# MATHEMATICS SYLLABUS SECONDARY 7th YEAR

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1 Europe Schools Office of the Secretry-Geerl Pedgogicl developmet Uit Ref.: D-41-e-2 Orig.: DE MATHEMATICS SYLLABUS SECONDARY 7th YEAR Stdrd level 5 period/week course Approved y the Joit Techig Committee o 9, 10 d 11 Ferury 2011 i Brussels Etry ito force i Septemer D-41-e-2 1/14

2 ALGEBRA (for guidce: 15 periods) TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY Complex Numers I dditio to the cocepts met i the 6th yer pupils must e le to d/or uderstd: represet complex umer geometriclly (Argd digrm) determie the modulus d rgumet of complex umer d its iverse determie the mgitude d rgumet of the product d quotiet of two complex umers the three differet forms of complex umer: o x iy o r cos i si i o r e covert from oe form to other for simple cses. determie the th root, 0 of complex umer give i oth forms z r cos i si i d z r e solve equtios of the type z, 1,2,3 d represet the solutios grphiclly The rgumet,of complex umer, i ll clcultios without techologicl tool is limited to the followig: k or k where k 6 4 Pupils must e le to d/or uderstd: check clcultios d solutios usig CAS ( computer lger system) i the techologicl tool write complex umer i ech of the three forms: o x iy o r cos i si i o r e solve, step y step, equtios of the form z,, D-41-e-2 2/14

3 ANALYSIS (for guidce: 55 periods) The techologicl tool i sequeces c e used to ot oly cofirm wht hs ee studied without clcultor ut lso to mirror, step y step, this process for sequece ot limited to sequeces from the sic set. It my lso e pproprite t times to use the tool to immeditely give solutio. Sequeces (10 periods) TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY I dditio to the cocepts met i the 6th yer pupils must e le to d/or uderstd: Pupils must e le to d/or uderstd: oserve the ehviour of sequece grphiclly s the reltio u, (time plot) determie whether sequece is icresig or decresig o Give the explicit th term the let d study f ' o Give recurrece reltio u u u the let f u 1 d study the sig of u 1 f fid M upper d m lower oud of sequece: u M d u m pply the theorem of covergece/divergece : o icresig sequece ouded y M coverges o decresig sequece ouded y m coverges for the sic set of recurrece reltios i the form u f u where, 1 o f : x x o f : x x o : x f x, c 0 cx d u eter sequece oth explicit d recurrece reltio i pproprite pplictio (spredsheet, CAS, grph etc.) give the explicit th term or recurrece reltio clculte prticulr term give the first terms of sequece fid the explicit th term d or the recurrece reltio where pproprite ivestigte the ehviour of sequece grphiclly s the reltio u, (time plot) crete we digrm of sequece of the form u f u d fid from the digrm, 1 possile limits ivestigte the ehviour of sequece y mipultig the iitil term fid y solvig the equtio f L coverget sequece L, the limit L of D-41-e-2 3/14

4 grphiclly represet such sequeces i we digrm; usig this digrm to oserve whether sequece is covergig d to wht limit Clculte limit of sequece from the sic set of recurrece reltios for simple cses The techologicl tool i expoetils d logrithms c e used to ot oly cofirm wht hs ee studied without clcultor ut lso to mirror, step y step, the sme process. It my lso e pproprite t times to use the tool to immeditely give solutio. TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY Expoetil d Logrithms I dditio to the cocepts met i the 6th yer pupils must e le to d/or uderstd: the properties of idices d logrithms : m m log u v log u log v m m m log u log u 1 1 log log m u u where u v,,, \ 0,1, m, the meig of the umer e Pupils must e le to d/or uderstd: verify the properties of logrithms d powers solve equtios d iequlities ivolvig logrithms d/or expoetils, kowig whe to reject ivlid solutio D-41-e-2 4/14

