Module Contact: Dr Steven Hayward, CMP Copyright of the University of East Anglia Version 1

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1 UNIVERSITY OF EAST ANGLIA School of Computing Sciences Main Series UG Examination MATHEMATICS FOR COMPUTING B CMP-4005Y Time allowed: 2 hours Answer ANY SIX questions out of SEVEN. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. CMP-4005Y Module Contact: Dr Steven Hayward, CMP Copyright of the University of East Anglia Version 1

2 Page 2 1. Consider the following systems of linear equations: x y + z = 0 2x + y z = 0. x + 5y 5z = 0 a) State the coefficient matrix. Call that matrix A. b) Does the product AA exist? Justify your answer! c) Compute AA t. d) State the augmented matrix for the above system. e) Compute the matrix minor for entry a 23 of A. f) Compute the cofactor for entry a 23 of A. g) Use Sarrus rule to compute the determinant of A. h) Is A invertible? Justify your answer! i) Is A a singular matrix? Justify your answer! j) Compute det(a t ). k) Do the 3 planes represented by the system of linear equations intersect in a plane? Justify your answer!

3 Page 3 2. Consider the points A = (2, 1, 0) and B = (3, 0, 4). a) Find the position vectors OA and OB. b) Compute the following quantities where a is the vector OA and b is the vector OB : i) 3a 2b. ii) 3a 2b. iii) 2b 3a. c) Are the vectors 3a and 3b orthogonal to each other? Justify your answer! d) Compute the magnitude of the vector 3a + 2b. e) Compute the cosine of the angle α between a and b. PLEASE TURN OVER

4 Page 4 3. a) Differentiate with respect to x: i) ii) y 2 x x e. x y. ln( x) 3 2 iii) y x cos( y). b) Perform the following integrals: i) x cos( x) dx. ii) 2 x x 1 dx iii) cos(2x )sin (2x) dx. c) Find the general solution of the following: dr r 2 ) d sin(. (5 marks)

5 Page 5 4. a) Simplify the following to the x + iy form: 3 2 i) 3i i 1. ii) 1. i 1 b) Find the complex solutions to the equation: 2 z 2z 2 0. c) For the complex numbers z1 1 i and z2 3 2i find: i) z1 z2. ii) z z 1 2. iii) z 1. z 2 d) Convert the following polar form to rectangular form (x + iy): 2 cos 3 i sin. 3 e) Convert the following rectangular form to polar form: 1 i. PLEASE TURN OVER

6 Page 6 5. a) The 21 st term in an arithmetic series is 20 and the 41 st term is 170. i) Determine the first term and the common difference. ii) Write the sum of the first 100 terms symbolically in sigma notation. iii) Evaluate the sum of the first 100 terms. b) Consider the following series: i) Give the 100 th term and calculate the sum of the first 100 terms. ii) What is the first term in the series that exceeds 1,000,000? (Show all working.) c) i) Find the limit of the sequence ii) State the preliminary test. 2n 3 n 2 3 5n 3 4 n 6 as n. iii) What can one say about the convergence of the infinite series 2 3 n 5n? 3 6 n 1 2n 3 4 n

7 Page 7 6. a) A bag contains 10 apples of which 3 are bad. If 3 apples are withdrawn at random and without replacement, find the probability that: i) all are good. ii) all are bad. iii) two are good and one is bad. b) A biased coin, which has a probability of ¾ of landing heads and a probability of ¼ of landing tails, is tossed 5 times. What is the probability of: i) exactly two heads? ii) at least three tails? c) If in part 6b) the biased coin is equally likely to have been swapped with a fair coin, what is the probability that the coin was fair if all heads were thrown? d) Describe under what conditions the Poisson distribution can be used and write down the probability, p(x), of x occurrences under the Poisson distribution. Define all terms. PLEASE TURN OVER

8 Page 8 7. a) Consider the word: MISSISSIPPI. i) How many distinct arrangements are there of the letters? ii) In how many of these arrangements are all the S s next to each other? b) A club consists of 50 members. i) If any four members are chosen to form the club committee, how many different selections are there? ii) If, of the 50 members, one is chosen for president, one for vicepresident, one for treasurer, and one for secretary, then how many different committee selections are there? iii) Explain the difference between the results in parts (i) and (ii) of this question. c) Find the coefficient of x 8 in the binomial expansion of (1 + x) d) Find the constant term in x. x END OF PAPER