Hi I need full solutions to Algebra and Combinatorics and Differential Equations please
School of Mathematics Level I Semester 2 Supplementary Examinations Instructions • Answer all questions for modules for which you are registered and which appear in this booklet. • You have a maximum period of time to complete and upload your solutions of 12+ 12q hours, where q is the number of full questions that you are required to complete. Any solutions submitted after your personal deadline will not be marked. • This assessment is not intended to need the full time period you have available; the time period indicates only the timescale within which you need to complete this assessment. The available time period automatically takes into account an adjustment for those stu- dents with Reasonable Adjustment Plans (RAPs) or individual personal circumstances. • Solutions must be neatly handwritten; typed solutions will not be marked. • Make sure that you read and follow the instructions for each module, in particular the page limit for your answers. Solutions will only be marked up to the specified page limit. • Solutions to each question should be uploaded to Canvas individually as a single PDF document and may be submitted at any point within the time period available. Solutions may be re-uploaded, however incorrectly submitted solutions, i.e. uploaded to the wrong question, may not be marked. It is your responsibility to ensure that your solutions have successfully uploaded to Canvas by your personal deadline. • Whilst you may use your notes and other materials available to you, submitted answers should be your own. The School of Mathematics has enhanced processes in place to identify any instances of plagiarism or collusion. There is no reduction in the threshold for plagiarism associated with this assessment. Appropriate University penalties will be applied to any individual found to be involved in any way with an act of plagiarism or collusion. • When your answers to each question are graded the marker will take into account a num- ber of factors to determine your awarded grade including: whether your answers are fully correct to all aspects of the question; the overall level of mathematical precision and rigor demonstrated in your arguments; the appropriateness of your mathematical explanation; and, your overall presentation of the mathematical material. The index uses hyperlinks to aid navigation and on each page there is a “Back to index” link. Page 2 Turn over Index 1 LI Algebra & Combinatorics 2 3 LI Differential Equations 5 LI Mechanics 7 LI Probability and Statistics 10 LI Statistics 2AC 06 25665 Level I LI Algebra & Combinatorics 2 Full marks may be obtained with complete answers to BOTH the following questions. Each answer must be no more than SIX sides of A4, any work in excess of this will not be marked. 1. You may apply results from the lecture notes, but you should clearly state which results you use and when you apply them. (a) Let ω = ei 2π 3 ∈ C and let R = Z[ω] = {a+ bω : a,b ∈ Z}. Let I = 〈2〉 be the principal ideal of R generated by 2. (i) Show that any element of R/I can be written in the form [c+ dω]I , where c,d ∈ {0,1}. (ii) Show that R/I is a field and deduce that I is a maximal ideal of R. (b) Let α = 3 √ 5 ∈ R. Let θ : Q[X ]→ R be the homomorphism defined by θ( f (X)) = f (α). Show that kerθ is the principal ideal of Q[X ] generated by X3−5. In (b) you may assume that there is no polynomial f (X) ∈ Q[X ] with f (X) 6= 0 and deg f (X)< 3="" such="" that="" f="" (="" 3="" √="" 5)="0." (c)="" let="" a,b="" ∈="" q="" with="" b="" 6="0" and="" let="" f="" (x)="" ∈="" z[x="" ]="" be="" a="" monic="" polynomial="" with="" deg="" f="" (x)="3." suppose="" that="" a+bi="" is="" a="" root="" of="" f="" (x).="" (i)="" show="" that="" x2="" +2ax="" +(a2="" +b2)="" is="" the="" minimal="" polynomial="" of="" a+bi="" over="" q.="" (ii)="" show="" that="" f="" (x)="" has="" a="" root="" in="" z.="" (d)="" let="" f2="{0,1}" be="" the="" field="" with="" two="" elements="" and="" let="" f2[x="" ]="" be="" the="" polynomial="" ring="" over="" f2.="" let="" i="〈X2+X" +1〉="" be="" the="" principal="" ideal="" of="" f2[x="" ]="" generated="" by="" x2+x="" +1="" ∈="" f2[x="" ]="" and="" let="" j="〈X4+X" +1〉="" be="" the="" principal="" ideal="" of="" f2[x="" ]="" generated="" by="" x4+x="" +1∈="" f2[x="" ].="" (i)="" prove="" that="" f2[x="" ]/i="" and="" f2[x="" ]/j="" are="" fields.