Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si ) + (cos + si )y = cos + si, [8 marks] a Fid the value of 1 lim( cot) 0 b Fid the iterval of covergece of the ifiite series [10 marks] ( + ) ( + ) ( + ) + + + 1 (i) Fid the Maclauri series for c l(1 + si ) up to ad icludig the term i Hece fid a series for l(1 si ) up to ad icludig the term i (iii) Deduce, by cosiderig the differece of the two series, that π π 16 l (1 + ) [1 marks] Give that + y ta = si, ad y = 0 = π whe, fid the maimum value of y [11 marks]
4a The diagram shows a sketch of the graph of y = 4 for > 0 By cosiderig this sketch, show that, for Z +, 1 r=+1 r 4 < < 4 1 r= r 4 Let 4b S = 1 r r=1 4 Use the result i (a) to show that, for, the value of S lies betwee [8 marks] 1 1 1 + r r=1 4 1 1 + r r=1 4 ad (i) Show that, by takig 4c = 8, the value of S ca be deduced correct to three decimal places ad state this value The eact value of S is kow to be π 4 where N N Z + Determie the value of N Now let 4d ( 1) r+1 T = r r=1 4 Fid the value of T correct to three decimal places [ marks]
5a (i) Sum the series r r=0 Hece, usig sigma otatio, deduce a series for (a) ; 1 1+ (b) arcta ; (c) π 6 [11 marks] 5b Show that 100! ( mod 15) =1 [4 marks] 6a Assumig the series for e, fid the first five terms of the Maclauri series for [ marks] 1 π e 6b (i) Use your aswer to (a) to fid a approimate epressio for the cumulative distributive fuctio of N(0, 1) Hece fid a approimate value for P( 05 Z 05), where Z N(0,1) A machie fills cotaiers with grass seed Each cotaier is supposed to weigh 8 kg However the weights vary with a stadard deviatio of 054 kg A radom sample of 4 bags is take to check that the mea weight is 8 kg 6c State ad justify a appropriate test procedure givig the ull ad alterate hypotheses 6d What is the critical regio for the sample mea if the probability of a Type I error is to be 5%? 6e If the mea weight of the bags is actually 81 kg, what would be the probability of a Type II error? [ marks] The fuctio f() is defied by the series (+) 1 (+) (+) f() = 1 + + + + 7a Write dow the geeral term [1 mark] 7b Fid the iterval of covergece [1 marks] Solve the differetial equatio 7c (u + v dv ) = v, givig your aswer i the form du u = f(v) [8 marks]
Fid the geeral solutio of the differetial equatio 8a (1 ) = 1 + y, for < 1 (i) 8b y = f() f(0) = π Show that the solutio that satisfies the coditio is f() = arcsi + π 1 Fid lim f() 1 (i) Show that the improper itegral 9a 1 is coverget 0 +1 1 =0 +1 Use the itegral test to deduce that the series is coverget, givig reasos why this test ca be applied (i) Show that the series 9b ( 1) is coverget =0 +1 If the sum of the above series is S, show that < S < 5 For the series 9c =0 +1 (i) determie the radius of covergece; determie the iterval of covergece usig your aswers to (b) ad (c) By evaluatig successive derivatives at 10a = 0, fid the Maclauri series for lcos up to ad icludig the term i 4 [8 marks] 10b Cosider l cos lim 0 R, where Usig your result from (a), determie the set of values of for which (i) (iii) the limit does ot eist; the limit is zero; the limit is fiite ad o-zero, givig its value i this case (i) Show that 11a d (secθ taθ + l(secθ + taθ)) = sec θ dθ Hece write dow sec θdθ
Cosider the differetial equatio 11b (1 + ) + y = 1 + give that y = 1 whe = 0 (i) Use Euler s method with a step legth of 01 to fid a approimate value for y whe = 0 Fid a itegratig factor for determiig the eact solutio of the differetial equatio (iii) Fid the solutio of the equatio i the form y = f() (iv) To how may sigificat figures does the approimatio foud i part (i) agree with the eact value of y whe = 0? [4 marks] 1 Cosider the differetial equatio = y for which y = 1 whe = 0 Use Euler s method with a step legth of 01 to fid a approimatio for the value of y whe = 04 1 (a) e e e (b) Assumig the Maclauri series for, determie the first three o-zero terms i the Maclauri epasio of The radom variable X has a Poisso distributio with mea μ Show that P (X 1( mod )) = a + be cμ where a, b ad c are costats whose values are to be foud 14 Cosider the ifiite series S = =1 ( 1) (a) Determie the radius of covergece (b) Determie the iterval of covergece [11 marks] 15 Cosider the differetial equatio + y ta = cos 4 give that y = 1 whe = 0 (a) y = f() Solve the differetial equatio, givig your aswer i the form (b) (i) By differetiatig both sides of the differetial equatio, show that [18 marks] d y + y = 10 si cos y Hece fid the first four terms of the Maclauri series for
