Question 1 [6 marks] Consider the recurrence Jnctien 1*! mg = 321" (4 )+2"logn Give an expression for the runtime TYLif the recurrence can be solved with the Master Theorem,___Assume that Rn) = I for n 5 I . [6] Question 2 [5 marks] Let n), gin) and 110?) be functions dened j: g , "lg: R+ > 12* where 2:" is the set of Positive integers and R=is the set of Real numbers Determine the truth of the following statement If 01) 2 90% (71)) . .9101) 2 9(9'2 (71)): and I710\") : 90b ('11)) then f1(n)g1 (n)h1(n) + f1 (")S1(n) + fit\") 2 902(n)g2 (Take (71)) [5] Question 3 [5 marks] By applying the principles of Modular Arithmetic (a) Examine the Addition and Multiplication Tables of 23, 29 and Zn), and describe the patterns of the Addition Table and Multiplication Table of Zn. [2] (b) Use the pattern described in part (a) to determine the values of the Addition and Multiplication Tables of Zn. [1] (c) Construct the Addition and Multiplication Tables of Zn, and determine the approximate percentage of the Addition and Multiplication Tables that may be determined from pattern matching. [2] Question 4 [5 marks] 320 In a geometric series, the sum of the fourth term and the seventh term is 820 E---- 280 Three consecutive terms of the same series are 29 81 2,1: 69%;: and 161 Ex," If x is equal to the fth term in the series, and the sum of the terms in the series is 9 :2: nd the number of terms in the series. [5] x, Question 5 [6 marks] Find the generating function for the sequence {105, 690, 2205, 5100, 9825, ...} [6] Question 6 [7 marks] Find a closed form expression for the following recurrence relation 1 l 4sn_1 + 215mg fern: 2 [7] tit: t7? Question 7 [9 marks] A coding system encodes messages using strings of base 7 digits. A codeword is considered valid if and only if it contains an odd number of zeroes. i. Find a recurrence relation for the number of valid codewords of length 1]. State initial conditions. [4] ii. Solve this recurrence relation using generating functions. [5] Question 8 [4 marks] A set of employees for a nancial institution was surveyed and provided a response on the Likert scale concerning whether mandatory vaccination should be instituted for the company_.___If the majority response is Strongly Agree, the probability of mandatory vaccination being instituted by the next month is B3%_.___If the majority response is Agree, the probability of mandatory vaccination the following month is 60%_.___If the majority response is Neutral, the probability of mandatory vaccination is 44%; however, if the majority response is Disagree, the probability is 23%,___If the majority response is Strongly Disagree, the likelihood of mandatory vaccination being instituted by the next month is only 7%. Concerning whether mandatory vaccination should be instituted for the company, the survey revealed the results were 10% of respondents Strongly Agree, 17% Agree, 3% Neutral, 41% Disagree, and 29% Strongly Disagree. 1. Given that the nancial institution actuallyinstltutes mandatory vaccination within the company the following month, what is the probability that the majority response of the survey was Disagree or Strongly Disagree? [2] ii. Draw the Probability Tree that represents the given scenario. [2] Question 9 [6 marks] Each integer in Zn represents a residue class or congruence class. a Define the set of congruence classes of (mod 13). [1] (b) List the congruence classes (mod 13). [1] (c) Amidst a certain pandemic, a certain country M has a National Stadium with seating capacity of approximately 28,000. As part of the government's effort to stem the spread of the virus, it has been designated that a maximum of 5,000 citizens will be allowed to attend recreational events at the National Stadium. For ease of reference, the Recreation Seating Ids for the allowed seats at the National Stadium are uniquely assigned and referred as Rec Seat Ids 0, 1, 2, 3, ..., 4999 Citizens wishing to be inside the National Stadium to attend a certain recreational event are required to purchase tickets through a ticket online portal (TOP). .In the ticket acquisition process, the citizen is assigned a unique 13-digit Ticket number that begins with one of the digits 2 to 9. The citizen is assigned one of the Recreation Seating Ids based on the unique 13- digit number and the result of the hashing function h. The ticket number is defined as T and the hashing function f is defined as AT) = T mod 5000. If Zinedine Zidane is a citizen of country M who accessed TOP and assigned Ticket number 6832974960124, by simplifying the Ticket number to a set of addition and multiplication of positive integers, apply the principles of modular arithmetic and illustrate to which RecSeatId, Zinedine Zidane would be initially assigned before adjustment for clashing. [4] Question 10 [4 marks] Determine the order of growth of Zk=1 (k -1)(n + k)m if m is a positive integer? [4]Question 11 [6 marks] Find a formula for the series c defined by Cn = > b; Where b is the sequence { 1, 2, 3, 3, 4, 4, 7, 9, 7, 8, 11, 27, ...} [6] 1=1 Question 12 [8 marks] Find the generating function for the sequence (1. 221, 2131, 8701, 24341, 54901, ...} [8] Question 13 [7 marks] Let f(n) be defined by f (1) = 3 f (n) = 81f (3) + 9n* logn if n > 1 and n =3k, where k is a positive integer. By using the principles of Recurrence Relation, find a general formula for f(n) [4] ii. Hence show that f(n) = 0(n+login). [3]Question 14 [7 marks] A biased coin is tossed c times (each toss is independent) with probability t for tails. 1 . Give a formula to determine the smallest c that lets the probability of at least one head to be x. [3] ii. Using the formula from (i), determine the smallest c that lets the probability of at least one tail to be in excess of 90%. [3] iii. State any assumptions made for parts i. and ii. [1] Question 15 [10 marks] Given the following concerning an arithmetic series and a geometric series: The second term of the geometric series is the same as the seventh term of the arithmetic series.Additionally, the eighth term of the arithmetic series less the sixth term of the said series is two more than the negation of the first term of the geometric series. - The first term of the geometric series exceeds the first term of the arithmetic series by 12-. The sum of the first nine terms of the arithmetic series, SAP- and the sum of the first three terms of the geometric series, Sop 3 are related by the formula SAP-9 - 4SGP-3 + 13 = 0 What is the total of the sum of the first two terms of the geometric series and the twelfth term of the arithmetic series? [10]