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Intro 15H Binomial Probabilities.notebook Intro 15H Binomial Probabilities notebook August 20 2014Consider a die which has 2 red faces and 4 black faces We rollthe die three times and record the resultsBinomial Probabilities If R represents the result is red and B represents the result isblack the possible outcomes are as shownCalculate probabilities usingBinomial ExpansionThe ratio of possibleout...

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ull marks may be obtained for answers to ALL questionsThe marks for the parts of questions are shown in round brackets e g 2There are 8 questions in this question paper The total mark for this paper is 75Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelledYou must show sufficient working to make your methods clear to the ExaminerAnswers without working m

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where un 2 32 26 x y 2x 4y 1 0a Determine the coordinates of the points of this circle when x 1 2 3b Determine whether 0 0 lies inside or outside the circle 37 a What is the maximum value of the function 5sin 2x 300 10b At what value s of x does this maximum occur given that 0 x 360 4c Solve the equation 5sin 2x 30 0 0 for 0 x 3600 68 A wire box is made such that the longest side of the box is 20c

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Binomial Expansion 251 6 Properties of Binomial coe cients 271 7 Some particular methods of proof 30Methods of proof by induction by contradiction necessary and su cientconditions1 8 Exercises 361 9 Hints and answers 392 Preliminary calculus 422 1 Di erentiation 42Di erentiation from rst principles products the chain rule quotientsimplicit di erentiation logarithmic di erentiation Leibniz theorem

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special casesThe Binomial Coef cientsThe coe cient of xk in the Binomial Expansion of 1 x nwhen n is a positive integer is denoted by n or n Ckkn n nk k n k n k0 1 n n n 1so for example 4 1 2 3 4The pattern of the coe cients is seen inPascal s triangle1 11 2 11 3 3 11 4 6 4 11 5 10 10 5 1nCk is the number of subsets with k elements that can bechosen from a set with n elements...

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SHARJAH INSTITUTE For comments corrections etc Please contact Ahnaf Abbas ahnaf mathyards comSharjah Institute of TechnologyBinomial Theorem Handout 71 Expansion OF BINOMIALSThe terms in any Binomial Expansion can be formed by tedious multiplication of termsfor examplex y 2 x y x yIf this is done the following table resultsx y 2 x2 2xy y2x y 3 x 3x2y 3xy2 y3...

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Chapter 9 Test Review Define the following terms and be able to give an example of eachSequence Series Arithmetic sequence Geometric sequence term explicit formula recursive formula sigmasummation notation Binomial Expansion and inductionProblems you should be able to do1 Write an explicit formula for the nth terma 0 3 8 15 24 b 1 -1 1 -1 1 c 12 10 8 6 4 d -13 5 9 -6 4 e f 1 3 1 3 12 Write the fir...

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OMMENDED QUESTIONS AND PROBLEMS Concentrate on the problemsfor sections 6 and 7 but have a look at the ones for 8 and 9 as well Problem 44 might bea good one to do On-line tutorials are available for Problems 13 21 23 31 and 39TURN IN Chapter 23 Problems 19 22 23 24 39NOTES AND ANNOUNCEMENTS1 This problem assignment is a challenging one Do not wait until the last moment tostart It may help to work

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t the end of the unit the learner will be able todetermine the modulus and an argument of the nth power of a nonzero complex numberdefined by their trigonometric formLesson objectivesBy the end of this lesson the learner should be able tocalculate the polar form of the nth power of a nonzero complex number defined in trigonometricformCalculate the polar form of the nth power of a nonzero complex n

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F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module M1 7 Series expansions and approximations1 Opening items 3 Infinite series1 1 Module introduction 3 1 Some standard infinite series1 2 Fast track questions 3 2 Convergence and divergence the ratio test1 3 Ready to study 3 3 Approximations to infinite series2 Finite series 4 Closing items2 1 Sequences 4 1 Module summary2 2 An ...

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JUST THE MATHS UNIT NUMBER19 6PROBABILITY 6Statistics for the Binomial distributionbyA J Hobson19 6 1 Construction of histograms19 6 2 Mean and standard deviation of abinomial distribution19 6 3 Exercises19 6 4 Answers to exercisesUNIT 19 6 - PROBABILITY 6STATISTICS FOR THE Binomial DISTRIBUTION19 6 1 CONSTRUCTION OF HISTOGRAMSElementary discussion on the presentation of data in the form of freque...

