Mathematics Tshwane South – South Africa Education ...



GAUTENG DEPARTMENT OF EDUCATIONSCHOOL BASED ASSESSMENT (SBA)MATHEMATICSGRADE 112017CONTENTS PageIntroduction3Informal or daily assessment4Formal assessment5Programme of assessment6Moderation Form7Assessment tasks 9 Test 1 Test 2Test 3Test 4Assignment 1 Project Investigation INTRODUCTION Assessment is a continuous planned process of identifying, gathering and interpreting information about the performance of learners, using various forms of assessment. It involves four steps: generating and collecting evidence of achievement; evaluating this evidence; recording the findings and using this information to understand and assist in the learner’s development to improve the process of learning and teaching. Assessment should be both informal (Assessment for Learning) and formal (Assessment of Learning). In both cases regular feedback should be provided to learners to enhance the learning experience.Although assessment guidelines are included in the Annual Teaching Plan at the end of each term, the following general principles apply: Tests and examinations are assessed using a marking memorandum.Assignments are generally extended pieces of work completed at home. Assignments can be collections of past examination questions, but should focus on the more demanding aspects as any resource material can be used, which is not the case when a task is done in class under strict supervision. At most one project or investigation and an assignment if this is the preferred option should be set in a year. The assessment criteria need to be clearly indicated on the project specification. The focus should be on the mathematics involved and not on duplicated pictures and regurgitation of facts from reference material. The collection and display of real data, followed by deductions that can be substantiated from the data, constitute good projects. A project, in the context of Mathematics, is an extended task where the learner is expected to select appropriate Mathematical content to solve a context-based problem. Investigations are set to develop the skills of systematic investigation into special cases with a view to observing general trends, making conjectures and proving them. To avoid having to assess work which is copied without understanding, it is recommended that while the initial investigation can be done at home, the final write up should be done in class, under supervision, without access to any notes. Investigations are marked using rubrics which can be specific to the task, or generic, listing the number of marks awarded for each skill: 40% for communicating individual ideas and discoveries, assuming the reader has not come across the task before. The appropriate use of diagrams and tables will enhance the investigation. 35% for the effective consideration of special cases; 20% for generalising, making conjectures and proving or disproving these conjectures; and 5% for presentation: neatness and visual RMAL OR DAILY ASSESSMENT The aim of assessment for learning is to collect continually information on a learner’s achievement that can be used to improve individual learning. Informal assessment involves daily monitoring of a learner’s progress. This can be done through observations, discussions, practical demonstrations, learner-teacher conferences, informal classroom interactions, etc., Informal assessment may be as simple as stopping during the lesson to observe learners or to discuss with learners how learning is progressing. Informal assessment should be used to provide feedback to the learners and to inform planning for teaching, it need not be recorded. This should not be seen as separate from learning activities taking place in the classroom. Learners or teachers can evaluate these tasks. Self-assessment and peer assessment actively involve learners in assessment. Both are important as these allow learners to learn from and reflect on their own performance. Results of the informal daily assessment activities are not formally recorded, unless the teacher wishes to do so. The results of daily assessment tasks are not taken into account for promotion and/or certification purposes. FORMAL ASSESSMENT All assessment tasks that make up a formal programme of assessment for the year are regarded as Formal Assessment. Formal assessment tasks are marked and formally recorded by the teacher for progress and certification purposes. All Formal Assessment tasks are subject to moderation for the purpose of quality assurance. Formal assessments provide teachers with a systematic way of evaluating how well learners are progressing in a grade and/or in a particular subject. Examples of formal assessments include tests, examinations, practical tasks, projects, oral presentations, demonstrations, performances, etc. Formal assessment tasks form part of a year-long formal Programme of Assessment in each grade and subject. Formal assessment tasks in Mathematics include tests, a June examination, a trial examination (for Grade 12), a project or an investigation. The forms of assessment used should be age- and developmental- level appropriate. The design of these tasks should cover the content of the subject and include a variety of activities designed to achieve the objectives of the subject. Formal assessment tasks need to accommodate a range of cognitive levels and abilities of learners as indicated in the CAPS document. Programme of Assessment: Learners are expected to have eight (8) formal assessment tasks for their school-based assessment, including end of year examinations. The weighting and number of tasks are listed below:TERMTASKWEIGHTDATETerm 1Project or InvestigationTest2010Term 2Test or AssignmentMid-year examination1030Term 3TestTest1010Term 4Test10School-based Assessment100School-based Assessment mark (as % of promotion mark)25%End-of-year Examinations75%Promotion mark100%NB: The school programme of assessment should indicate specific dates when tasks are to be administered during the year. In the event that teachers are not able to abide by the set dates due to unforeseen circumstances, minimal deviations are permissible. Although the project/investigation is indicated in the first term, it could be scheduled in terms 2 or 3. Only ONE project/investigation should be set per year. Tests should be at least ONE hour long and count at least 50 marks.ANNEXURE A635-63754000PRE-MODERATION OF SBA ACTIVITIES SET AT SCHOOL LEVELMATHEMATICS DISTRICTSUBJECTGRADENAME OF SCHOOLNAME OF EDUCATOR (S)NAME OF HODNAME OF MODERATORNAME OF SUBJECT ADVISORDATEMODERATIONFRONT PAGEYESNOCOMMENTName of schoolNames of moderator and examinerTime allocationTotal markSubject, e.g. Mathematics or Mathematical LiteracyGrade, e.g. Grade 10 or Grade 11 or Grade 12Assessment activity, e.g. Assignment or Investigation or ProjectDate, e.g. June 2017Are the instructions to candidates clearly specified and unambiguous?REST OF THE ACTIVITYYESNOCOMMENTSAll pages numberedMark totals indicated correctly per subsectionsMark totals indicated correctly per questionCorrelation between mark allocation, level of difficulty and time allocationIs “please turn over” indicated?Is the assessment activity complete with grid, memorandum, and diagram sheets?LAYOUT OF THE ACTIVITYYESNOCOMMENTSIs the appearance and typing consistent? e.g. font type and sizeAre sketches clear?Are sketches labelled and/or numbered?STANDARD OF ASSESSMENT TASKYESNOCOMMENTSDoes the task correspond with the programme of assessment?Is there a verbatim reproduction of questions from previous SBA activities?Are questions ordered from easy to difficult, e.g. Level 1 to Level 4 (different cognitive levels)?Are the subsections grouped by topics?Are questions concise and to the point (not ambiguous)?Are the assessment standards appropriately linked and integrated?Are the questions compliant with CAPS?Is the mark allocation/weighting for the task in accordance with CAPS?ASSESSMENT TOOLSYESNOCOMMENTSAre the assessment tools e.g. rubric, memoranda, checklists, etc. for the assessment task included?Are the tools on standard?AREAS OF GOOD PRACTICECHALLENGESRECOMMENDATIONS/FOLLOW-UPYESNOThe SBA activity is approved.The SBA activity is provisionally approved and requires some adjustments.The SBA activity is not approved and must be resubmitted on the following date:____________________ ___________________ ___________EDUCATOR SIGNATURE DATE____________________ ___________________ ___________HOD/ SUBJECT HEAD SIGNATURE DATE(MODERATOR)____________________ ___________________ ___________DISTRICT FACILITATOR SIGNATURE DATEASSESSMENT TASKSTEST 1INSTRUCTIONS AND INFORMATIONRead the following instructions carefully before answering the questions.This question paper consists of 3 questions.Answer ALL the questions.Number the answers correctly according to the numbering system used in this question paper.Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers.Answers only will not necessarily be awarded full marks.You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.If necessary, round off answers to TWO decimal places, unless stated otherwise.Diagrams are NOT necessary drawn to scale.Write neatly and legibly.QUESTION 1 Solve for 1.1.1 (4)1.1.2 (5)1.1.3 (4)1.2 Solve simultaneously for and in the following set of equations: (5)For which value(s) of will the expression be non-real? (3)1.4 Simplify, without the use of a calculator: (4) [25]QUESTION 2The sequence 4; 9; 37; . . . is a quadratic sequence.2.1 Calculate (3) 2.2 Hence, or otherwise, determine the term of the sequence.(4)[7]QUESTION 3In the diagram, the points P (-3; 5), S (1; -2), Q (5; 1) and R (5; 6) are given.M is the midpoint of RS.261937514033500243840020320000426953633020R (5; 6)00R (5; 6)991235109855P (-3; 5)00P (-3; 5)29527501790691704974321945466725010883910029502331662550S (1; -2)00S (1; -2)4270256636006Q (5; 1)00Q (5; 1)45720010560053.1 Calculate the gradient on PQ.(3)3.2 Determine the equation of the line RS in the form (4)3.3 Calculate the length of PQ (leave answer in surd form).(3)3.4 Calculate the coordinates of M, the midpoint of RS.(3)3.5 What relationship is there between the line segments PQ and RS? (2) Give a reason for your answer.3.6 Prove that PSQR is a Kite? Motivate your answer. (3) [18] TOTAL [50]TOETS 1INFORMASIE en INSTRUKSIESLees die volgende instruksie deeglik voordat jy die vrae beantwoord.Hierdie vraestel bestaan uit 3 vrae.Beantwoord ALLE vrae.Nommer alle antwoorde korrek en volgens die nommering gebruik in die vraestel Toon alle berekenings, diagramme, grafieke, ens. wat jy gebruik het om die vrae te beantwoord.Slegs ? antwoord sal nie noodwendig volpunte toegeken word nie.Jy mag slegs gebruik maak van ? goedgekeurde wetenskaplike sakrekenaar (nie-programmeerbaar en nie-grafies), tensy anders vermeld.Rond af tot TWEE desimale plekke indien nodig, tensy anders vermeld.Diagramme is nie noodwendig op skaal geskets nie.Skryf netjies en lessbaar.VRAAG 1 Los op vir 1.1.1 (4)1.1.2 (5)1.1.3 (4)1.2 Los x en y in die volgende gelyktydige vergelykings: (5)Vir watter waarde(s) van sal die volgende uitdrukking nie-re?el wees? (3)1.4 Vereenvoudig, sonder die gebruik van ? sakrekenaar: (4) [25]VRAAG 2Die ry 4; 9; 37; . . . is ? kwadratiese ry.2.1 Bereken (3) 2.2 Vervolgens, of andersins, bepaal die term van ry.(4)[7]VRAAG 3In die diagram, is die punte P (-3; 5), S (1; -2), Q (5; 1) en R (5; 6) gegee.M is die middelpunt van RS.261937514033500243840020320000426953633020R (5; 6)00R (5; 6)991235109855P (-3; 5)00P (-3; 5)29527501790691704974321945466725010883910029502331662550S (1; -2)00S (1; -2)4270256636006Q (5; 1)00Q (5; 1)45720010560053.1 Bereken die gradient/ helling van PQ.(3)3.2 Bepaal die vergelyking van die lyn RS in die vorm (4)3.3 Bereken die lengte van PQ (laat jou antwoord in wortelvorm).(3)3.4 Bereken die ko?rdinate van M, die middelpunt van RS.(3)3.5 Wat is die verwantskap tussen die lynsegment PQ en RS? (2) Gee ? rede vir jou antwoord.3.6 Bewys dat die figuur PSQR ? Vlie?r is. Motiveer jou antwoord. (3) [18] TOTAAL [50]TEST 1 MEMORANDUMQUESTION 11.1.1 or √ standard form√ factors√√ answers (4)1.1.2 or √ standard form√ substitution into formula√ 265√√ answers (5)1.1.3CV: or or √ standard form√ critical values√√ answers (4)1.2Substitute in or or OR or or √ substitution√ standard form√ factors√ both value√ both value (5) √ substitution√ standard form√ factors√ both values√ both values (5)1.3 will be non-real if:However, the expression will be undefined if . Therefore, the expression will be non-real if where √ non-real if √ √ (3)1.4√ √ √ applying exponential laws√ answer (4) [25]QUESTION 22.11762125170815126682517081511144251708157239001708154191001708151809751708154 9 3715525751035051114425160655800100160655419100160655 5 First difference: 5; Second difference: OR OR √ first differences √ second difference√ answer (3)√ equating√ manipulation√ answer (3)√ first differences√ equating√ answer (3)3.2116205098425962025984257239009842547625098425285750984254 9 20 3766675-635962025145415723900145415476250145415285750145415 5 11 17 6 6 √ √ √ √ (4) [7]QUESTION 33.1√ gradient formula√ substitution in gradient formula√ answer (3)3.2 = 2√ substitution in gradient formula√ √ substituting gradient and R(5;6) or S(1;-2)√ equation in correct form (4)3.3 = = = or 4√ formula√ substitution √ answer (3)3.4√ correct formula√ substitution in mid-point formula√ answer (3) 3.5Thus PQ RS√ PQ RS√ (2)3.6The diagonal PQ bisects diagonal RS at NOTE: For adjacent sides are equal this must be shown by calculations thatPS=PR= √ and SQ=QR=5√√ Length PS and PR√ PQ bisects RS√ (3) [18]TEST TERM 359499590170FORMAL TEST - QUESTION PAPER00FORMAL TEST - QUESTION PAPER2200910113030GRADE 1100GRADE 11gGR1126490164465MATHEMATICS TERM 3 TESTAUGUST 201600MATHEMATICS TERM 3 TESTAUGUST 2016 MARKS: 50TIME: 1 hourThis question paper consists of 5 pages including 1 diagram sheet.INSTRUCTIONS AND INFORMATIONRead the following instructions carefully before answering the questions.1.2.3.4.5.6.7.8.9.10.This question paper consists of 3 questions. Answer ALL the questions.Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers.An approved scientific calculator (non-programmable and non-graphical) may be used, unless stated otherwise.Answers only will not necessarily be awarded full marks.If necessary, answers should be rounded off to TWO decimal place, unless stated otherwise.Diagrams are NOT necessarily drawn to scale.ONE diagram sheet is attached at the end of this question paper. Write your name on this sheet in the space provided and insert it in your ANSWER BOOK.Number the answers correctly according to the numbering system used in this question paper.It is in your own interest to write legibly and to present your work neatly.QUESTION 11.1Consider the toy alongside made from a cylinder with a height of 14cm and a hemisphere (half of sphere) with a radius of 13cm. (Formulae: ; ; ; )11277605207013cm14cm13cm14cm1.1.1Calculate the volume of the toy.(6)1.1.2Calculate the total exterior surface area of the toy.(6) [12]QUESTION 22.1Complete: 2.1.1The opposite angles of a cyclic quadrilateral are…(1)2.1.2Equal chords subtend… (1)2.1.3The angle between a tangent and a chord is…(1)2675255155411OSDATMOSDATM2.2MAT is a tangent to circle O. Use the sketch and prove the theorem that states that .(6)2.3M is the midpoint of chord PT of a circle with centre O. OQ is a radius passing through M. PQ is produced to intersect tangent TA at A, such that TA PA. 170815232410P1QATOM21212300P1QATOM212123Prove that:2.3.1MTAQ is a cyclic quadrilateral.(4)2.3.2PQ = TQ(4)2.3.3(4)[21]QUESTION 33.1Two circles intersect each other in A and C. O, the centre of the smaller circle, lies on the circumference of the bigger circle. D is a point on the smaller circle and DC produced intersects the bigger circle in B. BO produced intersects AD in M. AO, OC, and AB are drawn. 703580-306705BAMDCOxBAMDCOx3.1.1If , give with reasons three more angles each equal to x.(6)3.1.2Express in terms of x. Give a reason.(2)3.1.3Hence, prove that BMAD.