5 Uless further specified, studets must e le to pply, without techologicl tool, the cocepts referred to i the secod colum kowledge d skills, for the followig sic fuctios where,, c, : polyomils fuctios Px of degree 3 Px Qx where Px d 2 x c; x x c cos x c; si x c; t x c l x ; x l x for 2, 1,0,1,2 Q x re polyomil fuctios of degree 2 x 2x x x x x 2 e ; ce de f; c e e ; e x x c The techologicl tool i followig lysis c e used to ot oly cofirm wht hs ee studied without clcultor ut lso to mirror, step y step, this process for fuctio which is ot limited to oly the sic oes. It my lso e pproprite t times to use the tool to immeditely give solutio. TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY Study of Rel Fuctios of Oe Rel Vrile I dditio to the cocepts met i the 6th yer pupils must e le to d/or uderstd: the cocept of iverse fuctio study the turl logrithmic d the expoetil fuctio of se e exmie the followig chrcteristics for ll the sic fuctios give ove: o domi o itersectio with the coordite xes o limits o symptotes o derivtive d how it my vry o tget t poit o extrem Pupils must e le to d/or uderstd: drw the grph of fuctio choose pproprite uits d widow so tht the essetil chrcteristics of grph c e viewed d studied use the techologicl tool to perform, step y step, the clcultios eeded for exmiig the chrcteristics referred to i the secod colum mipulte y grig d movig the grphs of turl logrithms d the expoetil to e le to oserve the essetil chrcteristics use sliders to ivestigte fmily of fuctios with oe or more prmeters D-41-e-2 5/14

6 o curvture d poits of iflexio o sketch the grph use the clcultio tool (cs) to determie the iverse of fuctio 1 grph f d f to oserve the symmetry of the curves study the iverse trigoometric fuctios rcsi x, rccos x, rctx study the fuctios \ 0,1 x log x d x x where Itegrtio Pupils must e le to d/or uderstd: Pupils must e le to d/or uderstd: uderstd the cocept of primitive determie the primitives of the followig fuctios: k 1 x x k \ 1,, si x, cos x, e x the cocept of itegrl o closed itervl,,, the cocept of improper itegrl f xdx, f xdx d f x dx where, (clculte these for simple cses) mootoicity properties & me vlue of fuctio d f x 0 o, the f x dx 0 o o d f x g x o, the f xdx g x dx o d m f x M o, the solve prolems with or without prmeters ivolvig o primitives o Itegrls o clcultio of res o clcultios of volumes o clcultios of legth of curves Study ot oly pure exmples ut lso pplictios from the fields of physics, iology, ecoomics d others D-41-e-2 6/14

7 m f x dx M the followig properties of itegrls: o f xdx 0 o f xdx f x dx c c o f x dx f x dx f x dx o f x g x dx f x dx g x dx d f x dx f x dx (Lierity of itegrls) clculte the itegrls y direct itegrtio for the fuctios: k 1 x x k \ 1,, si x, cos x, e x for the followig fuctios or fuctios tht c e reduced to them followig simple trsformtio (icludig sustitutio) o Polyomil fuctio o x cx d o x c Px of degree 3 o o x si cx, x cos cx o c l x ; x l x cos x c ; si x c ; t x D-41-e-2 7/14

8 x 2x x x x o e ce de f c e e x 2 e x x c for,, c, ; ; ; 2, 1,0,1,2 ud clculte the itegrls usig: o itegrtio y prts o Itegrtio y sustitutio itegrl iterpreted grphiclly s re pplictio of itegrtio to clculte o res i the ple o volumes of revolutio out the x-xis (I this prt, the im is ot to evlute the computtiol cpilities of studet ut uderstdig of itegrtio.) D-41-e-2 8/14

9 GEOMETRY (for guidce: 40 periods) The techologicl tool i geometry c e used to ot oly cofirm wht hs ee studied without clcultor ut lso to mirror, step y step, prolems which require hrder clcultios. It my lso e pproprite t times to use the tool to immeditely give solutio. TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY Alyticl Geometry i 3 All these cocepts should e resolved without the id of clcultor. However the im without clcultor is ot to ssess the computtiol ility of studet ut their uderstdig of sptil geometry. The etire work o geometry pplies oly to orthoorml coordite systems. The techologicl tool llows studets to solve geometric prolems umericlly usig sets of equtios derived from prolems with two or more ojects (for exmple: three or more ples, two spheres d ple etc.) I dditio to the cocepts met i the 6th yer pupils must e le to d/or uderstd: Pupils must e le to d/or uderstd: fid the crtesi equtio of sphere determie the reltive positio (poits of itersectio d / or equtios where pproprite) of two of the followig ojects: o poit o lie o ple o sphere determie the followig orthogol projectios: o poit oto ple o poit oto lie o lie oto ple fid the cetre d rdius of the circle formed y sphere with ple d two itersectig spheres clculte the followig shortest distces etwee: o two poits missig o poit d ple clculte the cross d dot product of two vectors D-41-e-2 9/14