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ive Frequency Distributions 404 Measures of Location 43Introduction 44Use of Measures of Location 45Means 46Median 51Quantiles 54Mode 57Choice of Measure 58Appendix Functions Equations and Graphs 60ABEiiChapter Title Page5 Measures of Dispersion 67Introduction 68Range 69Quartile Deviation 70Standard Deviation and Variance 72Coefficient of Variation 75Skewness 766 Index Numbers 79Introduction 80Sim

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0.dvi The asymptotic behavior of the Hurwitz binomialdistributionJim PitmanTechnical Report No 500Department of StatisticsUniversity of California367 Evans Hall 3860Berkeley CA 94720-3860June 16 1998AbstractHurwitz s extension of Abel s Binomial theorem de nes a probability distribu-tion on the set of integers from 0 to n This is the distribution of the number ofnon-root vertices of a fringe subtr...

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Jazmin Rodriguez MTH350-Project 2PART 1 The Central Binomial CoefficentSpecificsThe Binomial coefficients are defined recursively as follows1 for all integers2 for all integers3 for all integersPlot the central Binomial coefficient as a function of forFind upper bounds for or determine the rate of growth of as a functionofImagine walking on the grid below from the red dot the origin to any other d...

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value - equations include extraneous roots and inequalities 108 Rational Expressions - simplify simple and complex 4 basic operations equations extraneous roots and 11inequalities9 Radicals - simplify 4 basic operations equations extraneous roots rationalize denominator 1410 Complex Numbers - simplify 4 basic operations powers 1511 Functions - definition notations domain range 1-1 onto compositio

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quality to get some better bounds In 4 Rooinshows that for any non-negative real numbers x1 x2 xn we haven1 n k 1 11 5 An Gn An 1 xn An 1 k 0n n nnk 2in whichn1An xin i 1andnnGn xii 11991 Mathematics Subject Classi cation 11A41Key words and phrases Primes Inequalities AGM-Inequality12 MEHDI HASSANIApplying 1 5 on p1 p2 pn and using relations 1 2 and 1 3 for everyn 10 we obtainpn n n1 6 p 1 p 2 pn

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hal number Prime Jacobsthal number andFibonacci Jacobsthal number 1 are de ned ass sn 1nJn 1s 1p 1nnJPn ss2ps 1where pi is the i-th prime number p0 2 p1 3s fs 1nnJFn 3fs 1where fi is the i-th Fibonacci number f0 0 f1 1 f2 1 f3 2 f4 321Theorem 1 For all natural number m such that 2m 1 and 2m 3 both are prime numbersn 12m 3 JPsn m 1 Cx 2n x Jn xn mx 0Theorem 2s 2 s2fs 1 fs JFn fs 1 JFn2 Proof of The

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e field at from wavelet at isexp expd dwhere is the field strength per unit area and is assumed constant1 P a g ePHYS20312 Wave Optics - Section 4 DiffractionThe total at is thus given byexp expd4 1where the double integral is over the unobstructed area i e the apertureNote that1 The HFI does not consider polarisation2 It is not always solubleWe ll consider two special cases1 Fraunhofer diffractio

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Sample Questions for Mathematics Placement Diagnostic Test Elementary functions3x 4 A B1 If x2 x 6 x 3 x 2 then A2 13a 1 b 2 c d e None of these5 52 Let f x ax 2 and g x 3x b Find values for a and b so that f and g are inverse functions1 1 1 1a a b 6 b a b 6 c a b 6 d a b 63 3 3 33 The coe cient of x2 y 6 in the Binomial Expansion of x y 8 isa 0 b 28 c 56 d 360 e None of these34 If f x 2 x then th...