(4)3.1.4If it is further given that , prove that AD is a tangent to circle ABO.(5) [17]TOTAL: 50DIAGRAM SHEET 124860257810500LEARNER’S NAME & SURNAME:QUESTION 1.1QUESTION 2.2307721090805OSDATMOSDATM2266957556513cm14cm13cm14cm QUESTION 2.3QUESTION 33225800226695BAMDCOx00BAMDCOx-124358190425P1QATOM21212300P1QATOM212123-685803465195001154430149224FORMELE TOETS – VRAESTEL00FORMELE TOETS – VRAESTEL221170578740GRAAD 1100GRAAD 111126490164465WISKUNDE TERMYN 3 TOETSAUGUSTUS 201600WISKUNDE TERMYN 3 TOETSAUGUSTUS 2016 PUNTE: 50TYDSDUUR: 1 uurHierdie vraestel bestaan uit 5 bladsye insluitende 1 diagramvel.INSTRUKSIES EN INLIGTINGLees die volgende instruksies aandagtig deur voordat jy die vrae beantwoord.1.2.3.4.5.6.7.8.9.10.Hierdie vraestel bestaan uit 3 vrae. Beantwoord AL die vrae.Toon ALLE bewerkings, diagramme, grafieke, ensovoorts wat jy gebruik het om jou antwoorde te bepaal, duidelik aan.‘n Goedgekeurde wetenskaplike sakrekenaar (nie-programmeerbaar en nie-grafies) mag gebruik word, tensy anders vermeld.Antwoorde alleenlik sal NIE noodwendig volpunte verdien nie.Indien nodig moet antwoorde afgerond word tot TWEE desimale plekke, tensy anders vermeld.Diagramme is NIE noodwendig volgens skaal geteken nie.EEN diagramvel is aangeheg aan die einde van die vraestel. Skryf jou naam op hierdie bladsy in die spasie wat voorsien is en plaas dit binne-in jou ANTWOORDBOEK.Nommer die antwoorde korrek volgens die numeringstelstel wat gebruik is in hierdie vraestel.Skryf netjies en leesbaar.VRAAG 11.1Beskou die speeding hier langsaan. Dit bestaan uit ‘n silinder met ‘n hoogte van 14cm en ‘n hemisfeer (helfte van ‘n sfeer) met ‘n radius van 13cm. (Formules: ; ; ; )11277605207013cm14cm13cm14cm1.1.1Bereken die volume van die speelding.(6)1.1.2Bereken die totale buite-oppervlakte van die speeding.(6) [12]VRAAG 22.1Voltooi: 2.1.1Die teenoorstaande hoeke van ‘n koordevierhoek is…(1)2.1.2Gelyke koorde onderspan… (1)2.1.3Die hoek tussen ‘n raaklyn en ‘n koord is…(1)2675255155411OSDATMOSDATM2.2MAT is ‘n raaklyn aan sirkel O. Gebruik die skets en bewys die stelling wat beweer dat .(6)2.3M is die middelpunt van koord PT van ‘n sirkel met middelpunt O. OQ is ‘n radius wat deur M gaan. PQ is verleng om raaklyn TA te sny by A, so dat TA PA. 188595257810P1QATOM21212300P1QATOM212123Bewys dat:2.3.1MTAQ is ‘n koordevierhoek.(4)2.3.2PQ = TQ(4)2.3.3(4)[21]VRAAG 33.1Twee sirkels sny mekaar by A en C. O, die middelpunt van die kleiner sirkel, lê op die omtrek van die groter sirkel. D is ‘n punt op die kleiner sirkel en DC verleng sny die groter sirkel by B. BO verleng sny AD by M. AO, OC, en AB is geteken. 703580-306705BAMDCOxBAMDCOx3.1.1Indien , gee, met redes, drie ander hoeke elk gelyk aan x.(6)3.1.2Druk uit in terme van x. Gee ‘n rede.(2)3.1.3Vervolgens, bewys dat BMAD.(4)3.1.4Indien dit verder gegee is dat , bewys dat AD ‘n raaklyn is aan sirkel ABO.(5) [17]TOTAAL: 50DIAGRAMVEL 124323047861300LEERDER SE NAAM EN VAN:VRAAG 1.1VRAAG 2.2307721090805OSDATMOSDATM2266957556513cm14cm13cm14cm VRAAG 2.3VRAAG 33225800226695BAMDCOx00BAMDCOx-124358190425P1QATOM21212300P1QATOM212123-68580346519500115678038133FORMAL TEST - MEMORANDUM00FORMAL TEST - MEMORANDUM221170571755GRADE 1100GRADE 1111264906985MATHEMATICS FORMAL TEST 2AUGUST 2016MEMORANDUM00MATHEMATICS FORMAL TEST 2AUGUST 2016MEMORANDUMMARKS: 50This memorandum consists of 6 pages including the cover page.QUESTION 11.1.1Volume of cylinder:Volume of hemisphere (half sphere): Substitution answerSubstitution answerTotal Volume (6)1.1.2Surface area of cylinder:Surface area of hemisphere (half sphere): Substitution answerSubstitution answerTotal Area (6)[12]QUESTION 22.1.1Supplementary.2.1.2Equal angles.2.1.3Equal to an angle in alternate segment.2.270929595885OSDATMF1212OSDATMF1212 ConstructionStatement ReasonStatement and Reason Statement and ReasonConclusion (6) 2.3.1183565884143P1QATOM21212300P1QATOM212123Statement ReasonStatement Reason (4) 2.3.2Statement and ReasonStatement and Reason Statement and Reason Statement and Reason (4)2.3.3Statement Reason Statement Reason (4)[21]QUESTION 33.1.1Statement Reason Statement Reason Statement Reason (6)3.1.2 Statement Reason (2)3.1.3 Statement and ReasonAnswer Statement and ReasonAnswer(4)3.1.4 Statement and Reason Statement and Reason Answer Statement and ReasonReason(5)[17]Total: 50 Marks1154430149225FORMELE TOETS - MEMORANDUM00FORMELE TOETS - MEMORANDUM227838057784GRAAD 1100GRAAD 1111264906985WISKUNDE FORMELE TOETS 2AUGUSTUS 2016MEMORANDUM00WISKUNDE FORMELE TOETS 2AUGUSTUS 2016MEMORANDUMPUNTE: 50Hierdie memorandum bestaan uit 8 bladsye insluitende die voorblad.VRAAG 11.1.1Volume van silinder:Volume van hemisfeer (halwe sfeer): Substitusie antwoordSubstitusie antwoordTotale Volume (6)1.1.2Buite-oppervlakte van silinder:Buite-oppervlakte van hemisfeer (halwe sfeer): Substitusie antwoordSubstitusie antwoordTotale oppervlakte (6)[12]VRAAG 22.1.1Supplementêr.2.1.2Gelyke hoeke.2.1.3Gelyk aan ‘n hoek in die teenoorstaande sirkel segment.2.270929595885OSDATMF1212OSDATMF1212 KonstruksieBeweringRedeBewering en redeBewering en redeGevolgtrekking (6) 2.3.1183565884143P1QATOM21212300P1QATOM212123Bewering RedeBeweringRede (4) 2.3.2Bewering en RedeBewering en RedeBewering en RedeBewering en Rede (4)2.3.3Statement Reason Statement Reason (4)[21]VRAAG 33.1.1Bewering Rede Bewering Rede Bewering Rede (6)3.1.2 Bewering Rede (2)3.1.3 Bewering en RedeAntwoordBewering en RedeAnswer(4)3.1.4Bewering en Rede Bewering en Rede Antwoord Bewering en RedeRede(5)[17]Totaal: 50 Punte2364105126999GRADE 1100GRADE 1128765543180MATHEMATICSTERM 3 TESTSEPTEMBER 201600MATHEMATICSTERM 3 TESTSEPTEMBER 2016MARKS: 50TIME: 1 hourThis question paper consists of 6 pages.INSTRUCTIONS AND INFORMATIONRead the following instructions carefully before answering the questions.1.2.3.4.5.6.7.8.9.This question paper consists of 4 questions. Answer ALL the questions.Number the answers correctly according to the numbering system used in this question paper.Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers. Answers only will not necessarily be awarded full marks.You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.If necessary, round off answers to TWO decimal places, unless stated otherwise.Diagrams are NOT necessarily drawn to scale.Write neatly and legibly.QUESTION 1In the figure below, ACP and ADP are triangles with , , and PA bisects . Let and .1.1Show, by calculation, that .(3)1.21.3Calculate the length of AD.Determine y.(4)(3)[10]QUESTION 2In the figure, PQRS is a cyclic quadrilateral with units, units, units and where .2.1Write down an expression for the areas of and in terms of a trigonometric function value of .(4)2.2If the area of cyclic quadrilateral PQRS is equal to square units, calculate the value of .(6)[10]QUESTION 33.13.2Lethabo invested an amount of money six years ago. Now, after six years, it is worth R1?200?000. The interest rate for the savings period was 18% per annum compounded monthly. What was the amount that was originally invested six years ago?Johannes invests R24?000 at 14% per annum compounded quarterly for a period of twelve years.(4)3.2.1Calculate the future value of the investment using the nominal rate.(4)3.2.2Convert the nominal rate of 14% per annum compounded quarterly to the equivalent effective rate (annual).(4)3.2.3Use the annual effective rate to show that the same accumulated amount will be obtained as when using the nominal rate.(4)[16]QUESTION 4Mr Smith has a camera to give to one of his pupils after a recent competition. The names of his pupils, and their ages, are listed below.BOYSAGEGIRLSAGEDavid14Susan16Donovan15Denise14George16Beverley15Denis15Kristin15Andrew16Allison15Derrick15Kerri-Lee15Ryan15 4.1Write down n(S).(1)4.2Different events are defined as follows: : A girl will win : A 15-year old will win : The winner’s name starts with D : A 16-year old boy will winDetermine the following:4.2.1 P(E)4.2.2 P(E3)(1)(1) 4.2.3 P(E1 E4)(5)4.34.2.4 P(E E3)The probability that event A will occur is 0,3 and the probability that event Bwill occur is 0,7. The probability that both events A and B will occur, is 0,15.Are events A and B independent? Explain your answer.