16 (a) (i) Usig l Hôpital s rule, show that lim = 0; Z +, λ R + e λ Usig mathematical iductio o, prove that 0 e λ! = ; N, λ R + λ +1 (b) The radom variable X has probability desity fuctio Givig your aswers i terms of ad λ, determie (i) E(X) ; X (c) the mode of Customers arrive at a shop such that the umber of arrivals i ay iterval of duratio d hours follows a Poisso distributio with mea 8d The third customer o a particular day arrives T hours after the shop opes (i) Show that P(T > t) = e 8t (1 + 8t + t ) T (iii) T Fid a epressio for the probability desity fuctio of Deduce the mea ad the mode of λ +1 e λ! 0, Z +,λ R + f() = { otherwise 17 Use l Hôpital s rule to fid lim(csc cot ) 0
18a Differetiate the epressio ta y with respect to, where y is a fuctio of [ marks] 18b Hece solve the differetial equatio + si y = cos y give that y = 0 whe = 1 Give your aswer i the form y = f() Solve the followig differetial equatio 19 givig your aswer i the form y = f() ( + 1)( + ) + y = + 1 [11 marks] The fuctio f is defied by f() = l(1 + si ) Show that 0a f 1 () = 1+si [4 marks] Determie the Maclauri series for 0b f() as far as the term i 4 Deduce the Maclauri series for 0c l(1 si ) as far as the term i 4 [ marks] By combiig your two series, show that 0d lsec = + 4 + 1 [4 marks]
Hece, or otherwise, fid 0e l sec lim 0 [ marks] Whe a scietist measures the cocetratio μ of a solutio, the measuremet obtaied may be assumed to be a ormally distributed radom variable with mea μ ad stadard deviatio 16 He makes 5 idepedet measuremets of the cocetratio of a particular solutio ad correctly calculates the followig 0f cofidece iterval for μ [ 7, 61] Determie the cofidece level of this iterval He is ow give a differet solutio ad is asked to determie a 0g 95% cofidece iterval for its cocetratio The cofidece iterval is required to have a width less tha Fid the miimum umber of idepedet measuremets required Let S 1 k k=1 = Show that, for 1a, 1 S > S + [ marks] Deduce that 1b m S m+1 > S + Hece show that the sequece 1c { S } is diverget [ marks] Calculate the followig limit a 1 lim 0 [ marks] Calculate the followig limit b lim 0 (1+ ) 1 l(1+) Solve the differetial equatio + y = 1 + give that y = 1 whe = [9 marks]
The fuctio 4a f is defied by f() = e+ e (i) Obtai a epressio for f () (), the th derivative of f() with respect to Hece derive the Maclauri series for f() up to ad icludig the term i 4 (iii) Use your result to fid a ratioal approimatio to f ( ) 1 [1 marks] (iv) Use the Lagrage error term to determie a upper boud to the error i this approimatio 4b Use the itegral test to determie whether the series l is coverget or diverget =1 [9 marks] Usig l Hôpital s Rule, determie the value of 5 ta lim 0 1 cos Cosider the ifiite series ( 1) +1 1 =1 S = si ( ) Show that the series is coditioally coverget but ot absolutely coverget 6a 6b Show that S > 04 [ marks] The fuctio f is defied by f() = e cos Show that 7a f () = e si [ marks] 7b Determie the Maclauri series for f() up to ad icludig the term i 4 By differetiatig your series, determie the Maclauri series for 7c e si up to the term i [4 marks]
Cosider the differetial equatio + y sec = (sec ta), where y = whe = 0 8a Use Euler s method with a step legth of 01 to fid a approimate value for y whe = 0 8b (i) By differetiatig the above differetial equatio, obtai a epressio ivolvig d y Hece determie the Maclauri series for y up to the term i (iii) Use the result i part to obtai a approimate value for y whe = 0 [8 marks] 8c (i) Show that sec + ta is a itegratig factor for solvig this differetial equatio Solve the differetial equatio, givig your aswer i the form y = f() (iii) Hece determie which of the two approimate values for y whe = 0, obtaied i parts (a) ad (b), is closer to the true value [11 marks] Iteratioal Baccalaureate Orgaizatio 017 Iteratioal Baccalaureate - Baccalauréat Iteratioal - Bachillerato Iteracioal