Applied Algebra — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

1(a) Consider the production model x = Cx + dfor an economy with two sectors, where 2/andd =]. an inverse matrix to determine the production level necessary to. satisfy the final demand. 1(b) Find the 3 x 3 matrix that corresponds to the composite transformation of a scaling by .3, a rotation of 90°about the origin, and finally a translation that adds (-5,2) to each point of a figure. 1(c) Verify that 3k, (-3)K and 2Kare linearly independent signals. 2(a) Verify that the signal yk = k' satisfies the difference equation Yk+2 - 4Ук+1 + 3Ук = -4k for all k.Then find a description of all solutions of this equation. 2(b) Find a least-squares solution of the inconsistent system Ax = bfor A = 2,6-601 2(c) Find the equation y = Bo + Byxof the least-squares line that best fits the data points 3.(a) Define a Latin square and give example of a Latin square of order 6. Let L = [ay] be a matrix with aly = i + 5j (mod 5); 1 ≤ i,j ≤ 5. Is L a Latin square? 3(b) Construct and verify two orthogonal Latin squares for the group 7g with addition modulo 3. 3(c) For the graph below with four persons A, B,C, D and four jobs a, b,c, d, find a matching or explain there does not exist a matching 4.(a) For the graph below with vertices a, b,c, d,e, f,g, h, i, j,k,, find three different minimal sets of edges needed to link all the vertices. What is the cardinality of any such minimal set? 4(b) In a tournament, there are five teams and each team plays with three other teams. Draw the graph of the schedule of the tournament? 4(c) Suppose there are given three containers of water, of sizes 12 liters, 7 liters, and 6 liters. Initially the 12-liter container is full. Two other containers are empty. One can pour water from one container into another, pouring until the receiving pitcher is full or the pouring pitcher is empty. Is there a way to pour 3 liters of water in each of the 7 liter and 6-liter containers? If yes, find a minimal sequence of pourings to do so. 5 (a) Calculate(337 × 75) (mod 10). Is 10962801047 a valid US postal money order number according to its check digit scheme. A US Postal money order no. 27914009534 was incorrectly transmitted as 27914009543. Will this error be detected? When does this scheme fail to detect transposition of adjacent digits error? Justify. 5(b) In an ISBN number 03819Y5769, the digit at 6th place is not readable. Using the ISBN check digit scheme determine the missing digit Y. Give an example of single digit error in this number. Will this error be detected? Does this scheme detect all single digit errors? Prove it. What are the drawbacks of ISBN scheme? 5(c) In a 7 digit identification number ayazazazasaga the check digit ay is defined by Does this scheme detect all single digit errors? Prove it. If instead the check digit ay is defined by Does this scheme detect all single digit errors? Justify. A UPC number for a new product is 3-88207-29431-C. Calculate the check digit C