(3)(3)[14] TOTAL:50236410524765GRAAD 1100GRAAD 1128765519685WISKUNDE TERMYN 3 TOETSSEPTEMBER 201600WISKUNDE TERMYN 3 TOETSSEPTEMBER 2016PUNTE: 50TYDSDUUR: 1 uurHierdie vraestel bestaan uit 6 bladsye.INSTRUKSIES EN INLIGTINGLees die volgende instruksies aandagtig deur voordat jy die vrae beantwoord.1.2.3.4.5.6.7.8.9.10.Hierdie vraestel bestaan uit 4 vrae. Beantwoord AL die vrae.Toon ALLE bewerkings, diagramme, grafieke, ensovoorts wat jy gebruik het om jou antwoorde te bepaal, duidelik aan.‘n Goedgekeurde wetenskaplike sakrekenaar (nie-programmeerbaar en nie-grafies) mag gebruik word, tensy anders vermeld.Antwoorde alleenlik sal NIE noodwendig volpunte verdien nie.Indien nodig moet antwoorde afgerond word tot TWEE desimale plekke, tensy anders vermeld.Diagramme is NIE noodwendig volgens skaal geteken nie.EEN diagramvel is aangeheg aan die einde van die vraestel. Skryf jou naam op hierdie bladsy in die spasie wat voorsien is en plaas dit binne-in jou ANTWOORDBOEK.Nommer die antwoorde korrek volgens die numeringstelstel wat gebruik is in hierdie vraestel.Skryf netjies en leesbaar.VRAAG 1In die diagram hieronder is ACP en ADP driehoeke met , , en PA halveer . Laat en .1.1Toon met behulp van berekeninge aan dat.(3)1.21.3Bereken die lengte van AD.Bereken y.(4)(3)[10]VRAAG 2In die figuur hieronder is PQRS ‘n koordevierhoek met eenhede, eenhede, eenhede en met .2.1Skryf ‘n uitdrukking neer vir die oppervlaktes van en in terme van ‘n trigonometriese funksie waarde van .(4)2.2Indien die oppervlakte van koordevierhoek PQRS gelyk is aan vierkante eenhede, bereken die waarde van .(6)[10]VRAAG 33.13.2Lethabo het ‘n bedrag geld ses jaar gelede belê. Nou, na ses jaar, is die belegging R1?200?000 werd. Die rentekoers vir die beleggingsperiode was 18% per jaar, maandeliks saamgestel. Watter bedrag is oorspronklik belê ses jaar gelede?Johannes belê R24?000 teen 14% per jaar, kwartaalliks saamgestel, vir ‘n periode van twaalf jaar.(4)3.2.1Gebruik die nominale rentekoers om die toekomstige waarde van die belegging te bereken.(4)3.2.2Herlei die nominale rentekoers van 14% per jaar, kwartaalliks saamgestel, na die ekwivalente effektiewe koers (jaarliks).(4)3.2.3Gebruik die jaarlikse effektiewe koers om aan te toon dat dieselfde uiteindelike bedrag verkry word as die bedrag wanneer ons die nominale koers gebruik.(4)[16]VRAAG 4Mr Smith het ‘n kamera om vir een van sy leerders te gee na ‘n onlangse kompetisie. Die name van sy leerders, asook hulle ouderdomme, is hieronder gelys.SEUNSOUDERDOMDOGTERSOUDERDOMDavid14Susan16Donovan15Denise14George16Beverley15Denis15Kristin15Andrew16Allison15Derrick15Kerri-Lee15Ryan15 4.1Skryf n(S) neer.(1)4.2Verskillende gebeurtenisse word soos volg gedefinieer: : ‘n Dogter sal wen : ‘n 15-jarige sal wen : Die wenner se naam begin met ‘n D : ‘n 16-jarige seun sal wenBereken die volgende:4.2.1 P(E)4.2.2 P(E3)(1)(1) 4.2.3 P(E1 E4)(5)4.34.2.4 P(E E3)Die waarskynlikheid dat gebeurtenis A sal plaasvind is 0,3 en die waarskynlikheiddat gebeurtenis B sal plaasvind is 0,7. Die waarskynlikheid dat albei gebeurtenisse A en B sal plaasvind, is 0, 15.Is gebeurtenisse A en B onafhanklik? Verduidelik jou antwoord.(3)(3)[14]TOTAAL:50206883022859GRADE 1100GRADE 1128765522859MATHEMATICS MEMORANDUM CONTROL TEST TERM 3 SEPTEMBER 2016MEMORANDUM00MATHEMATICS MEMORANDUM CONTROL TEST TERM 3 SEPTEMBER 2016MEMORANDUMMARKS: 50This memorandum consists of 5 pages.NOTE:If a candidate answered a question TWICE, mark only the first attempt.Consistent accuracy applies in ALL aspects of the marking memorandum.Assuming values/answers in order to solve a problem is unacceptable.QUESTION 11.1.oOR o correct sine ratio 60o(3) correct sine ratio 60o (3)1.2 correct substitution into cosine rule (4)1.3OR correct substitution into sine rule sin y the subject 23,78o(3) correct substitution into cosine rule cos y the subject 23,82o (3) [10]QUESTION 22.1Area Area Area correct substitution into area formula (4)2.2Area PQRS = Area Area or adding the two areas substitution into formulasimplifying making sinthe subject 600 or 120o conclusion(6)[10]QUESTION 33.1 formula substitution P subject of the formula answer(4)3.2.1 formula i = answer(4)3.2.2 formulai = answer(4)3.2.3 formula i = answer(4)[16]QUESTION 44.113?answer???????(1)4.2.1P(E) ?answer (1) (1)4.2.2 P(E3) answer (1)4.2.3P(E1E4) = P (E1) + P (E4) – P ( E1 E4) = = formula?substitution?(5)4.2.4P (E E) = P (E) × P(E) = = ?formula??(3)4.3For events A and B to be independent P(A B) Here P(A B) 0,3 × 0,7 = 0,21 Thus the events are independent.formula0,3 × 0,7 = 0,21Conclusion(3)[14]TOTAL:502059305-141605GRAAD 1100GRAAD 1128765526670WISKUNDE MEMORANDUM TOETS TERMYN 3SEPTEMBER 2016MEMORANDUM00WISKUNDE MEMORANDUM TOETS TERMYN 3SEPTEMBER 2016MEMORANDUMPUNTE: 50Hierdie memorandum bestaan uit 5 bladsye.NOTA:Indien die leerder ‘n vraag TWEE KEER beantwoord het, merk slegs die eerste poging.Volgehoue akkuraatheid (CA) is van toepassing op ALLE aspekte van die memorandum.Veronderstelling van waardes/antwoorde om ‘n probleem op te los is onaanvaarbaar.VRAAG 11.1.oOF o korrekte sinus verhouding 60o(3) korrekte sinus verhouding 60o (3)1.2 korrekte substitusie in die cosinus re?l (4)1.3OF korrekte substitusie in die sinus re?l sin y die onderwerp 23,78o(3) korrekte substitusie in die cosinus re?l cos y die onderwerp 23,82o (3) [10]VRAAG 22.1Opp Opp Opp korrekte substitusie in die die formule (4)2.2Opp PQRS = Opp Opp of som van die twee oppervlaktes substitusie in die formule vereenvoudig sindie onderwerp 600 of 120o gevolgtrekking(6)[10]VRAAG 33.1 formule substitusie P onderwerp van formule antwoord(4)3.2.1 formule i = antwoord(4)3.2.2 formulei = antwoord(4)3.2.3 formule i = antwoord(4)[16]VRAAG 44.113?antwoord??????(1)4.2.1P(E) ?antwoord (1) (1)4.2.2 P(E3) antwoord (1)4.2.3P(E1E4) = P (E1) + P (E4) – P ( E1 E4) = = formule?substitusie?(5)4.2.4P (E E) = P (E) × P(E) = = ?formule??(3)4.3Gebeurtenisse A en B sal onafhanklik wees indien P(A B) P(A B) 0,3 Dus, die twee gebeurtenisse is onafhanklik.formule0,3 × 0,7 = 0,21gevolgtrekking(3)[14]TOTAAL:50237045533655GRADE 1100GRADE 112870208255MATHEMATICSTERM 4 FORMAL TEST 201600MATHEMATICSTERM 4 FORMAL TEST 2016MARKS: 50TIME: 1 hourThis question paper consists of 8 pages with 5 diagram sheets.INSTRUCTIONS AND INFORMATIONRead the following instructions carefully before answering the questions.1.2.3.4.5.6.7.8.9.This question paper consists of 8 questions. Answer ALL the questions.Number the answers correctly according to the numbering system used in this question paper.Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answers. Answers only will not necessarily be awarded full marks.You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.If necessary, round off answers to TWO decimal places, unless stated otherwise.Diagrams are NOT necessarily drawn to scale.Write neatly and legibly.QUESTION 11.1The data below shows the number of commuters travelling in the Gaubus on Saturday 15 November 2014.5 12 19 29 35 23 15 33 37 21 26 18 23 18 13 21 18 22 201.1.1Determine the mean of the given data. (2)1.1.2Calculate the standard deviation of the data. (2)1.1.3Determine the number of commuters that lies within ONE standard deviation of the mean. (4) 1.1.4Determine the interquartile range for the data. (1)[9]QUESTION 22.1The table below gives the weights of different rugby players from around the world.Weight (in Kg)Frequency Cumulative Frequency85≤ x<90490≤x<95695≤x<1006100≤x<1057105≤x<1108110≤x<11511115≤x<1209120≤x<1256125≤x<13022.1.1Complete the cumulative frequency for the data above. (Redraw the table in your answer book.)(3)2.1.2Draw a frequency polygon for the above data. (Use diagram sheet 1)(4)2.1.3Draw an ogive to represent the weights of the different rugby players (Use Diagram sheet 2).(4)2.1.4Find the first quartile, median and third quartile for the ogive.(3)2.1.5How many rugby players’ weights are in the top 25% of the weights?