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1. (a) Given A = {1, 4, a, H}, B = {4, u, 2, -2}, and C = {u}, determine each of the following sets: (i) (AUB)nc (ii) BUC (iii) A \\ B (b) Let f(x) = 0, x1 Evaluate the following: (i) f(g(0)) (ii) g(g(1/2)) (c) Consider the function h(x) = x2 for x E [-1, co). Explain why this func- tion does not have an inverse, and give the largest subset of the domain on which the inverse may be defined. (d) Show that n E(1-2)2 n 2 i=22. (a) A binary sequence is an ordered sequence of ones and zeros. (i) How many binary sequences are there of length 8? (ii) How many binary sequences of length 8 contain exactly 5 ones? (iii) What is the probability that a binary sequence of length 8 contains exactly 5 ones given that the rst digit is a one. (b) Consider the following discrete probability distribution a: '1 2 3 P(X=:c)l0.1 0.6 a for some unknown 0.. (i) What value of (1 makes X a valid discrete distribution? (ii) What is the expected value of X? (iii) What is the variance of X? (c) Central Adelaide comprises two electoral divisions, A and B. The p0pula- tion of division A is 107,000, and the p0pulation of division B is 105,000. At the 2019 Federal Election, the probability that a randomly selected voter voted for The Greens party was 0.16 in division A and 0.07 in division B. (i) What is the probability that a randomly chosen voter in Central Ade laide is from division A? (ii) What is the probability that a randomly chosen voter in Central Ade laide voted for The Greens party? (Use the law of total probability to nd the probability of selecting a Greens voter at random from central Adelaide.) (iii) What is the probability that a randomly chosen voter in Central Ade laide is from division A given that they voted for The Greens party? (a) Consider the following matrices 11 X=[_46 '32] Y: 3 2 5 3 If possible, calculate each of the following. If not possible, explain why. (i) X + Y (ii) X Y (iii) YX YT le (iv (V (b) Suppose that Alice buys packets of chips, nuts, and dried fruit for a party. Each packet of chips costs $3, each packet of nuts costs $5, and each packet of dried fruit costs $6. ) ) In total, Alice spends $74. She buys twice as many packets of chips as packets of nuts, and she buys one more packet of dried fruit than she buys packets of nuts. Formulate a system of linear equations which could be used to determine the number of each type of snack that Alice buys. You do NOT need to put the equations in matrix form or solve them, just give the equations with clear denitions for your variables. (c) Consider the matrix 0 0 2 A: l 05 oz 1 5 0 Find the determinant of A, showing all working, and thus determine the value(s) of oz for which A is invertible