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# Ib math hl portfolio ia the dice game/filling up the petrol tank/shadow functions/patterns from complex numbers help tutors sample example solution guide

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MATHEMATICS
Higher Level
For use in 2012 and 2013

While real zeros of polynomial functions may be easily read off the graph of the
polynomial, the same is not true for complex zeros. In this task, you will investigate the
method of shadow functions and their generators, which helps identify the real and
imaginary components of complex zeros from key points along the x-axis.
Consider the quadratic function 2 2
Write down the coordinates of the vertex.
Show that 1 y has zeros, where.

_ How can the findings with quadratics and cubics be applied to quartics?
PATTERNS FROM COMPLEX NUMBERS HL TYPE I
To the student: The work that you produce to address the questions in this task should be a report
that can stand on its own. It is best to avoid copying the questions in the task to adopt a “question

THE DICE GAME HL Type II
To the student: The work that you produce to address the questions in this task should be a report
that can stand on its own. It is best to avoid copying the questions in the task to adopt a “question
The aim of this task is to create a dice game in a casino and model this using
probability. It is important to examine how best to run the game from both the
perspective of a player and the casino. In doing so analyse the game to consider the
optimal payments by the player and payouts by the casino.
_ Consider a game with two players, Ann and Bob. Ann has a red die and Bob a white die.

_ Now consider other models for the game including cases such as where the player or the
bank rolls their dice multiple times, or where multiple players are involved in the game.
FILLING UP THE PETROL TANK HL Type II
To the student: The work that you produce to address the questions in this task should be a report
that can stand on its own. It is best to avoid copying the questions in the task to adopt a “question
In this task, you will develop a mathematical model that helps motorists decide which of
the following two options is more economical.
Option 1: to buy petrol from a station on their normal route at a relatively higher price.
Option 2: to drive an extra distance out of their normal route to buy cheaper petrol.
Consider a situation in which two motorists, Arwa and Bao, share the same driving route but own
different sized vehicles. Arwa fills up her vehicle’s tank at a station along her normal route for
US\$ 1 p per litre. On the other hand, Bao drives an extra d kilometres out of his normal route to fill
up his vehicle’s tank for US\$ p2 per litre where p2 _ p1 .

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### Ib math hl portfolio ia the dice game/filling up the petrol tank/shadow functions/patterns from complex numbers help tutors sample example solution guide

3. 3. KUNMING, SANYA, HARBIN, TIANJIN, URUMQI,CHENGDU, CHONGQING, GUANGZHOU, QINGDAO, NANJING, HANGZHOU, XIAN, Denmark; Copenhagen,Arhus, Odense, Aalborg, Esbjerg, Greece; Athens,Peloponnese, Cyclades, Crete, Hungary; Budapest,Pécs, Szeged, Sopron, Debrecen, Indonesia; Jakarta , Makassar ,Medan, Banda Aceh, Surabaya, Bandung, Denpasar, Ireland;Dublin, Cork, Limerick, Kilkenny, Galway, Waterford,Wexford, Sligo, Kenmare, Tralee, Italy; Rome, Venice,Florence, Milan, Naples, Verona, Turin, Bologna, Perugia,Genova, Japan; Tokyo, Yokohama, Osaka, Nagoya,Sapporo, Kobe, Kyoto, Fukuoka, Kawasaki, Saitama,Hiroshima, Sendai, Kitakyushu , Malaysia; KualaLumpur, Johor Bahru, Ipoh, Kuching, KualaTerengganu, Kuantan, Kota Bahru, Seremban,Georgetown, Kota Kinabalu, Sandakan, Taiping,Kajang-Sungai Chua, Alor Setar, Sibu, Tawau,Kluang, Sungai Petani, Miri, Melaka, BandarPenggaram, Mexico; Mexico City, Guadalajara , Ecatepec,Nezahualcoyotl , Puebla , Monterrey , Tijuana , Ciudad Juárez,Leon , Zapopan, Guadalupe, Querétaro, Cuernavaca, Tampico,Veracruz, Naucalpan, Toluca, Morelia, Aguascalientes,Netherlands; Amsterdam, Rotterdam, Utrecht, Den Haag(TheHaugue), Eindhoven, Tilburg, Groningen, Nijmegen, Leiden,Zwolle, Venlo, New Zealnd; Auckland, Hamilton, Nelson,Dunedin, Napier, Rotorua, Taupo, Tauranga, Blenheim,Invercargill, Kaikoura, Peru; Lima (and Callao), Arequipa,Trujillo, Chiclayo, Piura, Iquitos, Cusco, Chimbote, Huancayo,Tacna, Philippines; Manila, Cebu, Baguio, Boracay, Palawan,Albay, Davao, Zamboanga, General Santos City, Poland;
4. 4. Cracow, Gdansk, Poznan, Szczecin, Warsaw,Wroclaw, Poznari, Tricity, Russia; Moscow, SaintPetersburg, Kazan, Yekaterinburg, Volgograd, Irkutsk,Vladivostok, Nizhny Novgorod, Saudi Arabia; Abha,Dhahran, Jubail, Qatif, Al-Ahsa, Hail, Madinah,Riyadh, Al-khobar, Jeddah, Makkah, Tabouk, Baha,Jizan, Najran, Taif, Dammam, Jouf, Qassem,Yanbu, Singapore; Bedok, Seletar, Woodlands, Toa Payoh,Ang Mo Kio, Choa Chu Kang, Tampines, South Africa;Alberton, Benoni, Cape Town, Carltonville, Durban,East London, Johannesburg, Kimberley, Klerksdorp,Krugersdorp, Mhluzi, Midrand, Msunduzi, Newcastle, Paarl,Pretoria, Soweto, Port Elizabeth, Spain; Madrid, Barcelona,Valencia, Sevilla , Saragoza, Malaga, Zaragoza, Murcia, Palmade Mallorca, Las Palmas, Bilbao, Switzerland; Geneva, Zürich,Bern, Lausanne, Base, Luzern, Thailand; Phuket, Chiang Rai,Khon Kaen, Chiang Mai, Lampang, Pak Kret, Hat Yai,Phitsanulok, Trang, Udon Thani, Yala, Samut Prakan, Yala,Songkhla, Nakhon, Pathom, Surat Thani, Ubon Ratchathani,Nakhon Ratchasima, Nonthabur Rayong, Nakhon Sawan,Nakhon Si Thammarat, Nakhon Si Ayutthaya, Bangkok, ChiangMai, UAE; Dubai, Abu Dhabi, Sharjah, Al Ain, Ajman, UK;Birmingham, Bristol, Edinburgh, Glasgow, Leeds,Liverpool, London, Manchester, United States; New York,Los Angeles California, Chicago Ill, Houston, Tex, Philadelphia,Phoenix, San Antonio, San Diego, Dallas, San Jose,Jacksonville, Indianapolis, San Francisco, Austin, Columbus,Fort Worth, Charlotte, Detroit, El Paso, Memphis, Baltimore,Boston, Seattle, Washington, DC, Nashville-Davidson, Denver,Louisville-Jefferson , Milwaukee, Portland, Las Vegas,
5. 5. Oklahoma City, Albuquerque, Tucson, Fresno, Sacramento,Long Beach, Kansas City, Mesa, Virginia Beach, Atlanta,Colorado Springs, Omaha Nebr, Raleigh N.C, Miami, Fla,Cleveland, Ohio, Tulsa, Okla, Oakland, Calif, Minneapolis,Minn, Wichita, Kans, Arlington, Tex, Vietnam; Ho Chi Minhcity, Hanoi, Hai Phong, Da Nang, Can Tho, Nha Trang, Hue, HaLong, Da Lat, Vung Tau, Can Thi, Sapa, etc.Get Expert Top Class Help by the Best IB Tutors/ Expert IBTutors/ 100% Guaranteed help on IB Mathematics/ IBMaths Studies SL IA/ Math portfolio (SL/HL Type 1/I Type2/II) IA (Internal Assessment) Circles/ Lacsap’s Fractions/Gold Medal Heights/ Fish Production/ Shadow Functions/Patterns from Complex Numbers/ The Dice Game/ Fillingup the Petrol Tank/ Extended Essays (EE) Help TutorsHL/SL Design Technology Design Projects RA/PA IA SL IBDP IA Samples Examples Solution Tutors Guidance/ TOKEssay/ TOK Presentation/ English IOP/IOC, etc.MATHEMATICSStandard LevelThe portfolio – IA tasksFor use in 2012 and 2013CIRCLES SL TYPE IAim: The aim of this task is to investigate positions of points inintersecting circles.