10 TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY o two prllel ples o lie prllel to ple o two lies o poit d sphere o lie d sphere clculte the cute gle if it exists mde y: o two lies o two ples o lie d ple fid the equtio of the lie of commo perpediculr etwee two lies fid the equtio of the ple of symmetry of two poits fid the equtio(s) of the ple(s) of symmetry of two prllel ples d two itersectig ples D-41-e-2 10/14

11 PROBABILITY AND STATISTICS (for guidce: 40 periods) The techologicl tool i proility ot oly replces pper versios of sttisticl tles ut llows pupils the opportuity to explore d pply the cocepts itroduced i the colum kowledge d skills. It my lso e pproprite t times to use the tool to immeditely give solutio. A theoreticl pplictio or proof of the formule cotied i this sectio c ever form prt of cclurete exm questio. I the proility sectio the cquired kowledge d skills should oly e pplied i simple cses wheever techologicl support is ot used. TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY Coditiol Proility I dditio to the cocepts met i the 6th yer pupils must e le to d/or uderstd: tht two idepedet evets is specil cse of coditiol proility pply up to three idepedet evets : P A B C P A P B P C Ideticl repetitios of three Beroulli trils kow d pply, for 2 : o the lw of totl proility: o P B P A P B A P A P B A Byes Theorem: P A B k k P B Ak PB P A, k 1; 2 Pupils must e le to d/or uderstd: pply up to idepedet evets pply the lw of totl proility, coditiol proility d Byes theorem for D-41-e-2 11/14

12 TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY Discrete Distriutios I dditio to the cocepts met i the 6th yer pupils must e le to d/or uderstd: to uderstd the followig cocepts: o discrete rdom vrile o proility distriutio of discrete rdom vrile d the proility distriutio fuctio of discrete rdom vrile o cumultive distriutio fuctio of discrete rdom vrile F x P X x p xi xi x o expected vlue, vrice d stdrd devitio of discrete rdom vrile i i, i 1 E X x P X x 2 2 Vr X E X E X where 2 2 i i E X x P X x i 1 determie d pply the proility desity fuctio of iomil rdom vrile k k P X k p 1 p k kow d pply the formuls for the me, vrice d stdrd devitio of rdom vrile which is distriuted iomilly p d Vr 2 p 1 p All clcultios usig proility distriutios will e doe usig techologicl tool Pupils must e le to d/or uderstd: clculte proilities for iomilly distriuted rdom vrile step y step usig k k P X k p 1 p k fid cumultive proilities for iomilly distriuted rdom vrile grphiclly represet proilities clculte the vrice d stdrd devitio of rdom vrile tht follows iomil distriutio (Studets will e le to use spredsheet to develop the ove ides of iomil vrile step y step) Apply d use the Poisso distriutio for differet sized itervls D-41-e-2 12/14

13 TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY Cotiuous Distriutios Pupils must e le to d/or uderstd: the followig cocepts: o cotiuous rdom vrile o proility desity fuctio (p.d.f) of cotiuous rdom vrile d its reltio to itegrl clculus: 0 f d f xdx 1 o cumultive distriutio fuctio for cotiuous rdom vrile d its reltio to itegrl clculus: P X f xdx d k F k P X k f x dx, where f( x ) is the p.d.f. o me (expected vlue), vrice d stdrd devitio of cotiuous rdom vrile: E X x f x dx d 2 2 V X E X E X, where 2 2 E X x f x dx the cocept of the orml distriutio d the x stdrdised vrile : z Pupils must e le to d/or uderstd: use the orml distriutio stdrdise y orml distriutios x z D-41-e-2 13/14

14 TOPIC KNOWLEDGE & SKILLS USE OF TECHNOLOGY Modellig Pupils must e le to d/or uderstd: the coditios whe vrile c e modelled y the followig distriutios: o discrete uiform o cotiuous uiform o iomil o Poisso questio the suitility of model y checkig tht the coditios of distriutio hve ee me Pupils must e le to d/or uderstd: ppropritely model situtios y the followig distriutios o discrete uiform o cotiuous uiform o iomil o Poisso o Norml ivestigte dt give i tle or digrm to determie correspodig orml distriutio recogise the coditios for Poisso pproximtio to the iomil 50 d p 0.1 recogise the coditios for orml pproximtio to the iomilpq 9 d use this pproximtio icludig the cotiuity correctio x 0.5 z D-41-e-2 14/14

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