(3)[17]QUESTION 33.1Given below is a table with different African countries and the percentage of children below the age of 5 years that are underweight first measured between 1993 and 2000 and measured a second time between 2005 and 2008 taken from the data collected by the World Health Organisation.Country1993 - 20002005-2008DifferenceAngola37%15.6%Botswana15.1%11.2%Cameroon17.8%16.6%Central African Republic20.4%28%DRC30.7%28.2%Ethiopia42%34.6%Ghana25.1%13.9%Kenya17.6%18.4%Lesotho18.9%16.6%Malawi 26.5%15.5%Mozambique28.1%18.3%South Africa8.0%8.7%Tanzania19.6%14.9%Zambia26.9%16.7%Zimbabwe11.5%14.0%3.1.1Determine the difference between the first measurement and the second measurement. (Use diagram sheet 3 attached)(5)3.1.2Draw a box and whisker plot to represent the measurements between 1993 and 2000. (Use diagram sheet 4 supplied) (3)3.1.3Draw a box and whisker plot below the box and whisker plot in question 3.1.2 to represent the measurements between 2005 and 2008. (Use the same diagram sheet 2 supplied for 3.1.2)(3)3.1.4Compare the two box and whisker plots and state your conclusions for the plots. (5)3.1.5Are there any outliers for either of the data? State the outlier should there be one.(2)[18]QUESTION 4The table below represents the number of people infected with HIV in a certain region from 2002 to 2007.YearNumber of people infected with HIV2002118200312320041312005134200613620071384.1Use the data above to draw a scatter plot. (Use diagram sheet 5)(3)4.2Explain whether a linear, quadratic or exponential curve would be a line or curve of best fit.(2)4.3If the same trend continues,estimate, by using your graph, the number of people that will be infected in 2008.(1)[6]TOTAL:50 DIAGRAM SHEET 1 2.1.21276357048500DIAGRAM SHEET 22.1.3()DIAGRAM SHEET 33.1 Country1993 - 20002005 - 2008DifferenceAngola37%15.6%Botswana15.1%11.2%Cameroon17.8%16.6%Central African Republic20.4%28%DRC30.7%28.2%Ethiopia42%34.6%Ghana25.1%13.9%Kenya17.6%18.4%Lesotho18.9%16.6%Malawi 26.5%15.5%Mozambique28.1%18.3%South Africa8.0%8.7%Tanzania19.6%14.9%Zambia26.9%16.7%Zimbabwe11.5%14.0%DIAGRAM SHEET 43.1.2 & 3.1.3DIAGRAM SHEET 54.1 Scatter Plot:2371060119438GRAAD 1100GRAAD 1128707940935WISKUNDETERMYN 4 FORMELE TOETS 201600WISKUNDETERMYN 4 FORMELE TOETS 2016PUNTE: 50TYD: 1 uurHierdie vraestel bestaan uit 8 bladsye met 5 diagrammeINFORMASIE en INSTRUKSIESLees die volgende instruksies deeglik voordat jy die vrae beantwoord.Hierdie vraestel bestaan uit 4 vrae.Beantwoord ALLE vrae.Nommer alle antwoorde korrek en volgens die nommering gebruik in die vraestelToon alle berekenings, diagramme, grafieke, ens. wat jy gebruik het om die vrae te beantwoord.Slegs ? antwoord sal nie noodwendig volpunte toegeken word nie.Jy mag slegs gebruik maak van ? goedgekeurde wetenskaplike sakrekenaar (nie-programmeerbaar en nie-grafies), tensy anders vermeld.Rond af tot TWEE desimale plekke indien nodig, tensy anders vermeld.Diagramme is nie noodwendig op skaal geskets nie.Skryf netjies en lessbaar.VRAAG 11.1Die data hieronder stel die aantal pendelaars wat gebruik maak van die Gaubus op Saterdag 15 November 2014.5 12 19 29 35 23 15 33 37 21 26 18 23 18 13 21 18 22 201.1.1Bepaal die gemiddeld van die gegewe data. (2)1.1.2Berekan die standard afwyking van die data. (2)1.1.3Bepaal die aantal pendelaars wat binne EEN standaard afwyking van die gemiddeld lê. (4) 1.1.4Bepaal die interkwartiel waarde van die data. (1)[9]VRAAG 22.1Die table stel die gewigte van veskillende rugby spelers van verskillende lande in die wêreld.Gewig (in Kg)Frekwensie Kumulative Frekwensie85≤ x<90490≤x<95695≤x<1006100≤x<1057105≤x<1108110≤x<11511115≤x<1209120≤x<1256125≤x<13022.1.1Voltooi die kumulatiewe frekwensie vir die data hierbo. (Teken die table injou antwoord boek.)(3)2.1.2Teken die frekwensie poligoon vir die data. (Gebruik diagram 1)(4)2.1.3Teken die ogief om die gewig van die verskillende rugby spelers (Gebruik Diagram 2).(4)2.1.4Vind die eerste kwartiel, die median en die derde kwartiel vir die ogief.(3)2.1.5Hoeveel rugby spelers se gewigte is in die boonste 25% van die gewigte?(3)[17]VRAAG 33.1Die tabel hieronder toon die persentasie kinders onder die ouderdom van 5 jaar, in verskillende Afrika lande, wie ondergewig is. Die data is gedoen vir die periode 1993 en 2000 die eerste keer en toe weer tussen 2005 en 2008 soos aangeteken deur die Wereld Gesondheidsorganisasie.Land1993 - 20002005-2008DifferenceAngola37%15.6%Botswana15.1%11.2%Cameroon17.8%16.6%Central African Republic20.4%28%DRC30.7%28.2%Ethiopia42%34.6%Ghana25.1%13.9%Kenya17.6%18.4%Lesotho18.9%16.6%Malawi 26.5%15.5%Mozambique28.1%18.3%South Africa8.0%8.7%Tanzania19.6%14.9%Zambia26.9%16.7%Zimbabwe11.5%14.0%3.1.1Bepaal die verskil tussen die eerste en tweede opnames. (Gebruik diagram 3 aangeheg)(5)3.1.2Gebruik die data van 1993 en 2000. Stel dit voor as ?n kat-en-snor diagram (houer-en-punt diagram) om die kritiese data voor te stel (Gebruik diagram 4) (3)3.1.3Teken `n kat-en-snor diagram (houer-en-punt diagram) onder die stelsel gebruik in vraag 3.1.2 van die getal waardes vir 2005 en 2008. (Gebruik dieselfde diagram 4 gebruik in 3.1.2)(3)3.1.4Vergelyk die twee periode voorstellings in jou kat-en-snor en dui aan wat jou waarnemeings is. (5)3.1.5Is daar enige uitskieters in enige een van die diagrmme? Wat is die uitskieter sou daar een wees?(2)[18]VRAAG 4Die tabel hieronder verteenwoordig die aantal mense wat met MIV besmet is in ?n sekere area van 2002 tot 2007.JaarAantal mense besmet met MIV2002118200312320041312005134200613620071384.1Gebruik die data om ?n puntediagram te skets. (Gebruik diagram 5)(3)4.2Verduidelik of die kurwe van beste pas lineêr, Kwadraties of ?n eksponenti?le kurwe sal wees.(2)4.3As dieselfde tendens voortgaan, skat, deur van jou grafiek gebruik te maak, hoeveel mense besmet sal wees in 2008.(1)[6]TOTAAL:50 DIAGRAM 1 2.1.23028953429000DIAGRAM 32.1.3DIAGRAM 33.1 Land1993 - 20002005 - 2008VerskilAngola37%15.6%Botswana15.1%11.2%Cameroon17.8%16.6%Central African Republic20.4%28%DRC30.7%28.2%Ethiopia42%34.6%Ghana25.1%13.9%Kenya17.6%18.4%Lesotho18.9%16.6%Malawi 26.5%15.5%Mozambique28.1%18.3%South Africa8.0%8.7%Tanzania19.6%14.9%Zambia26.9%16.7%Zimbabwe11.5%14.0%DIAGRAM 43.1.2 & 3.1.3DIAGRAM 54.1 Puntediagram:206883019685GRADE/GRAAD 1100GRADE/GRAAD 1128765524765MATHEMATICS P1/WISKUNDE V1MARCH 2015MEMORANDUM00MATHEMATICS P1/WISKUNDE V1MARCH 2015MEMORANDUMMARKS/PUNTE: 50TIME/ TYD: 60 minutesThis memorandum consists of 5pages.Hierdie memorandum bestaan uit 5 bladsye.NOTE:If a candidate answered a question TWICE, mark only the first attempt.Consistent accuracy applies in ALL aspects of the marking memorandum.Assuming values/answers in order to solve a problem is unacceptable.LET WEL:As 'n kandidaat 'n vraag TWEE keer beantwoord het, merk slegs die eerste poging.Volgehoue akkuraatheid is DEURGAANS in ALLE aspekte van die memorandum van toepassing.Aanvaarding van waardes/ antwoorde om 'n problem op te los, is onaanvaarbaar.QUESTION/VRAAG 15, 12, 13, 15, 18, 18, 18, 19, 20, 21, 21, 22, 23, 23, 26, 29, 33, 35, 37 1.1.13661919,26 (2)1.1.2 σ2 = i=1n(xi-x)2n(2)1.1.314 commuters19,26-8,0219,26+8,02(11,24;26,26)14 (4)1.1.437-1819 (3) [9]QUESTION/VRAAG22.1.1Weight (in Kg)Frequency Cumulative Frequency85≤ x<904490≤x<9561095≤x<100616100≤x<105723105≤x<110831110≤x<1151142115≤x<120951120≤x<125657125≤x<1302591M for everyCorrect hreeanswers(3)2.1.2952514414500(85; 0) all points correctjoining Pointscorrectvariables(4)2.1.3(0; 0)all points correct joining pointscorrect variables (4)2.1.4First Quartile = ± 100Median = ± 109Third Quartile = ±116 Q1 Q2 Q3(3)2.1.5Top 25% will be the weight of more than 116;Thus 59 – 42 = 17 rugby players are in the top 25% of weights.11659-4217 (3)[17]QUESTION/VRAAG33.1.1Country1993 - 20002005-2008DifferenceAngola37%15.6%21.4%Botswana15.1%11.2%3.9%Cameroon17.8%16.6%1.2%Central African Republic20.4%28%-7.6%Democratic Republic of Congo30.7%28.2%2.5%Ethiopia42%34.6%7.4%Ghana25.1%13.9%11.2%Kenya17.6%18.4%-0.8%Lesotho18.9%16.6%2.3%Malawi 26.5%15.5%11%Mozambique28.1%18.3%9.8%South