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MAT9004 Practice Exam

tag 1 Download 4 Pages / 867 Words tag 13-11-2020
  • Course Code: MAT9004
  • University: Monash University
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  • Country: Australia

Question:

where x ∈ [−2, 3]. (a) What is f 0 (x)? [1] (b) What is f 00(x)? [1] (c) Find all the stationary points of f. Find the value of f(x) at each stationary point. [2] (d) For each stationary point of f, find whether it is a local minimum, a local maximum or neither. [2] (e) Find all the global minima and maxima of f in the interval [−2, 3]. [2] 2. Let M =  1 6 1 0  (a) Find the eigenvalues of M. [3] (b) Find one eigenvector of M with respect to each eigenvalue. [3]
 
(c) Diagonalize M, i.e., write M in the form M = NDN −1 , where D is a diagonal matrix. [2] 3. X is a continuous random variable with density f where f(x) = x + x 2 + x 5 for x ∈ [0, 1] and f(x) = 0 elsewhere. What is the probability of the event {X > 1/2}? [6] 4. Suppose you want to launch your startup in some innovative area. Currently, there exist two enterprises A and B competing for this market. You want to find out their investment strategies (which are given by real numbers a, b) prior to commiting into the competition.
 
A friend who works in A told you that their profit is given by p(a, b) = ab2 − a 2 + b 2 + 6b − 9a + 10, but unfortunately he does not know the values of a, b. He also mentioned that both enterprises are apparently not very good in keeping secrets, so the function f(x) = p(x, b) achieves the maximum value at point x = a, while x = b minimises g(x) = p(a, x). Use this information to find a and b [8] 5. A positive integer is called prime-like if it is not divisible by 2 and not divisible by 5.
 
(a) How many prime-like numbers with 4 digits (from 1000 to 9999)? [2] (b) How many prime-like numbers with 4 digits have at least one even digit? [3]
 
(c) How many prime-like numbers with 4 digits are not divisible by 3? [4] 6. Let Y be a random variable uniformly distributed on the set {−1, 0, 1, 2}. Let U1, U2 be the random variables defined by U1 = Y 2 − Y and U2 = Y 2−1 Y 3−2 . (a) Find the expected values of U1 and U2. [2] (b) Find variances of U1 and U2. [2] (c) Is U1 independent of U2? Justify your answer. [2] 7. A fair 6-sided die is rolled three times. (a) Find the probability that the sum of all three outcomes equals 8.
 
[4] (b) Find the probability that the first roll was 1 given that the sum of outcomes equals 8. [3] 8. Eight students Alice, Bob, Casey, Drew, Eva, Francis, Glen, Hunter entered Monash University this year. Alice, Bob and Casey are from Canberra and Drew, Eva, Francis are from Newcastle.
 
Glen and Hunter are international students from different countries. Alice, Casey and Hunter practice tennis, while Drew, Francis, Glen and Hunter practice soccer. Any two students from the same city or practicing the same sport know each other. (a) Draw a graph corresponding to the student acquaintances. Find its number of edges.
 
[2] (b) Write down the adjacency matrix for this graph. Does it contain a spanning tree? [3] (c) Is it possible to seat all eight students at a round table in such a way that any of them knows both neighbours. Justify your answer. [3]
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