="" (ii)="" define="" θ="" :="" f2[x="" ]/i→="" f2[x="" ]/j="" by="" θ([a+bx="" ]i)="[a+bX" +bx2]j="" .="" prove="" that="" θ="" is="" a="" homomorphism.="" li="" algebra="" &="" combinatorics="" 2="" turn="" over="" page="" 1="" back="" to="" the="" index="" 2.="" you="" may="" apply="" results="" from="" the="" lecture="" notes="" or="" the="" examples="" sheets,="" provided="" that="" you="" state="" them="" clearly="" and="" explain="" how="" you="" apply="" them.="" (a)="" use="" the="" rule="" of="" double="" counting="" to="" show="" that="" there="" are="" (n−1)!="" permutations="" of="" the="" num-="" bers="" 1,="" .="" .="" .="" ,n,="" for="" n≥="" 2,="" with="" exactly="" one="" cycle.="" (b)="" show="" that="" the="" number="" of="" walks="" from="" (0,0)="" to="" (n,k),="" where="" k=""> 0 and n+k is even, which reach at least r, for r > k, is equal to the total number of walks from (0,0) to (n,2r− k). (c) Let G = (V,E) be a graph with exactly one vertex having degree equal to ∆(G), where ∆(G) is the maximum degree in G. Show that χ(G)≤ ∆(G). (d) Let G = (V,E) be a graph on n ≥ 1 vertices with independence number α(G) ≤ n1/3. Prove that χ(G)≥ n2/3. (e) Identify the algorithm that outputs in increasing order the keys that are stored in a binary search tree and show that it is correct. End of LI Algebra & Combinatorics 2 Page 2 Back to the index 2DE 06 25670 Level I LI Differential Equations Full marks may be obtained with complete answers to BOTH the following questions. Each answer must be no more than SIX sides of A4, any work in excess of this will not be marked. 1. (a) Find the general nontrivial separable solution to the following system: ∂ 2u ∂x2 =−∂ 2u ∂y2 , 0 < x="">< π,="" 0="">< y="">< π,="" (1)="" u(0,y)="0," ux(π,y)="0," 0="">< y="">< π,="" (2)="" u(x,0)="sin" (x="" 2="" )="" +="" sin="" (="" 3x="" 2="" )="" ,="" u(x,π)="0" 0="">< x="">< π.="" (3)="" (b)="" (i)="" by="" multiplying="" the="" fourier="" sine="" series="" (with="" coefficients="" bn)="" for="" a="" general="" function="" f="" (x)="" on="" 0≤="" x≤="" l="" by="" f="" (x),="" show="" that="" 2="" l="" ∫="" l="" 0="" (="" f="" (x))2="" dx="∞" ∑="" n="1" b2n.="" (4)="" hint:="" consider="" the="" definition="" of="" bn.="" (ii)="" find="" the="" fourier="" sine="" series="" of="" f="" (x)="1" on="" 0="">< x="">< π="" .="" (iii)="" using="" parts="" (b)(i)="" and="" (b)(ii),="" obtain="" a="" series="" approximation="" for="" π2.="" li="" differential="" equations="" turn="" over="" page="" 3="" back="" to="" the="" index="" 2.="" (a)="" consider="" the="" following="" two-dimensional="" system="" in="" r2,="" ẋ="−xy−" y+="" y2,="" ẏ="−x+4x2," where˙means="" ddt="" .="" i)="" determine="" the="" position="" and="" nature="" of="" the="" equilibrium="" points="" of="" the="" system.="" ii)="" sketch="" the="" phase="" portrait="" of="" the="" system="" -="" you="" may="" assume="" that="" a="" centre="" for="" the="" linear="" system="" remains="" a="" centre="" for="" the="" nonlinear="" one.="" (b)="" consider="" the="" dynamical="" system="" ẋ="x2" +2x−="" c+2,="" x="" ∈="" r,="" where="" c="" ∈="" r,="" constant,="" is="" a="" control="" parameter="" of="" the="" system.="" i)="" determine="" the="" number="" and="" location="" of="" the="" equilibrium="" points="" of="" the="" above="" system="" for="" all="" values="" of="" the="" control="" parameter="" c.="" ii)="" using="" linear="" stability="" analysis,="" determine="" the="" stability="" of="" the="" equilibrium="" points="" that="" you="" have="" found="" in="" part="" (i).="" if="" linear="" stability="" analysis="" fails,="" use="" a="" graphical="" argument="" to="" deter-="" mine="" stability.="" iii)="" sketch="" the="" bifurcation="" diagram="" of="" the="" system.="" end="" of="" li="" differential="" equations="" page="" 4="" back="" to="" the="" index="" 1mech2="" 06="" 27345="" level="" i="" li="" mechanics="" full="" marks="" may="" be="" obtained="" with="" a="" complete="" answer="" to="" the="" following="" question.="" your="" answer="" must="" be="" no="" more="" than="" six="" sides="" of="" a4,="" any="" work="" in="" excess="" of="" this="" will="" not="" be="" marked.="" li="" mechanics="" turn="" over="" page="" 5="" back="" to="" the="" index="" 1.="" (a)="" if="" the="" position="" vector="" of="" a="" particle="" is="" given="" by="" r="at3i+bt2j+" ctk,="" where="" t="" is="" time="" and="" a,="" b,="" c="" are="" constants,="" find="" the="" velocity="" and="" acceleration.