Midterm Preparation Problems The following are practice problems for the Math 1200 Midterm Exam. Some of these may appear on the exam version for your section. To use them well, solve the problems, then discuss and compare your methods and answers. See if you can identify what particular knowledge and skills were required. 1. Consider the following proof fragment. There exists an integer k such that n = 3k+1. Then n 2 = (3k+1) 2 = 9k 2 + 6k + 1 = 3(3k 2 + 2k) + 1. For each of the statements, (a), (b), (c), below, answer the following. Does the fragment provide a proof of the statement? If yes, explain why. If no, explain why not. The letter n denotes an integer. (a) If n is odd, then n 2 is odd. (b) If n 2 is divisible by 3, then n is divisible by 3. (c) If n leaves remainder 1 on division by 3, then so does n 2. 2. Compare the two numbers: 0.7621 and 141/185. 3. Decide if the following statements are true or false. For false statements explain your thinking. (a) {1} {1, 2, 3} (b) {1, 2, 3} (c) {1, 2, 3} {1, 2, 3} 4. How many positive integers less than 1000 are (a) are divisible by 7? (b) are divisible by 11? (c) are divisible by both 7 and 11? (d) are divisible by either 7 or 11? (e) are divisible by neither 7 nor 11?
5. Prove that if xy is irrational then at least one of x and y is irrational. 6. This exercise provides a proof that 3 + 2 is an irrational number. You may take as given (no proof required) that 3 and 2 are irrational numbers. (a) Define what it means for a real number to be rational, and for a real number to be irrational. (b) Prove that the sum and that the product of two rational numbers is rational. (c) Verify that if 3 + 2 is rational so is 3 2. Hint: What is their product? (d) Verify that if 3 + 2 is rational, so is 3. (e) Given that 3 is not rational, what can you conclude about 3 + 2? Explain your argument. (f) Generalize. If x and y are irrational, what condition on x 2 y 2 ensures that x + y be irrational? 7. Consider the rational function where a represents some real number. R(x) = N(x) D(x) = (a + 4)x2 + 6x 1 x + 3 (a) Find the domain of the function R(x). (b) Find all values of a such that the quadratic equation N(x) = 0 has a unique solution. (c) Find all values of a such that the quadratic equation N(x) = 0 has solution x = 3. (d) Find all values of a such that the equation (a + 4)x 2 + 6x 1 x + 3 has a unique solution. Find this solution. = 0 8. Prove that 2 n > n for all non-negative integers n. 9. Prove by induction that n 3 + 2n is divisible by 3 for every non-negative integer n.
10. Prove that 1 + 2 + 3 + + n = n(n + 1)/2 for every positive integer n. 11. Prove that for all positive integers n, 7 n 2 n is divisible by 5. 12. (a) Prove by induction that if b is an odd number and n a positive integer, then b n is also odd. (b) Using the conclusion of the first part of this question, show that the equation x 19 + x + 1 = 0 has no integral solutions. (c) Using the conclusion of the first part of this question, show that the equation x 19 + x + 1 = 0 has no rational solutions. 13. A totem pole has representations of n animals arranged in a vertical column. If there are 10 animals that can be used on a totem pole, show that there are 10 n possible totem poles. 14. Consider the sequence of numbers a 0, a 1, a 2, a 3,... given by a 0 = 2, a 1 = 3, and for any positive integer k 2, a k+1 = 3a k 2a k 1. (a) Evaluate a 2, a 3, a 4, a 5. Show your work. (b) Prove by mathematical induction that for all positive integers n, a n = 2 n + 1. 15. Let N represent the set of positive integers. Find all functions f : N N such that f(1) = 2, and f(xy) = f(x)f(y) f(x + y) + 1 for all x, y N. Hint: Start by determining f(2), f(3), f(4),.... Make a conjecture and then use Mathematical Induction to prove your conjecture is correct. 16. (a) Carefully present a proof by mathematical induction that, for all positive integers n, (2n 1) 2 1 is divisible by 8. (b) Consider the statement, for all odd positive integers n, n 2 1 is divisible by 8,
and the following suggested proof. Carefully read the proof and answer the questions which follow. Suggested Proof: The smallest positive odd number is 1, and 1 2 1 = 0 is divisible by 8. Assume that for some positive odd number k, k 2 1 is divisible by 8. Then (k + 2) 2 1 = (k 2 1) + 4k(k + 1). Since k is odd, k + 1 is even and 4k(k + 1) is divisible by 8. As both k 2 1 and 4k(k + 1) are each divisible by 8, so is (k + 2) 2 1. Questions: i. The structure of the proof follows that of a proof by mathematical induction. In what way does it vary from a standard proof by mathematical induction? ii. The second paragraph of the proof (from Assume until the end) proves an implication. Clearly state that implication using a complete ordinary language sentence. iii. Is the sum of two numbers divisible by 8, also divisible by 8? Provide a simple direct proof or counterexample to justify your answer. iv. Is the suggested proof a valid proof of the statement. Explain. 17. Use these conversion formulas for the following. z = x + iy z = x iy x = 1 (z + z) 2 y = 1 (z z). 2i (a) (Revised Dec 7) Verify that z + z = 1 is the equation of a vertical line in the complex plane by converting to xy coordinates, i.e., setting z = x + iy, z = x iy and simplifying. (b) Verify that (1 2i)z + (1 + 2i) z = 6 is the equation of a line.
(c) Write the equation of the line 3x + 2y = 6 in z z coordinates. (d) Is 2z + i z = 1 + i the equation of a line? If not, what (geometric object) is its graph? 18. Determine the set of complex numbers z which satisfy z 3 z + 3 = 2 and represent the set graphically. 19. Find a complex number z such that z 3 = 2i 2. 20. (a) Describe the solutions of the equation ( ( z 4 + 1 )) k 2 i = 1 in terms of roots of unity. (b) Describe the solutions of the equation ( ( z 4 + 1 )) k 2 i = 2 21. Express in similar terms. 1 + i 3 + i in the form x + iy, where x, y R. By writing each of 1 + i and 3 + i in polar form, deduce that cos π 12 = 3 + 1 2 2 and sin π 12 = 3 1 2 2. 22. (a) Find a complex number in the form x + yi whose square is 5 + 12i. (b) Extend your method in (a) to prove that every complex number has a square root. 23. The vertices of ABC in the complex plane are given by the points 1 + 2i, 4 2i, and 1 6i. (a) Represent ABC graphically. (b) Prove that ABC is an isosceles triangle and determine the length of each of its sides.