Linear Algebra - Midterm Exam Questions with Solutions | Math 6, Exams of Linear Algebra

Download Linear Algebra - Midterm Exam Questions with Solutions | Math 6 and more Exams Linear Algebra in PDF only on Docsity! MidTerm: Th. May. 13, 3.30pm - 4.50pm Discrete Mathematics ICS 6A Instructor: Max Welling • This exam is closed book • Spend your time wisely: get a shot at each question. • Try the bonus question last, if you get it right it will compensate one mis- take. If you get it wrong it will have no impact on your score. • If you didn’t bring your calculator, you may simplify your answer as much as possible and leave it at that. • Good Luck ! 1. (2 points) Truth Tables Let p and q be 2 propositions. a. Construct truth tables for the following compound propositions: (i) (p ∧ ¬q)→ q, (ii) (p ∨ q)↔ (p ∧ q). answer: If we use the following ordering in the truth table: p: T T F F, q: T F T F, then answers are (i): T F T T, (ii) T F F T. b. Assume we have n propositions p1, ..., pn , with n a positive integer and we construct a compound proposition which includes all n propositions above. How many entries would the truth table have for arbitrary n (i.e. how many different assignments can the string of propositions p1, ..., pn have) ? answer: 2n. 2. (4 points → 2 for each item) Sets and Logic Let U (the universal set) be equal to all integers Z. Define the sets A,B,C as follows: A = {x|(x ∈ Z) ∧ (0 ≤ x ≤ 1)} B = {x|(x ∈ Z) ∧ (x odd} C = {x|(x ∈ Z) ∧ (x even) ∧ (x > 0)} a. Determine the following sets (any set notation treated in class is ac- ceptable in answering your questions): (i) A ∪ C, (ii) (B ∪ C) ∩ C, (iii) B̄, (iv) A−B. answer: (i) {0, 1, 2, 4, 6, ...}, (ii) C, (iii) all even integers, (iv) {0}. b. Are the following propositions true? ∀x [((x ∈ A) ∧ (x ∈ B))→ (x > 0)] ∃x [(x ∈ (A−B))↔ (x = 2)] answer: (i) T, (ii) T (e.g. x = 1). 3. (2 points+bonus) Bottles of wine A store sells three kinds of wine: red, white and port. Of each kind the store has 10 indistinguishable bottles. a. 3 different customers arrive. Customer 1 buys 1 bottle, customer 2 buys 2 bottles, customers 3 buys 3 bottles. In how different many ways can the three customers buy their wine? answer: C(3, 1)× C(4, 2)× C(5, 3) = 3× 6× 10 = 180 b. Next, 4 identical (indistinguishable) brothers walk into the store. Each brother buys 1 bottle. In how many ways can they buy their bottles of wine? answer: C(6, 4) = 15 Bonus. Finally, 3 identical (indistinguishable) sisters walk into the store, each buying 2 bottles of wine. In how many ways can they buy their bottles (hint: imagine that pairs of bottles are sold in a bag. There are as many different bags as there are different pairs of bottles. Each sister buys 1 bag.) ? answer: C(8, 3) = 56. 4. (4 points) The Biased Die A gambler is confronted with a biased die. Let X be the random variable which value is equal to the number of “eyes” on the die (“eyes” are mark- ings/indentations on the die, the number of which represents the value of that side/face of the die). a. The probability of i eyes coming up is P (X = i) = i × p, with p a parameter. Compute the value of p? answer: p = 21. b. What is the expected value E(X) ? answer: E(X) = 4 13 . c. What is the standard deviation σ(X) ? answer: σ(X) = 1.4907. d. We throw the die 3 times. What is the probability of finding two 3’s and one 6, in any order? answer: 3× 32 × 6/213 = 0.0175. 5. (3 points) Checking in Hats At a party 10 people checking in their hats. As it turns out 5 of them are indistinguishable black hats, while the other 5 are distinguishable colored hats. The person at the counter gets utterly confused and forgets all about which hat belongs to who. In despair he decides to return them at random when people collect their hats. a. What is the probability that someone who checked in a colored hat recollect the correct hat again? answer: 110 . b. Assuming that a person will not notice when he/she receives an indis- tinguishable hat from his own, what is the probability that a person who checked a black hat will go home with the “correct” hat? answer: 12 . c. Let Xi be the random variable that takes value Xi = 1 if person i receives the correct hat. Compute the expected number of correctly returned hats E[X1 + ...+X10]. answer: E[X1 + ...+X10] = 3. 6. (4 points) Students and Grades Consider the set S of 30 students that currently take the ICS 6A class in discrete mathematics and the set G of 4 possible grades A,B,C,D.