In my case R=1

Question 1 (25 marks) (a) Let A and B be two events. Given the following probabilities for these events: P(A) =0.75, P(B|A) =0.6 and P(BA) =0.1. (i) What is the probability of P(BriA) ? (ii) What is the probability of P(B) ? (iii)What is the probability of P(A UB) ? (iv) What is the probability of P(A B)? (v) What is the probability of P( A B)? (10 marks) (b) Let x,,x, and x, be a positive integer. + + = 20, Consider a system of linear equations + 2x, + 5x, = 80. Using Gauss-Jordan elimination and performing elementary row operations to solve the above system of linear equations and determine ALL the possible combinations of X, X, and x; . (7 marks) + (c) Consider the system of linear equations Wi + where f and m are real + 4W, = m-R, constant. (i) Suppose ? : 1 ma] [8 45 2 2] Use elementary row operations to determine a and & in terms of t and m. (2 marks) (ii) Hence, find all the value(s) of / and m, if any, such that the given system of linear equations has 1) unique solution no solution 3) infinitely many solutions. (6 marks)Question 2 (25 marks) (a) Let f(x) = 2x -1 for x * -3. 3x +9 (i) Determine, with reason, whether the function /(x) is odd, even or neither. (3 marks) (ii) Determine the inverse function of /(x) and state the domain and range of /(x). (4 marks) (iii) Find the function g such that (fog)(x) =e" . (4 marks) (iv) Find the function h such that (ho f) (x) = x. (4 marks) (b) Let a and b be real constants and f : IR -> R be a function defined by 2, if XS- 1, f (x) = ax' +bx+1, if -1