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S228/318 DUBLIN INSTITUTE OF TECHNOLOGY KEVIN STREET, DUBLIN 8. _____________ BSc Computer Science Year 3 ______________ SUPPLEMENTAL EXAMINATIONS 2009 ______________ Computational Mathematics John Gilligan Dr D. Lillis Marking Scheme Autumn 2009 Time allowed 2 Hours Answer question 1 and 2 other questions. 1: a) Find the distance between the points i) (0, 2) and (3, 6) and ii) (-2, 3) and (4, -5) (4 marks) Solution/Question Rationale Simple distance formula Marking Scheme 2 marks for each distance. b) Find the values of a and b if i) (4, 1) is the midpoint (a, b) and (-1, 5) ii) (3, b) is the midpoint of (a, 2) and (0, 0). (6 marks) Solution/Question Rationale Algebraic manipulation of midpoint formula Marking Scheme 3 marks for each calculation c) Sketch the image of △abc under an axial symmetry in K. a K b c (6 marks) Solution/Question Rationale I introduced various transformations such as axial symmetry. Can they realise this graphically. Marking Scheme 2 marks for knowing what axial symmetry is 4 marks for correctly drawing image d) For each of the following letters, state whether it has a centre of symmetry and indicate where it is. H T A Z (4 marks) Solution/Question Rationale Do they know what the centre of symmetry is and where it is on these symbols Marking Scheme 1 mark by 4 e) What is [2]5 X [6]5 [12]5 = [2]5 (6 marks) Solution/Question Rationale Are they aware of the multiplication rule for modulo arithmetic and can it be applied? Marking Scheme 4 marks for multiplication rule. 2 marks for reduction to modulo 5 f) Using modulo 5 and the number 13 for illustration, define what is meant by the equivalence class modulo n containing m (6 marks) {….,-7,-2,3,8,13,18,23,28,…..} Marking Scheme 6 marks for correctly listing equivalence class. g) Prove that A(2, 5), B(-4, 5) and C(-1, 2) are the vertices of a right-angled isosceles triangle. (6 marks) Solution/Question Rationale Can the students apply Pythagoras and Line distance formula. Marking Scheme 3 marks for correctly using line distance formula and 3 marks for Pythagoras 2: a) Given Vectors A = (1,1,1) and B = (1,1,0) and C = (1,0,0), Find the scalars α, β, and γ if αA + βB + γC = (2,-3,4) (5 marks) Solution/Question Rationale Do the students understand scalar multiplication and can they solve linear equations. Marking Scheme 3 marks for correctly doing scalar multiplication and 2 marks for solving linear equations Let A = (1,3,6) B = (4, -3, 3) C = (2,1 ,5) be three vectors in V3 Determine the components of the following vectors i) -3A +2B + 6C (3 marks) Solution/Question Rationale This is basic vector addition Marking Scheme 3 marks for correct answer. ii) 2A + B – 3C (3 marks) Solution/Question Rationale This is basic vector addition Marking Scheme 3 marks for correct answer. iii) Draw these Geometrically. (4 marks) Solution/Question Rationale This is just to remind them of graph plotting Marking Scheme 2 marks for each graph. iv) || A + B + C|| (A + B + C) = (7,1,14) || A + B + C|| = 2 7 1 14 2 (4 marks) Solution/Question Rationale Can they calculate vector length Marking Scheme 4 marks for correct answer. v) Calculate A.B and B X C 1*4 + 3*-3 + 6*3 = 13 -3*5 – 1*3 = -18 3*2 – 5*4 = -14 4*1 – 2*3 = -2 (-18,-14,-2) (6 marks) Solution/Question Rationale These are basic vector operations of dot and cross product. Marking Scheme 3 marks for correct answer. X2 vi) Prove for any vectors u, v and w that: (u+v)+w = u + (v+w) (5 marks) Solution/Question Rationale Associativity is important within vector algebra. Marking Scheme 4 marks for good attempt 1 marks for correct proof 3 a) Suppose 3 7 A 2 5 1 1 3 B 2 1 4 1 1 C 2 3 Either calculate each of the following or explain why the calculation cannot be done. i) 3A-C ii) AB ii) BA iv) A-1 v) Bt (15 marks) Solution/Question Rationale These are basic Matrix Operations. Marking Scheme 3 marks for each X 5. b) Using Gaussian elimination, solve the system X+2Y-Z=6 2X+3Y+Z=11 X+3Y-2Z=2 (15 marks) Question Rationale This a major algorithmic technique for solving simultaneous equations. Marking Scheme 3 marks for augmented matrix 8 marks for correctly applying row operations 4 marks for correct solution. 4: a) Compute the greatest common divisor of 7,276,500, and 3,185,325 Gcd(7,276,500, 3,185,325) = gcd(3,185,325, 905,850) = gcd(905,850 , 467,775) = gcd(467,775, 438,075) = gcd (438,075, 29,700) = gcd(29,700, 22,275) = gcd(22,275,7,425) = gcd(7,425, 0) Ans = 7,425 (10 marks) Solution/Question Rationale Can they compute GCD using Euclid Marking Scheme 7 marks for Euclid 3 for correct answer b) Encrypt “We will not win the All Ireland” using a columnar transposition Cipher with key “Meath” (10 marks) Solution/Question Rationale Do the students know how apply the basic columnar transposition. Marking Scheme 6 marks for algorithm 2 for correct answer c) Encrypt Plaintext = 1107300 using the RSA algorithm with two primes; p= 563 and q = 2357. (10 marks) Solution/Question Rationale Can the students apply RSA algorithm. Will they compute n, z,, d and e Marking Scheme 7 marks for algorithm 3 for correct answer