.,')

7 Powers Series lly-Ln2 r-rr -ir")Rln{r +tr- -n2+l-r+l I lr":l"I \r 2 )

(ii)ii0

8 DifferentiationStationary points t k'lI ,4 = 2, G i" in"."*i,uI Equations and lnequalities r) Calorics in tlol dog = 72. Calories in bun = 41,lalo.ies in Juice = 72. No ofGlasses = 4'10 lurve Sketching11 Vectors ,),=,,0,..,",o I i ],r,,=, *,1? Applicataons of Differentiation 41;444 = 1"-7' t p - 3{),f2cn

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