6. 6. The following diagram shows a circle with centre O and radius r, and any point P. 1 CThe circle has centre P and radius OP. Let A be one of the points of intersection of and . Circle has centre A, and radius r. The point is the intersection of with (OP). This is shown in the diagram below.Let . Use an analytic approach to find , when , and . Describewhat you notice and write a general statement to represent this.Let . Find , when , and . Describe what you notice and write ageneral statement to represent this. Comment whether or not thisstatement is consistent with your earlier statement.Use technology to investigate other values of r and OP. Find thegeneral statement for . OPTest the validity of your general statement by using differentvalues of OP and r.Discuss the scope and/or limitations of the general statement. Explain how you arrived at the general statement.LACSAP’S FRACTIONS SL TYPE IAim: In this task you will consider a set of numbers that arepresented in a symmetrical pattern. Consider the five rows of numbers shown below.Describe how to find the numerator of the sixth row.Using technology, plot the relation between the row number, n,and the numerator in each row. Describe what you notice fromyour plot and write a general statement to represent this.Find the sixth and seventh rows. Describe any patterns you used.
7. 7. Let be the element in the nth row, starting with . Example: .Find the general statement for.Test the validity of the general statement by finding additionalrows.Discuss the scope and/or limitations of the general statement.Explain how you arrived at your general statement.FISH PRODUCTION SL TYPE IIAim: This task considers commercial fishing in a particularcountry in two different environments – the sea and fish farms(aquaculture). The data is taken from the UN Statistics DivisionCommon Database.The following table gives the total mass of fish caught in thesea, in thousands of tonnes (1 tonne = 1000 kilograms).Define suitable variables and discuss any parameters/constraints.Using technology, plot the data points from the table on a graph.Comment on any apparent trends in your graph and suggestsuitable models.Analytically develop a model that fits the data points. (You mayfind it useful to consider a combination of functions.)On a new set of axes, draw your model function and the originaldata points. Comment on any differences. Revise your model ifnecessary.The table below gives the total mass of fish, in thousands oftonnes, from fish farms.
8. 8. Plot the data points from this table on a graph, and discusswhether your analytical model for the original data fits the newdata.Use technology to find a suitable model for the new data. On anew set of axes, draw both models.Discuss how trends in the first model could be explained bytrends in the second model.By considering both models, discuss possible future trends inboth types of fishing.GOLD MEDAL HEIGHTS SL TYPE IIAim: The aim of this task is to consider the winning height forthe men’s high jump in the Olympic Games.The table below gives the height (in centimeters) achieved bythe gold medalists at various Olympic Games.Note: The Olympic Games were not held in 1940 and 1944.Using technology, plot the data points on a graph. Define allvariables used and state any parameters clearly. Discuss anypossible constraints of the task.What type of function models the behaviour of the graph?Explain why you chose this function. Analytically create anequation to model the data in the above table.On a new set of axes, draw your model function and the originalgraph. Comment on any differences. Discuss the limitations ofyour model. Refine your model if necessary.Use technology to find another function that models the data. Ona new set of axes, draw both your model functions. Comment onany differences.