Africa8.0%8.7%-0.7%Tanzania19.6%14.9%4.7%Zambia26.9%16.7%10.2%Zimbabwe11.5%14.0%-2.5%1M for every Three CorrectValues (5)3.1.2&3.1.322834602565400017767305080000127381046990001407160133286500845185118999000114046011899900032131013423900027368520891500156781425780900Q3.1.2min & maxbox(3)Q3.1.3min & maxbox (3)3.1.4The second box and whisker plot (2005 – 2008) has a much smaller spread of data than the first box and whisker plot (1993 – 2000). More than 50% of the second set of data lies within the first 25% of the first set of data.The second set of data has one outlier, which is still less than the maximum of the first set of datasmaller spread of data More than 50% of the second set of data lies within the first 25% of the first set of data second set of data has one outlierwhich is still less than the maximum of the first set of data. (5)3.1.5Yes, in the 2005 – 2008 set of data there is one outlier – 34.6yes– 34.6 (2)[18] QUESTION/VRAAG44.1?both variables?1st three points?last three points(3)4.2A linear line of best fit would be suitable to describe this relation Points indicate a positive linear growth?linear line of best?positive linear growth (2)4.3There would probably be an estimate of 145 HIV infections in 2008?145(1)[6]TOTAL/TOTAAL:50ASSIGNMENT ( FUNCTIONS – TERM 2)QUESTION 1[10]The accompanying diagram shows the graph of f(x)=axWrite down the coordinate of A; explain.(2)How can we tell that: 0<a<1?(1)Determine a if B is the point (4; 116).(3)Determine the equation of the graph obtained if the graph above is reflected about the y-axis .(2)Write down the coordinates of the point of intersection of the two graphs.(2)QUESTION 2 [17]The sketch below; which is not drawn to scale, represents the graphs of f(x)= 1x-2-1 and gx=mx+cThe graphs intersect at points R and T 2.1 Write down the equations of the asymptotes of f.(2)2.2 Write down the value of a.(1)2.3.1 Calculate the value of m , the gradient of the straight line g (to the nearest integer) if g makes an angle of 63,43°with the x-axis.(1)2.3.2 Hence, determine the value of c.(1)2.4 Determine the coordinate of R and T.(5)2.5 For what value of x is gx ≥fx.(4)2.6 Determine an equation for the axis of symmetry of f which has a positive slope.(3) QUESTION 3 [14]Given: fx=3(x-1)2-12 and gx=-4x+2+13.1 Calculate the coordinates of the x – intercept and the y – intercept of g.(3)3.2 Calculate the coordinates of the x – intercept and the y-intercept of f.(3)3.3 What is the minimum value of f(x) ?(1)3.4 On the same set of axes, sketch the graphs of f and g . Indicate all intercepts with the axes and the coordinates of the turning point of f.(5)3.5 For which values of x will both fx and gx increase as x increases?(2)QUESTION 4[10]Given: 4.1Determine the and intercepts of .(2)4.2Write in the form: .(2)4.3Draw the graph of , clearly show the intercepts with the axes and the asymptotes.(4)4.4Give the equations of the asymptotes of .(2)TOTAL: [50]WERKOPDRAGVRAAG 1[10]Die diagram hieronder toon die grafieke van f(x)=axSkryf die ko?rdinate van A neer; verduidelik.(2)Hoe weet jy dat: 0<a<1?(1)Bepaal a as B die punt (4; 116) is.(3)Bepaal die vergelyking van die grafiek as bogenoemde grafiek ? refleksie om die y-as.(2)Skryf die ko?rdinate van die snypunt van die twee grafieke. (2)VRAAG 2 [17]Die skets hieronder; wat nie op skaal geteken is nie; is die grafieke van: f(x)= 1x-2-1 and gx=mx+cDie snypunte van die grafieke is R and T 2.1 Skryf die vergelykings van die asymptote van f neer.(2)2.2 Skryf die waarde van a neer.(1)2.3.1 Bereken die waarde van m , die gradient/ helling van die reguitlyn g (tot die naaste heelgetal); as g ? inklinasie hoek van 63,43° met die positiewe x-as vorm.(1)2.3.2 Vervolgens, bepaal die waarde van c.(1)2.4 Bepaal die ko?rdinate R and T.(5)2.5 Vir watter waarde van x is gx ≥fx.(4)2.6 Bepaal ? vergelyking vir die simmetrie-as van f wat ? positiewe helling voorstel.(3) VRAAG 3 [14]Gegee: fx=3(x-1)2-12 and gx=-4x+2+13.1 Bereken die ko?rdinate van die x – afsnit en die y – afsnit van g.(3)3.2 Bereken die ko?rdinate van die x – afsnit en die y – afsnit van f.(3)3.3 Wat is die minimum waarde van f(x) ?(1)3.4 Skets die grafieke van f and g op dieselfde assestelsel. Toon die afsnitte met die asse en die draaipunt van f duidelik aan.(5)3.5 Vir watter waardes van x sal beide fx and gx toeneem as x toeneem?(2)VRAAG 4[10]Gegee: 4.1Bepaal die en afsnitte van .(2)4.2Skryf in die vorm: .(2)4.3Skets die grafiek van . Toon alle afsnitte met die asse en die asymptote duidelik aan.(4)4.4Gee die vergelyking van die asymptote van .(2)TOTAL: [50]ASSIGNMENT MEMORANDUMQUESTION 1[9]1.1A0;1Because y=ax =a0 =1 A(0;1)=a0 (2)1.2The sketch indicates a decreasing graph QUOTE √ decreasing (1)1.3B(4;116) y= ax 116=a4 (12)4=(a)4 ∴a=12116=a4 (12)4=(a)4a=12 (3)1.4y=(12)-xy= (12)-x (1)1.5(0;1) (0;1) (2)QUESTION 2[17]2.1x=2y=-1x intercepty intercept (2)2.2a=2a=2 (1)2.3.1m=tan(63,43)°∴m=2m=2 (1)2.3.2y=mx+cS2;00=22+c∴c=-4 y=2x-4c=-4 (1)2.41x-2-1=2x-41x-2=2x-31=2x2-4x-3x+62x2-7x+5=02x-5x-1=0x=52 or x=1y=2x-4y=252-4 or y=21-4y=5-4 or y=2-4y=1 or y=-21x-2-1=2x-42x2-7x+5=02x-5x-1=0x=52 or x=1y=1 or y=-2 (5)2.5x∈1;2∪52;∞ or 1≤x<2 or 52≤x<∞[1; 2); [52; ∞) (4)2.6y=x or y=x+cy=x-2-1 or -1=2+cy=x-3 or y=x-3 y=xy=x-2-1y=x-3 (3)QUESTION 3[14]3.1 gx=-4x+2+1gx=-4x+2+1 g0=-40+2+1 0 =-4x+2+1 g0=-2+1 4x+2=1 g0=-1 x+2=4∴x=2using g0& x=0g0=1 or y=1x=2 (3)3.2f(x)=3(x-1)2-12 0=3(x-1)2-12 123=(x-1)2±4=(x-1)2 x-1=2or x-1=-2 x=3or x= -1f(0)=3(0-1)2-12 f0=-9x=3x=-1f0=-9 or y=-9 (3)3.3Minimum Value: -12:min=-12 or y=-12 (1)3.4DiagramPARABOLAShapex - interceptsy – interceptASYMPTOTESHYPERBOLA correct quadrants (5)3.5y∈-∞;1∪1;∞ or y∈R;y≠1y∈-∞;1) y∈(1;∞) (2)QUESTION 4[10]4.1y: y=-10-2=5x : 0=4x-10x-2 0=4x-10 10=4x52=x y=5 x=212 (2)4.2fx=4x-10x-2=-2+4(x-2)(x-2)=-2x-2+4 factorise -2+4x-2answer (2)4.389090569215006115053175y00y2014855930275x00x274955168275f00f16065553975001466857238900 x-intercept y-intercept asymptote form (4)4.4x=2y=7 answeranswer (2)PROJECT Marks: [50]-3053715-205740000Human Mathematics (Golden Ratio)The table below presents the following information: Height (H) from top of head to floor (m) Height (h) from belly button to floor (m)LearnerABCDEFGHIJKLMNOH (Head to floor)h (Belly button to floor)PART A [ 23 marks ]Question 1Use the values in the table to complete the calculation for all the values in the table. (Row 3) (1)1.2 Use the following scale: y-axis: 0,1; 0,2; 0,3; 0,4; 0,5; …………..2,0 and x-axis: 0,1; 0,2; 0,3; 0,4; 0,5; ………1,5 to do a freehand point-by-point graph (scatter plot) of the values for H (y-axis) and h (x-axis) on blocked or graph paper. (5)1.3 What kind of relationship do you notice after plotting these points? What kind of function is represented? (2)Choose any six points plotted, group them 3 groups of 2(two) each and calculate the gradient for each of the three groups. What do you notice about the gradients you calculate? (4)If this information could be translated to an equation, what would the equation be? (2)Based on your equation in 1.5 and points on the graph, make the following readings/ predictions (indicate where the readings on the graph is made and/ or how you calculated the answers):The belly button height if: A learner is 2.0 m long. (Use an A on your graph) (2)The belly button height if: A learner is between 0,5m and 1,4 m in length. (Use B) (3)How long would a learner be if his/ her bellybutton is 0,2m from the floor? (Use C) (2)What would the length of a learner be if his bellybutton is 1,5m from the floor?Do you think there could be a learner in your class with these lengths? Motivate your answer. (2)PART B [27]In the following questions the values for H (Measures Head to floor) should be used:Question 2Calculate: the mean (2) the mode (1)2.3 the median (1)Question 33.1 Summarise the values of H in the following Frequency Table: (6)Frequency TableClass (meters)FrequencyCumulative Frequency0.8 – 1.001.01 – 1.21.21 – 1.41.41 – 1.61.61 – 1.81.81 – 2.0 3.2 Use the table to draw a point-by-point frequency polygon. (Your polygon should commence at 0.5) (4)3.2.1 Would you consider your graph skewed, and if so what would you think the reason could be? (2) Using the same table draw a cumulative frequency polygon (ogive). (8)3.3.1 Indicate the mean, median and mode on your ogive. (3)TOTAL: [50]PROJEK Punte: [50]-3053715-205740000Die goue verhouding. (Menslike Wiskunde)Die table hieronder stel die volgende informasie voor: Hoogte (Lengte) (H) vanaf jou kroontjie tot die vloer (m) Hoogte (Lengte) (h) van jou naeltjie tot die vloer (m)LeerderABCDEFGHIJKLMNOH (Kroontjie tot vloer)h (Naeltjie tot vloer)AFDELING A[23 punte]Vraag 1 Gebruik die waardes in die table en voltooi die volgende berekenings vir alle waardes in die tabel. (3de ry)(1) Gebruik die volgende skaal: y-axis: 0,1; 0,2; 0,3; 0,4; 0,5; …………..2,0 en x-axis: 0,1; 0,2; 0,3; 0,4; 0,5; ………1,5 om ? vryhand punt-vir-punt skets (spreiding skets) vir die waardes van H (y-as) en h (x-as) op blokkies papier of grafiek papier te skets. (5) Watter tipe verhouding neem jy waar nadat jy die punte geplot het? Watter tipe funksie word hier verteenwoordig?