="" what="" form="" must="" the="" applied="" force="" be="" to="" cause="" this="" motion?="" (b)="" consider="" a="" particle="" of="" mass="" m="" attached="" to="" a="" spring="" (spring="" constant="" k,="" fixed="" at="" the="" other="" end)="" and="" surrounded="" by="" a="" liquid="" (which="" is="" at="" rest).="" the="" liquid="" acts="" as="" a="" drag="" force="" which="" opposes="" the="" motion="" proportional="" to="" the="" velocity="" of="" ball="" through="" the="" liquid,="" with="" constant="" of="" proportionality="" µ="" .="" (i)="" what="" are="" the="" dimensions="" of="" µ?="" (ii)="" derive="" an="" ordinary="" differential="" equation="" for="" the="" motion="" of="" the="" particle.="" you="" may="" ne-="" glect="" gravity.="" (note="" that="" you="" are="" only="" asked="" to="" derive="" the="" equation,="" not="" solve="" it!).="" (c)="" a="" particle="" of="" mass="" m="" slides="" on="" the="" surface="" of="" a="" smooth="" bowl="" with="" equation="" z="−a2/ρ" in="" cylindrical="" polar="" coordinates="" ρ="" ,="" θ="" and="" z,="" with="" the="" z-axis="" pointed="" vertically="" upwards.="" (i)="" briefly="" explain="" why="" 1="" 2="" m="" (="" ρ̇2="" +ρ2θ̇="" 2="" +="" ż2="" )="" +mgz="constant," ρ2θ̇="constant." (ii)="" the="" particle="" is="" initially="" at="" ρ="a" and="" projected="" horizontally="" along="" the="" bowl’s="" surface="" with="" speed="" ρθ̇="v." show="" that="" ρ̇2="" +ρ2θ̇="" 2="" +="" ż2="" +2gz="v2" −2ga,="" ρ2θ̇="av." (iii)="" hence="" deduce="" that="" ρ̇2="" (="" 1+="" a4="" ρ4="" )="v2" (="" 1−="" a="" 2="" ρ2="" )="" +2ga="" (="" a="" ρ="" −1="" )="" .="" (iv)="" using="" an="" expression="" for="" ρ̈="" ,="" find="" the="" condition="" on="" v="" for="" the="" particle="" to="" initially="" rise="" or="" fall.="" (v)="" if="" v2="">< 2ag show that the particle moves between two values of ρ and find them. end of li mechanics page 6 back to the index statistical tables 1ps2 06 26709 level i li probability and statistics full marks may be obtained with a complete answer to the following question. your answer must be no more than six sides of a4, any work in excess of this will not be marked. li probability and statistics turn over page 7 back to the index statistical tables 1. (a) let x be a continuous random variable with probability density function fx , given by fx(x) = ax+0.05 for x ∈ [0,4]; 0 for x /∈ [0,4]. (i) find a. (ii) what is p(1≤ x ≤ 5)? (iii) calculate e[x ]. (b) a bag contains 12 tiles. one letter appears on each tile and together the tiles spell the word arrangements. two tiles are selected randomly without replacement from the bag. (i) what is the probability that both letters on the selected tiles are vowels? (ii) what is the probability that the letters on the selected tiles are two different conso- nants? (c) a coloured 2ag="" show="" that="" the="" particle="" moves="" between="" two="" values="" of="" ρ="" and="" find="" them.="" end="" of="" li="" mechanics="" page="" 6="" back="" to="" the="" index="" statistical="" tables="" 1ps2="" 06="" 26709="" level="" i="" li="" probability="" and="" statistics="" full="" marks="" may="" be="" obtained="" with="" a="" complete="" answer="" to="" the="" following="" question.="" your="" answer="" must="" be="" no="" more="" than="" six="" sides="" of="" a4,="" any="" work="" in="" excess="" of="" this="" will="" not="" be="" marked.="" li="" probability="" and="" statistics="" turn="" over="" page="" 7="" back="" to="" the="" index="" statistical="" tables="" 1.="" (a)="" let="" x="" be="" a="" continuous="" random="" variable="" with="" probability="" density="" function="" fx="" ,="" given="" by="" fx(x)="" ="" ="" ax+0.05="" for="" x="" ∈="" [0,4];="" 0="" for="" x="" ∈="" [0,4].="" (i)="" find="" a.="" (ii)="" what="" is="" p(1≤="" x="" ≤="" 5)?="" (iii)="" calculate="" e[x="" ].="" (b)="" a="" bag="" contains="" 12="" tiles.="" one="" letter="" appears="" on="" each="" tile="" and="" together="" the="" tiles="" spell="" the="" word="" arrangements.="" two="" tiles="" are="" selected="" randomly="" without="" replacement="" from="" the="" bag.="" (i)="" what="" is="" the="" probability="" that="" both="" letters="" on="" the="" selected="" tiles="" are="" vowels?="" (ii)="" what="" is="" the="" probability="" that="" the="" letters="" on="" the="" selected="" tiles="" are="" two="" different="" conso-="" nants?="" (c)="" a="">