10. 10. method of shadow functions and their generators, which helpsidentify the real andimaginary components of complex zeros from key points alongthe x-axis.Part A (Quadratic Polynomials)Consider the quadratic function 2 2Write down the coordinates of the vertex.Show that 1 y has zeros, where.The “shadow function” to 1 y is another quadratic 2 y whichshares the same vertex as.However, 2 y has opposite concavity to that of 1 y and its zerosare in the form.Use various values for a and b to generate pairs of functions 1 yand.Hence, or otherwise, express 2 y in terms of 1 y and m y , wherem y is called the “shadowgenerating function”._ On a labelled diagram, illustrate how the zeros of 2 y may behelpful in the determinationof the real and imaginary components of the complex zeros of 1y.Part B (Cubic Polynomials)Now consider the cubic function 1 _ _ _ _ y _ (x _ 2) x _ (3_2i) x _ (3_ 2i) .The shadow function in this case is another cubic function 2 ywhich shares two points with 1 y ,has opposite concavity and its zeros are _2 and 3_ 2 as depictedin the figure below. The shadow
11. 11. generating function in this case passes through the two pointsof intersection.Write down an expression for 2 y and hence find the points ofintersection between 1 y and 2 y ._ Hence, or otherwise, determine the equation of the shadowgenerating function m y in thiscase._ Once again, express 2 y in terms of 1 y and m y . Usetechnology to investigate similarcubic functions._ Write a general statement about the functions 1 y , 2 y and m yand prove your statement._ On a labelled diagram, illustrate how the zeros of 2 y may behelpful in the determinationof the real and imaginary components of the complex zeros of 1y._ How can the findings with quadratics and cubics be applied toquartics?PATTERNS FROM COMPLEX NUMBERS HL TYPE ITo the student: The work that you produce to address thequestions in this task should be a reportthat can stand on its own. It is best to avoid copying thequestions in the task to adopt a “questionand answer” format.Part AUse de Moivre’s theorem to obtain solutions to the equation z3_1 _ 0 .
13. 13. The aim of this task is to create a dice game in a casino andmodel this usingprobability. It is important to examine how best to run the gamefrom both theperspective of a player and the casino. In doing so analyse thegame to consider theoptimal payments by the player and payouts by the casino._ Consider a game with two players, Ann and Bob. Ann has ared die and Bob a white die.They roll their dice and note the number on the upper face. Annwins if her score is higherthan Bob’s (note that Bob wins if the scores are the same). Ifboth players roll their diceonce each what is the probability that Ann will win the game?_ Now consider the same game where Ann can roll her die asecond time and will note thehigher score of the two rolls but Bob rolls only once. In this casewhat is the probabilitythat Ann will win?_ Investigate the game when both players can roll their dicetwice, and also when bothplayers can roll their dice more than twice, but not necessarilythe same number of times._ Consider the game in a casino where the player has a red dieand the bank has a white die.Find a model for a game so that the casino makes a reasonableprofit in the case where theplayer rolls the red die once and the bank rolls the white dieonce. (When creating yourmodel you will need to consider how much a player must pay toplay a game and how
14. 14. much the bank will pay out if the player wins. Do this from theperspective of both theplayer and the casino and consider carefully the criteria forwhether the game can beconsidered worthwhile for both the player and the casino.)_ Now consider other models for the game including cases suchas where the player or thebank rolls their dice multiple times, or where multiple playersare involved in the game.FILLING UP THE PETROL TANK HL Type IITo the student: The work that you produce to address thequestions in this task should be a reportthat can stand on its own. It is best to avoid copying thequestions in the task to adopt a “questionand answer” format.In this task, you will develop a mathematical model that helpsmotorists decide which ofthe following two options is more economical.Option 1: to buy petrol from a station on their normal route at arelatively higher price.Option 2: to drive an extra distance out of their normal route tobuy cheaper petrol.Consider a situation in which two motorists, Arwa and Bao,share the same driving route but owndifferent sized vehicles. Arwa fills up her vehicle’s tank at astation along her normal route forUS\$ 1 p per litre. On the other hand, Bao drives an extra dkilometres out of his normal route to fillup his vehicle’s tank for US\$ p2 per litre where p2 _ p1 .
