M06/5/MATHL/HP2/ENG/TZ0/XX MATHEMATICS HIGHER LEVEL PAPER 2. Thursday 4 May 2006 (morning) 2 hours INSTRUCTIONS TO CANDIDATES

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1 IB MATHEMATICS HIGHER LEVEL PAPER DIPLOMA PROGRAMME PROGRAMME DU DIPLÔME DU BI PROGRAMA DEL DIPLOMA DEL BI Thursday 4 May 006 (morig) hours INSTRUCTIONS TO CANDIDATES Do ot ope this examiatio paper util istructed to do so Aswer all the questios Uless otherwise stated i the questio, all umerical aswers must be give exactly or correct to three sigificat figures pages

2 Please start each questio o a ew page Full marks are ot ecessarily awarded for a correct aswer with o workig Aswers must be supported by workig ad/or explaatios I particular, solutios foud from a graphic display calculator should be supported by suitable workig, eg if graphs are used to fid a solutio, you should sketch these as part of your aswer Where a aswer is icorrect, some marks may be give for a correct method, provided this is show by writte workig You are therefore advised to show all workig 1 [Maximum mark: 1] Let A be the poit (, 1, 0), B the poit (, 0, 1 ) ad C the poit (1, m, ), where m, m < 0 (a) (i) Fid the scalar product BA g BC Hece, give that ABC $ = arccos, show that m = 1 [6 marks] Determie the Cartesia equatio of the plae ABC [4 marks] (c) Fid the area of triagle ABC [ marks] (d) (i) The lie L is perpedicular to plae ABC ad passes through A Fid a vector equatio of L The poit D( 6, 7, ) lies o L Fid the volume of the pyramid ABCD [8 marks] [Maximum mark: 1] π π Let z = cosθ + i siθ, for < θ < 4 4 (a) (i) Fid z usig the biomial theorem Use de Moivre s theorem to show that cos 4cos θ = θ cosθ ad siθ = siθ 4si θ [10 marks] Hece prove that siθ siθ = taθ [6 marks] cosθ + cosθ (c) Give that siθ = 1, fid the exact value of taθ [5 marks]

3 [Maximum mark: ] Particle A moves i a straight lie, startig from O A, such that its velocity i metres per secod for 0 t 9 is give by v = 1 A t + t + Particle B moves i a straight lie, startig from O B, such that its velocity i metres per secod for 0 t 9 is give by v B t = e 0 (a) Fid the maximum value of v A, justifyig that it is a maximum [5 marks] Fid the acceleratio of B whe t = 4 [ marks] The displacemets of A ad B from O A ad O B respectively, at time t are s A metres ad s B metres Whe t = 0, s A = 0, ad s B = 5 (c) Fid a expressio for s A ad for s B, givig your aswers i terms of t [7 marks] (d) (i) Sketch the curves of s A ad s B o the same diagram Fid the values of t at which s = s [8 marks] A B Tur over

4 4 4 [Total mark: 1] Part A [Maximum mark: 1] The time, T miutes, required by cadidates to aswer a questio i a mathematics examiatio has probability desity fuctio 1 ( 1t t 0), for 4 t 10 f ( t) = 7 0, otherwise (a) Fid (i) µ, the expected value of T ; σ, the variace of T [7 marks] A cadidate is chose at radom Fid the probability that the time take by this cadidate to aswer the questio lies i the iterval [ µ σ, µ ] [5 marks] Part B [Maximum mark: 19] Adrew shoots 0 arrows at a target He has a probability of 0 of hittig the target All shots are idepedet of each other Let X deote the umber of arrows hittig the target (a) Fid the mea ad stadard deviatio of X [5 marks] Fid (i) P( X = 5 ) ; P( 4 X 8) [6 marks] Bill also shoots arrows at a target, with probability of 0 of hittig the target All shots are idepedet of each other (c) (d) Calculate the probability that Bill hits the target for the first time o his third shot Calculate the miimum umber of shots required for the probability of at least oe shot hittig the target to exceed 099 [ marks] [5 marks]

5 5 5 [Maximum mark: 4] x Cosider the system of equatios T y =, where T = 0 r z 4 r 0 s (a) Fid the solutio of the system whe r = 0 ad s = [4 marks] The solutio of the system is ot uique (i) Show that s = 9 r Whe r = ad s =18, show that the system ca be solved, ad fid the geeral solutio [11 marks] (c) Use mathematical iductio to prove that, whe r = 0, T ( ) ( ) = s, + [9 marks]

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physicsandmathstutor.com physicsadmathstutor.com 5. Solve, for 0 x 180, the equatio 3 (a) si( x + 10 ) =, 2 (b) cos 2x = 0.9, givig your aswers to 1 decimal place. (4) (4) 10 *N23492B01028* 8. (a) Fid all the values of, to 1 decimal