(2)Kies ses punte aandgedui op jou grafiek, groepeer 3 groepe van 2 punte elk en bereken die helling (gradi?nt) vir elk van die groepe. Wat merk jy van die hellings wat jy bereken het?(4)Sou ons die informasie herlei na ? vergelyking, wat sal die vergelyking wees?(2)As die vergelyking in 1.5 en die punte op die grafiek gebruik word, maak die volgende lesings/ voorspellings (dui op jou grafiek aan waar die lesings gemaak is en/ of hoe jy die berekenings gemaak het om jou antwoorde te kry):Die hoogte van jou naeltjie as jy 2,0 m lank. (Gebruik ? A op jou grafiek)(2)Die moontlike hoogte van jou naeltjie as jy tussen 0,5m en 1,4 m lank.(Gebruik ? B)(3)Hoe lank kan ? leerder moontlik wees as sy/ haar naeltjie 0,2m van die vloer meet? (Gebruik C)(2)Wat sal die lengte van ? leerder wees as sy naeltjie 1,5m van die vloer is?Dink jy daar kan ? leerder in jou klas wees met hierdie lengtes? Motiveer jou antwoord.(2)AFDELING B[27 punte]In die volgende vrae moet die waardes van H (Lengte van kop tot die vloer) gebruik word: Vraag 2.Bereken:1.1 Die Gemiddelde Die Modus Die Mediaan(4)Vraag 3.3.1 Voltooi die volgende frekwensie tabel vir H:(6)Frekwensie TabelKlas (meters)FrekwensieKumulatiewe Frekwensie0.8 – 1.001.01 – 1.21.21 – 1.41.41 – 1.61.61 – 1.81.81 – 2.0 Gebruik die data van die tabel om ? punt-vir-punt frekwensie poligaan te skets (verspreidingsdiagram). (Jou skaal behoort by 0,5 te begin) (4)Sou jy jou grafiek as ‘skeef’ beskryf? Indien so, wat sou jy die rede daarvoor beskryf. (2)Gebruik dieselfde tabel om 'n kumulatiewe frekwensie poligoon (Ogief) te skets. (8)Dui jou Gemiddelde, Modus en Mediaan aan op die Ogief.(3) TOTAAL: [50]PROJECT: Memorandum Mark: [50]-3053715-205740000Human Mathematics (Golden Ratio)Note to Educator:For AUTHENTICITY purpuses it is important that the measurements are actually made and not made-up.Before the project is done the following information should be collected from the class:Select 15 learners to do the following measurements under Educator’s supervision. They can be grouped 5 learners at a time and bring them during break or after school hours to do the following measurements:Measure and record in the table below before printing and distributing the task to learners:1.1Height (H) from top of head to floor (m) (Use a ruler to get Parallel height to the floor)1.2Height (h) from belly button to floor (m)2.Recording table: [This table should be completed by the Educator based upon the readings done by learners and the completed table must be supplied to the learners on their copy of the TASK.] Measure and record:Height (H) from top of head to floor (m) (Use a ruler to get Parallel height as floor)Height (h) from belly button to floor (m)Recording table: LearnerABCDEFGHIJKLMNOHhThe following indicates a template for the memorandums and should be completed by the Educators based on the measurements recorded. PART A (Functions)[23 marks]Measurements: To establish the accuracy of the measurements, Educators can test that the ratio H/h should be ± 1,668 (≈ 1,7)1.1 r = 1,6668 (≈ 1,7)r = 1.668 is the exact accurate value.Since the measurements may be inaccurate allow 1,58 ≤ r ≤ 1,71 mark for more than 10 correct ratios = 1 markGraph: (1.2)1.2 Appropriate scales used on each axes:y-axis: 0,1; 0,2; 0,3; 0,4; 0,5; …………..2,0x-axis: 0,1; 0,2; 0,3; 0,4; 0,5; ………1,5Plotting:15 points accurately from the table2 marks for the origin and correct spacing 1 mark per 5 points = 3 marks1.3 Linear relationship; straight line1 mark each = 2 marks1.4 Calculation 1: Possible answers (; ; )Calculation 2:Calculation 3:ALL gradients should be more or less equal.Correct formula used in each calculation. Incorrect formula = 0 mark.1 mark each for calculations = 3 marksACCURATE: = 1.668Deduction: 1 mark1.5 Restrictions: x > 0 and y > 0; y = 1,6668 x (Value of m: 1,66; 1,7; 1,6; 1,59)Restrictions: 1 markEquation: 1 markNB. The graph does NOT have a y-intercept.1.6 [These are the actual values, based on previous rounding. Educators should adjust the values as per measurements]1.6.1 Belly button height: 1,199m (at A)1 mark for answer; 1 mark for reading/ calculationas insdicated on graph1.6.2 Belly button height: 0,299m < x < 0,83m (at B)1 mark each for upper and lower answer = 2 marks; 1 mark for reading/ calculationon graph1.6.3 Learner will be 0,3336m long. (at C)Readings that will not make sense within the class, but it is fine1 mark for answer; 1 mark for reading/ calculation on graph1.6.4 Bellybutton: 1,5m means he/ she will be 2,5002m long.NO/ YES; no if no learner is this tall, yes if there is a learner of this length in the class. 1 mark for answersYes/ No and viable reason: 1 marksPART B: (Statistics)[27 marks]The values for H should be used for the following Memorandum:In the following questions the values for H (Measures Head to floor) should be used:Question 2.(4)Caculate: the mean – Educator to calculate based on measurements supplied to learners.1 mark for formulae 1 mark for correct answer2.2 the mode – Educator to calculate based on measurements supplied to learners.mark correct value2.3 the median – Educator to calculate based on measurements supplied to learners.1 mark for formulae and correct answerQuestion 33.1 Summarise the values of H in the following Frequency Table:Frequency TableClass (meters)FrequencyCumulative Frequencyinequalities again1.01 – 1.21.21 – 1.41.41 – 1.61.61 – 1.81.81 – 2.0Frequency:1 mark for each 2 correct values (only 1 value instead of 2 = no marks) = 3 marksCumulative frequency:1 mark for each 2 correct values (one value incorrect = no marks for the rest of the values) = 3 marks (6)3.2 Use the table to draw a point-by-point frequency polygon. Learners should use the same scaling as in 1.2.EDUCATOR: may provide an axis, but no scaling and naming1Marks for naming of both axes1 mark for at least 3 points plotted correctly = 1marks2 marks for extending the beginning and ending to be anchored to the x-axis. (4)3.2.1 Would you consider your graph skewed, and if so what would you think the reason could be? 1 YES/ NO1 Possible reasonEducator to establish based on own calculations and then allocate marks accordingly. (2)3.3 Using the same table, draw a cumulative frequency polygon (Ogive).1 Marks for naming of both axes and using the values as per frequency table.1 mark each for actual points plotted = 6 marksShape correct = 1 mark (8)2.3.1 Indicate the mean, median and mode on your ogive. 1 mark each for correct indication and labelling = 3 (3) Total = 50INVESTIGATION Marks: [50]This investigation has a surprising unexpected outcome. Investigating the relationship between the width and the area of rectangles with a constant perimeter of 24 cm Isaac made all these rectangles with 24 cm lengths of string. This implies that the perimeters of all the rectangles are equal. 76835014224000 1.1Draw 11 rectangles with a perimeter of 24 cm. For the width, use natural numbers, starting at 1. Indicate the length, the width, as well as the area on the sketch of each rectangle.