Hi, I need full solutions to the Algebra and Combinatorics and Differential Equations questions on the attached problem sheets please. You can ignore the statistics questions I don't need those ones.
School of Mathematics Level I Semester 2 Week 6 Supplementary Assessment 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 4 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 THREE 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 R be a ring, let S be a subring of R and let I be an ideal of R. Prove that I∩S is an ideal of S. (b) Let I be an ideal of Z[i] with I 6= {0}. (i) Show that there exists m ∈ N such that I∩Z is the principal ideal of Z generated by m. (ii) Consider the case where I is the principal ideal of Z[i] generated by 3−4i. Determine m ∈ N such that I∩Z is the principal ideal of Z generated by m. (c) Define R = {( a b 2b a ) ∈M2(Z) : a,b ∈ Z } ⊆M2(Z). You are given that R is a subring of M2(Z). Define θ : R→ R by θ ( a b 2b a ) = a+b √ 2. Show that θ is an injective homomorphism. LI Algebra & Combinatorics 2 Turn over Page 1 Back to the index 2. (a) For n > 2, let S be a family of subsets of the set [n] := {1, . . . ,n} each having size at most k, where 1 ≤ k < n/2,="" such="" that="" no="" member="" of="" s="" is="" a="" proper="" subset="" of="" another="" member="" of="" s="" .="" show="" that="" |s="" |="" ≤="" (n="" k="" )="" .="" (b)="" determine="" the="" number="" of="" permutations="" on="" the="" set="" {1,="" .="" .="" .="" ,n}="" with="" n≥="" 3="" with="" one="" cycle="" of="" length="" 3="" and="" all="" other="" cycles="" having="" length="" 1.="" justify="" your="" answer.="" (c)="" let="" n="" ≥="" 2="" be="" even.="" find="" the="" number="" of="" permutations="" on="" the="" set="" {1,="" .="" .="" .="" ,n}="" that="" are="" in-="" volutions,="" that="" is,="" they="" satisfy="" σσ="id," where="" id="" is="" the="" identity="" permutation.="" justify="" your="" answer.="" (d)="" calculate="" the="" number="" of="" walks="" from="" (0,0)="" to="" (2n,0),="" where="" n="" ∈="" n,="" that="" hit="" 0="" exactly="" once="" at="" x="n." 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="" 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.="" 1.="" (a)="" (i)="" by="" seeking="" a="" solution="" in="" the="" form="" y(x)="xα" where="" α="" ∈="" r,="" obtain="" two="" linearly="" inde-="" pendent="" solutions="" to="" the="" following="" homogeneous="" equation="" in="" y="y(x):" x2y′′−2xy′+2y="0," x=""> 0. (1) (ii) Use the Reduction of Order method (not Variation of Parameters) with each linearly independent solution derived above to obtain the general solution to the following inhomogeneous equation: x2y′′−2xy′+2y = x2, x > 0, (2) (i.e. you must provide two separate derivations of the general solution). How should the two general solutions to (2) that you have derived compare? (b) (i) Obtain a power series solution to the following homogeneous equation in y = y(x): y′′−2xy′+λy = 0, (3) where λ ∈R. Please include the first three terms in each component of the general solution to (3). (ii) What can you say about the solution to (3) if λ is an even natural number? (c) Determine the interval on which it is guaranteed that a unique solution exists to the follow- ing initial value problem for y = y(x): ( x− π 4 ) y′′′− x2y′+ xy = tan(x), (4) y′′′(0) = π, y′′(0) = 0, y′(0) = 0, y(0) = π. (5) You do not need to solve the equation. End of LI Differential Equations Page 3 Back to the index 2S 06 25671 Level I LI 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. 1. (a) Let (Ω,F ,P) be a probability space and let B ∈ F be such that P(B) > 0. For any A ∈F let PB(A) = P(A∩B) P(B) . Show that the function PB : F → R is a probability measure on (Ω,F ). (b) Let Ω = {1, . . . ,6} and let F = {Ω,∅,{1,2,3},{4,5,6}} be a σ -algebra on Ω. Let P be the probability measure on (Ω,F ) such that for any A ∈F we have P(A) = |A|/6. Let X : Ω→ R be the function which is X(ω) = 0 if ω is odd, 1 if ω is even. Is X a random variable on (Ω,F ,P)? Justify you answer. (c) Let Xn be binomially distributed with parameters n and p, where p ∈ (0,1) does not depend on n. Let µn = E(Xn) and σ2n = Var(Xn). Let X̂n = Xn µn . Show that X̂n converges in probability to 1 as n→ ∞. (d) Let (Xn)n∈N be a sequence of random variables defined on a probability space (Ω,F ,P) such that E(Xn) = 0 and Var(Xn)≤ n, for all n ∈ N. Show that P(Xn ≥ n3/2 infinitely often) = 0. End of LI Statistics Page 4 Back to the index School of Mathematics Level I Semester 2 Week 11 Supplementary Assessment 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 4 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 THREE 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 I = 〈X2−X〉 be the principal ideal of Q[X ] generated by X2−X and let J = 〈X−1〉 be the principal ideal of Q[X ] generated by X−1. Define θ : Q[X ]→Q[X ]/I by