Problems 1: pages 2 and 4Problems 2: pages 2, 3 and 4Problems 3: pages 3, 4, 7 and 8
School of Mathematics Level I Semester 1 & 2RCA 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 1 2 LI Linear Algebra & Linear Programming 3 LI Multivariable & Vector Analysis 4 LI Real & Complex Analysis 1AC2 06 27363 Level I LI Algebra & Combinatorics 1 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. (a) Let a,b,c ∈ Z. (i) Suppose that a | b and a | c. Prove that a2 | b3−ac. (ii) Prove that hcf(2a+b,a+b) | b. (b) Let a,b,c ∈ Z. Suppose that 3a−5b = 1 and that a | bc. Prove that a | c. (c) Let a,b ∈ Z and n ∈ N. Suppose that a≡ b+1 mod n. Prove that a2−2a≡ b2−1 mod n. (d) Let p ∈ N be prime with p > 5. Prove that p4 ≡ 1 mod 10. 2. (a) Do there exist sets A and B whose power sets P(A) and P(B) are disjoint? Please explain your answer. (b) How many positive factors of 3600 are there? (c) How many positive factors of 3600 are divisible by either 8 or 15? (d) I place 19 stones on squares of a chessboard (i.e. an 8x8 square grid). Prove that however I do this, there must somewhere be a 3x3 grid which contains at least 3 stones. End of LI Algebra & Combinatorics 1 Page 1 Back to the index 2LALP 06 25765 Level I LI Linear Algebra & Linear Programming 2LA 06 15552 Level I Linear Algebra 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) Recall that P2 is the space of all real polynomial functions f (x) of degree up to 2. Consider the map α : P2→ P2 given by f 7→ xd f dx +2 f (0). (i) Verify that α is a linear map. (ii) Find the matrix of α with respect to the standard basis of P2. (iii) Find ker(α) and state its dimension. (iv) Is α bijective? Explain. (b) Suppose that U and V are vector spaces over a field F, of finite dimensions n = dimU and m = dimV , where n < m.="" explain="" why="" no="" linear="" map="" ψ="" :="" u="" →v="" can="" be="" surjective.="" (c)="" for="" the="" matrix="" a="(" 1="" −1="" −4="" 1="" )="" ,="" (i)="" find="" its="" characteristic="" polynomial="" and="" eigenvalues;="" (ii)="" for="" each="" eigenvalue,="" find="" the="" corresponding="" eigenspace;="" (iii)="" diagonalise="" the="" matrix,="" that="" is,="" find="" an="" invertible="" matrix="" t="" such="" that="" d="TAT−1" is="" diagonal.="" end="" of="" li="" linear="" algebra="" &="" linear="" programming="" page="" 2="" back="" to="" the="" index="" 2mva="" 06="" 25667="" level="" i="" li="" multivariable="" &="" vector="" analysis="" 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)="" for="" the="" function,="" f="" (x,y)="ln" |="" x2="" +="" y3="" |,="" calculate="" ∂="" f∂x="" ,="" ∂="" f="" ∂y="" ,="" ∂="" 2="" f="" ∂x∂y="" and="" ∂="" 3="" f="" ∂x∂y2="" at="" the="" point="" (1,="" 2).="" (b)="" using="" the="" method="" of="" lagrange="" multipliers,="" find="" the="" maximum="" and="" minimum="" values="" of="" the="" function="" f="" (x,y,z)="x+" z,="" where="" (x,y,z)="" is="" on="" the="" following="" surface="" x2="" +="" y2="" +="" z2="1." (c)="" calculate="" the="" following="" integral="" using="" polar="" coordinates="" i="∫" 2="" √="" 2="" dx="" ∫="" x="" 0="" dy="" (x2="" +="" y2)3/2="" .="" (d)="" integrate="" the="" function="" f="" (x,y,z)="y" over="" the="" volume="" enclosed="" by="" the="" planes="" z="0" and="" z="x+" y,="" and="" the="" surfaces="" x="y2" and="" x="2y−" y2.="" end="" of="" li="" multivariable="" &="" vector="" analysis="" page="" 3="" back="" to="" the="" index="" 2rca="" 06="" 25666="" level="" i="" li="" real="" &="" complex="" analysis="" 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="" α="" ∈="" n,="" and="" f="" :="" r−→="" r="" be="" the="" function="" given="" by="" f="" (x)="" ="" ="" |x|α="" cos="" (1="" x="" )="" ,="" if="" x="" 6="0," 0,="" if="" x="0." (i)="" show="" that="" f="" is="" continuous="" at="" 0="" for="" all="" α="" ∈="" n.="" (ii)="" show="" that="" f="" is="" not="" differentiable="" at="" 0="" when="" α="1." (iii)="" justify="" why="" f="" is="" differentiable="" at="" x="" 6="0" when="" α="1," and="" give="" the="" value="" of="" f="" ′(x)="" for="" all="" x="" 6="0." (b)="" let="" (="" fn)∞n="3" be="" the="" sequence="" of="" functions="" defined="" by:="" fn(x)="" ="" ="" nx,="" 0≤="" x≤="" 1="" n="" ,="" 1,="" 1="" n="">< x="">< 2="" n="" ,="" n="" n−2(1−="" x),="" 2="" n="" ≤="" x≤="" 1.="" (i)="" find="" the="" pointwise="" limit="" of="" the="" sequence="" of="" functions="" (="" fn)∞n="3." (ii)="" does="" (="" fn)∞n="3" converge="" uniformly="" on="" [0,1]="" to="" its="" pointwise="" limit?="" justify="" your="" answer.="" (c)="" let="" a=""> 0 and b ∈ R. Use the Intermediate Value Theorem to show that the equation x3 +ax+b = 0 has at least one real solution. End of LI Real & Complex Analysis Page 4 Back to the index School of Mathematics Level I Semester 1 & 2RCA Week 12 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 1 2 LI Linear Algebra & Linear Programming 3 LI Multivariable & Vector Analysis 4 LI Real & Complex Analysis 1AC2 06 27363 Level I LI Algebra & Combinatorics 1 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. (a) (i) Find y,z ∈ Z such that 44y+34z = 2. (ii) Solve the simultaneous congruences x≡ 7 mod 34 x≡ 19 mod 44 giving your solution in the form x≡ s mod n, where s ∈ Z and n ∈ N. You should give sufficient explanation to justify that you have found all solutions in (a)(ii). (b) Let f = (14263) ∈ S6 and g = (13562) ∈ S6. (i) Calculate f ◦g. (ii) Find h ∈ S6 such that f ◦h = g. (c)