(5)1.2Copy and complete the table to show the, width (w) and area(a) of all the rectangles drawn.(3)Length (l)1110Width (w)16Area(a)2.Which sentence best describes Isaac’s rectangles? (1)A As the width of the rectangle increases, so does the area.B As the width of the rectangle increases, the area remains the same.C As the width of the rectangle increases, the area first Increases and then decreases.3. What do you observe about the sum of the length and the width? (Give a reason for your answer). (2)4.1Write down all the ordered number pairs in the form (width; area) that represent the relationship. (1)4.2Let the width and the length be real numbers. The perimeter remains a constant 24cm. Draw the graph for Label the axes appropriately.(5)5.1Name the type of graph that represents this relationship. (1)5.2State whether this is a quadratic, exponential, hyperbolic or linear type of relationship.(1)6Consider the domain. Classify this relationship as either discrete or continuous. Give a reason for your answer.(2)7.1Use the formula y = a(x + p)2 + q to determine the equation of the graph.(5)7.2Use the values in the table and write down the areas as a number sequence from 11 to and including 36. Determine the nth term of the sequence.(7)7.3Does this answer support or contradict the answer of 7.1?(1)8.1For which values of w is the graph increasing?(2)8.2For which values of w is the graph decreasing?(2)9.Write down the equation of the axis of symmetry of the graph.(1)10.Calculate the average rate of change of area with respect to width on the graph between the points where w = 2 and w = 6.(3)11.1Use the graph to determine a possible area for a width of 5,5cm. Indicate where the answer could be read off with an A.(2)11.2Use the graph to determine which possible width, correct to one decimal place, will yield an area of approximately 18,55cm2. Indicate where the answer could be read off with a B.(2)plete: from the investigation it is clear that the relationship (width ; area) is a …………… function. This is indicated by the highest ……… of the width that is two. The area is the ………….variable, while the width is the ……………variable.(4) TOTAL: [50]ONDERSOEKTOTAAL: [50]Hierdie ondersoek het ‘n verrassende, onverwagte uitkoms. Ondersoek na die verwantskap tussen die wydte en die oppervlakte van reghoeke met ‘n konstante omtrek van 24 cm Isaac het al hierdie reghoeke gemaak van 24 cm lange stukkies tou. Dit impliseer dat die omtrekke van al die reghoeke gelyk is aan mekaar. 76835014224000 1.1Teken 11 reghoeke met ‘n omtrek van 24 cm. Gebruik natuurlike getalle vir die wydte. Begin by 1. Dui die lengte, die wydte, sowel as die oppervlakte op die skets van elke reghoek aan. (5)1.2Kopieer en voltooi die tabel om die lengte(l), wydte(w) en oppervlak(o) van al die reghoeke wat geteken is aan te toon. (3)Lengte (l)1110Wydte (w)16Oppervlak(o)2Watter sin beskryf Isaac se reghoeke die beste?(1)A Soos wat die wydte toeneem, so doen die oppervlakte.B Soos wat die wydte van die reghoeke toeneem, bly die oppervlakte dieselfde. C Soos wat die wydte van die reghoeke toeneem, neem die oppervlakte eers toe en dan begin dit afneem. 3.Wat neem jy waar in verband met die som van die lengte en wydte?Gee ‘n rede vir jou antwoord. (2)4.1Skryf al die geordende getallepare wat die verwantskap verteenwoordig in die vorm (wydte; oppervlak) neer.. (1)4.2Laat die wydte en die lengte re?le getalle wees. Die omtrek bly ‘n konstante 24cm lank. Teken die grafiek vir .Benoem die asse op ‘n gepaste wyse. (5)5.1Benoem die tipe grafiek wat die verwantskap verteenwoordig. (1)5.2Sê of hierdie ‘n kwadratiese, eksponensi?le, hiperboliese of lineêre tipe verwantskap is. (1)6.Beskou die definisieversameling. Klassifiseer hierdie verwantskap as of diskreet of kontinu. Gee ‘n rede vir jou antwoord. (2)7.1Gebruik die formule y = a(x + p)2 + q om die vergelyking van die grafiek mee te bepaal. (5)7.2Gebruik die waardes in die tabel en skryf die oppervlaktes neer as ‘n getallery vanaf 11 tot en met 36. Bepaal die nde term van die ry. (7)7.3Bevestig of weerspreek hierdie antwoord die antwoord van 7.1?(1)8.1Vir watter waardes van w is die grafiek stygend?(2)8.2Vir watter waardes van w is die grafiek dalend?(2)9.Skryf die vergelyking van die simmetrie-as van die grafiek neer.(1)10.Bereken die gemiddelde koers van verandering van oppervlakte met betrekking tot wydte tussen die punte waar w = 2 en w = 6.(3)11.1Gebruik die grafiek om ‘n moontlike oppervlak te bepaal vir as ‘n wydte van 5,5cm. Dui die plek waar die antwoord afgelees kan word met ‘n A aan.(2)11.2Gebruik die grafiek om te bepaal watter moontlike wydte, korrek tot een desimale plek, wat ‘n oppervlak van ongeveer 18,55cm2 gee. Dui die plek waar die antwoord afgelees kan word met ‘n B aan. (2)12Voltooi: vanuit die ondersoek is die duidelik dat die verwantskap (wydte ; oppervlak) ‘n …………… funksie is. Dit word aangedui deur die hoogste ……… van die wydte wat gelyk is aan twee. Die oppervlakte is die ………….veranderlike, terwyl die wydte die ……………veranderlike is. (4) TOTAAL:[50] INVESTIGATION : Memorandum Marks: [50] Investigate the relationship between the width and the area of rectangles with a constant perimeter of 24 cm 1.1 01460521020cm211111cm221020cm23927cm284432cm235cm2576636cm25735cm24832cm23927cm211111cm20021020cm211111cm221020cm23927cm284432cm235cm2576636cm25735cm24832cm23927cm211111cm2 1.1 Use increasing Natural numbers for the measure of the widths. Start at 1.1.11 mark for every two rectangles with all the information. The last group will consist of 3 rectangles. Subtract 2 marks if the areas or are not indicated. No marks if neither areas nor dimensions are indicated. Subtract one mark if all the information is there but very untidy drawings were made. (5)1.2One mark per row with correct entries. (3)2As the width of the rectangle increases, the area first increases and then decreases. Correct sentence(1)3The sum stays a constant 12cm because the perimeter has to stay equal to 24 cm. The sum stays a constant 12cm because the perimeter stays a constant 24cm. (2)4.1(1 ; 11) (2 ; 20) (3 ; 27) (4 ; 32) (5 ; 35) (6 ; 36) (7 ; 35)(8 ; 32) (9 ; 27) (10 ; 20) (11 ; 11) One mark for all the ordered number pairs(1)4.2(5)48641006985Correct shapeCorrect plotting of 6 pointsCorrect plotting of other 5 pointsCorrect domainBoth axes labeled appropriately00Correct shapeCorrect plotting of 6 pointsCorrect plotting of other 5 pointsCorrect domainBoth axes labeled appropriately-13970047625w(1 ; 11)x(2 ; 20)x(3 ; 27)x(4 ; 32)x(5 ; 35)x(6 ; 36)xx (7 ; 35)x (8 ; 32)x (9 ; 27)x (10 ; 20)x (11 ; 11)aWIDTHAREABA00w(1 ; 11)x(2 ; 20)x(3 ; 27)x(4 ; 32)x(5 ; 35)x(6 ; 36)xx (7 ; 35)x (8 ; 32)x (9 ; 27)x (10 ; 20)x (11 ; 11)aWIDTHAREABA5.1It is a parabola.Answer(1)5.2It is a quadratic relationship.Answer(1)6It is a continuous relationship because the domain consists of real numbers: ContinuousDomain consists of real numbers(2)7.1 Use the turning point (6 ; 36) and (2 ; 20) or any other point. So: Correct substitution of the turning point and one other point16a = -16AnswerSubstitute backFinal equation(5)7.2The sequence is: 74422017589500 11 ; 20 ; 27 ; 32 ; 35 ; 36 9 7 5 3 1 -2 -2 -2 -2 c = 0 Correct sequence2nddifference = 2aAnswer for a Answer for b c = 0Final equation(7)7.3The two equations are exactly the same so they support each other. The one proves that the other is correct. (1)8.1lhs of inequalityrhs of inequality(2)8.2lhs of inequalityrhs of inequalityt(2)9w = 6Answer(1)10Use (2 ; 20) and (6 ; 36) = = = Correct formulaCorrect substitutionAnswer(3)11.135,7cm2 read off at AAllow 0,2 cm off upwards or downwardsFor indicating A(2)11.21,8cm read off at BAllow 0,2 cm off upwards or downwardsFor indicating B(2)12From the investigation is clear that the relationship (width ; area) is a quadratic function. This is indicated by the highest power of the width that is a 2. The area is the independent variable, while the width is the dependent variable.(4)TOTAL50 ................
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