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rom a Set of n Such ElementsWe use n n factorial for such arrangementsThe number of n different elements from a set of n suc elements isn Where n n n 1 n 2 n 3 3 2 1Ordered Selections of r Different Elements from a Set of n Such ElementsIf we select r different elements from a set of n such elements there would benn r Ordered selectionsThis is donated by nPr permutationThe number of ordered sets e

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avitational force between them is so small compared to the electric force andsince acceleration is proportional to the total force the di erence in acceleration will be negligibleWeek 1 August 20 2011 1 compiled August 20 2013c Let s say that the electron is orbiting the proton in a radius of r 0 53 10 10 m What would thevelocity of the electron be if it moves in a perfect circleSolution1 q1 q2Fe

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Microsoft Word - mod8lesson1.doc Grades 9-12 Module 8Polynomials Polynomial FunctionsCombinations and PermutationsClick Here forModule OverviewClick Here forPre-TestLesson 8 1MA 912 A 4 12 Apply the Binomial TheoremBinomial ExpansionExample 1x 4 3x 4 x 4 x 4x2 8x 16 x 4x3 4x2 8x2 32x 16x 64x3 12x2 48x 64Let s look at an alternative way to expand binomials It s faster and easier to usennDefinition ...

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ted on 5 May 2014Binomial Theoremn notation nCr notation n nCrBinomial TheoremBinomial Expansion- nCr n r n-r 20C18- Binomial Expansion x- General Term--Exponential and Logarithmic Functions- ax logaxExponential SeriesMathematical InductionLimitM1 M2 M1 M264L 21 7 Mon 12 30-1 40pm 740 1414SH1AL4L 22 7 Tue 12 30-1 40pm 740 1424SH2AL4L 23 7 Wed 1 55-3 05pm 650 14B4SH3AL4L 24 7 Thu 1 55-3 05pm 650 14

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placement evaluationCONTENT Functions linear systems graphs exponentials Binomial Expansion andsummation notationSTUDENTLEARNINGOUTCOMESMath 1531 Manipulate mathematical expressions at an intermediate level2 Solve mathematical equations at an intermediate level3 Demonstrate the qualitative behavior of graphs at an intermediate level4 Apply mathematical tools and concepts in solving word situation

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eneral case 895 2 Key properties of di erentials 915 3 Applications of di erentiation 935 4 The Newton Raphson method 996 Introduction to Mathematical Economics 1046 1 Supply and demand 1046 2 Excise tax 1056 3 Sales tax 1076 4 Revenue costs and pro t 1107 Series 1167 1 Arithmetic series 1167 2 Geometric series 1177 3 The Binomial Expansion 1207 4 The Binomial series 1227 5 Maclaurin Series 1258 I

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ay be obtained for answers to ALL questionsThere are 10 questions in this question paper The total mark for this paper is 75Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelledYou must show sufficient working to make your methods clear to the Examiner Answerswithout working may gain no creditN23558A This publication may only be reproduced in accordance w

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g GCE Maths teacher There are obviously many other ways oforganising the work and there is absolutely no requirement to use these schemesScheme 1 For A Level Maths with 2 TeachersIt is assumed that there are a total of 4 4 5 hours per week of contact time and around 36 weeks in theacademic year available for teaching It is also assumed that there is some teaching time after the ASexams in Y12 thou

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2 9 1 E1 wwwOR x 4-y 9 4sec 4-9tan 9sec -tan 1 35 i dx du 2u dy du 6u2 B1 both 2u and 6u2dy dy du 6u 2 M1dx dx du 2u3u A13 2 1 2OR y 2 x-1 dy dx 3 x-1 3u 3 B1 y f x M1 differentiation A1ii At 5 16 u 2 M1dy dx 6 A1 cao2124754 Mark Scheme June 20081 31 M1 Binomial Expansion with p -1 26 i 1 4 x21 2 2 2 4 x 2 221 4 x2 21 2 x 2 6 x 4 A1 1 2x2A1 6x4M1A1Valid for 1 4x2 1 - x5ii 1 x2 M1 substitituting th

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26 ALGEBRA 2 TRIGONOMETRY ALGEBRA 2 TRIGONOMETRY 27 Test 3 Test 3Part I9 Solve for x x x l 5Answer all 27 questions in this part Each correct answer will receive 21 3 8 2 -3 8 3 3 4 8 9credits No partial credit will be allowed For each question write in the spaceprovided the numeral preceding the word or expression that best completes the10 What is the third tenn in the Binomial Expansion of x - 2...

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On the Hamiltonian of a Multi-Electron Atom Austen GroenerDrexel UniversityPhiladelphia PAOctober 29 20101 Introduction Let s now look in more depth at the terms that camefrom this Expansion By multiplying the factor ofIn this paper we will exhibit the process of achieving me c2 through the bracket the rst term whichthe Hamiltonian for an electron gas Making the sim- appears is the rest energy of ...

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