15. 15. _ Choose suitable parameters for Arwa’s and Bao’s vehicles.Justify your choices._ Suppose 1 p _ US\$1.00 per litre, 2 p _ US\$0.98 per litre and d_10 km. Which motorist is getting the better deal? Show allcalculations and justify any assumptions you make._ On a spreadsheet, choose several sets of values for 1 p , 2 pand d to investigate further._ Define a set of variables that would be relevant in the abovesituation.“Effective litres” is a method of comparing the cost of petrolbought under the two optionsdescribed above. Effective litres, for a given vehicle, are thoselitres used when the vehicle travelsits normal route._ Use your variables and parameters to write algebraicexpressions for 1 E and 2 E whichrepresent the cost per effective litre under options 1 and 2respectively._ Write a model that helps motorists decide on the moreeconomical option for theirvehicles.In the remainder of this task, you need to consider the twovehicles you have chosen for Arwaand Bao._ Use your model to find the farthest distance that Bao shoulddrive to obtain a 2 % pricesaving._ Investigate the relationship between d and 2 p when 2 E iskept constant (e.g. US\$0.90,US\$1.00, … etc.). Use technology to draw a family of curves forArwa’s vehicle. Repeat
16. 16. for Bao’s vehicle._ For 2 E _ US\$0.90 , provide Arwa with information on threedifferent stations that yieldthis same cost per effective litre. Discuss how such informationmay be useful to Arwa._ For 2 E _ US\$1.00 and 2 p _ US\$0.80 , compare themaximum distance that each motoristshould drive and still save money.Arwa is a busy person and wonders whether the saving inmoney would be worth the time shewould lose in extra driving._ Modify your model to account for the time taken to drive toand from an off-route station.Clearly justify any assumptions you make.IB DP Math SL type 2 IA portfolio task Fish production, GoldMedal heightsIB DP Math HL type 2 IA portfolio task The dice game, Fillingup the petrol tank.IB DP Math HL type 1 IA portfolio task shadow function,Patterns from complex number.IB DP Math SL type 1 IA portfolio task Lacsap’s fraction,Circles IB ToK Essay Tutor IB ToK Essay Topics 2012/13 IB ToK Essay Help: Choosing the Right Essay Title IB ToK Essay Help: Planning and Writing the Essay IB ToK Essay Help: Marking and Assessing the Essay
17. 17. IB ToK Essay Help: Grammar and Punctuation Check IB TOK Essay on Common Sense IB Sample TOK essays Examiners mark essays against the title as set. Respond to the title exactly as given; do not alter it in any way. Your essay must be between 1200 and 1600 words in length, double spaced and typed in size 12 font.1. In what ways may disagreement aid the pursuit of knowledge in the natural and human sciences? 2. “Only seeing general patterns can give us knowledge. Onlyseeing particular examples can give us understanding.” To what extent do you agree with these assertions? 3. “The possession of knowledge carries an ethical responsibility.” Evaluate this claim.4. The traditional TOK diagram indicates four ways of knowing. Propose the inclusion of a fifth way of knowing selected from intuition, memory or imagination, and explore the knowledge issues it may raise in two areas of knowledge. 5. “That which can be asserted without evidence can be dismissed without evidence.” (Christopher Hitchens). Do you agree? 6. Can we know when to trust our emotions in the pursuit of knowledge? Consider history and one other area of knowledge. IB TOK ESSAY WRITERS HELP SOLUTION. IB COMMENTARY ESSAY HELP; IB ECONOMICS COMMENTARIES, IB TOK, IB EXTENDED ESSAY, IB IB TOK Essay Help, Mathematics, Physics, Economics, Chemistry