(from previous offerings – both tests, and relevant questions on old exams)

Term Test #3 March 14, 1999

1. For each of the following numbers, state whether it is constructible and justify your answer:

a) 1/ Root of ( 3+ 4(Root 6))

b) 1/ p

c) 0.67893

d) Root of (p ² +4)

e) cos(15 degrees)

2. Is there a line in the plane such that every point on it is constructible? Justify your answer

3.a) Find a polynomial p with rational coefficients such that p( 3- root of(2 + 7(root of5)) =0.

3 b) Suppose that x satisfies x⁴ + 2x + root of(3) = 0. Must x be algebraic? Justify your answer.

4. Define "surd".

5. Does x³ - 3x² - 2x + 6 =0 have a constructible root? Explain your answer (Hint: Yes it does)

6. Does x³ + 4x + 1 = 0 have a constructible root? Justify your answer (Hint: No it doesn't)

7. State which of the following angles are constructible and briefly explain why:

a) 120 degrees

b) 75 degrees

c) 7.5 degrees

d) 35 degrees

3) 80 degrees

8. Let Q(root 2)) = {a + b (root 2): a,b Î Q}, and let F = {c + d(root 3): c,d Î Q(root 2). Prove that every element of F is the root of polynomial of degree 4 with rational coefficients.

9. Prove that the cube can't be "tripled", in the sense that, starting with an edge of a cube of volume 1, an edge of a cube of volume 3 can't be constructed with straightedge and compass.

Term Test #3 March 16, 1999 (Make Up Test)

1. Does the polynomial p(x) = 5x⁹ – 2x⁸ + x² – x + 1 have a real root? Justify your answer.

2. Find all the complex roots of the equation z⁷ – z =0.

3. State whether each of the following angles can be trisected with straightedge and compass and justify your answers:

a) an angle of 15 degrees

b) an angle of 30 degrees.

4. State whether the following angles can be constructed and justify your answers.

a) an angle of 7.5 degrees

b) an angle of 160 degrees.

5. For each of the following number, state whether or not it is constructible and justify your answer

a) Eighth root of 2

b) Sixth root of 2

6. Find ( 1/(root 2) + i/(root 2) )¹⁰⁶

7. Prove ( you can quote without proof theorems we've proven in class) that the cube root of a natural number is not constructible unless it is an integer.

8. Say that the complex number at a +bi is constructible if the point (a,b) is constructible (equivalently, if a and b are both constructible real numbers). Are any of the cube roots of ½ + i (root3/2) constructible? Justify your answer. (Hint: It is not)

9. Say that the complex number is algebraic if it is the root of a polynomial with integer coefficients. Is the set of all algebraic complex numbers countable? Prove your answer. (Hint: Yes they are)

10. Does the equation x³ – 2x –1 =0 have a constructible root? Prove that your answer is correct.

Test 3 Questions from 1998 Exam

5. State which of the following angles are constructible, and justify your answer.

a) 10 degrees

b) 30 degrees

c) 15 degrees

d) 75 degrees

e) 5 degrees

6. Determine whether the equation

z³ – z² + z + 1 = 0 has a constructible root, and justify your answer.

7. Suppose that S and T each have cardinality c (the cardinality of R). Show that

S U T has cardinality c.

10. a) State whether each of the following numbers is constructible and justify.

i) 1/ root of (3 + root(2))

ii) 7 to the power of 2/3

iii) p ¹⁰

iv) sin20 degrees

v) 0.37219

b) Prove that the acute angle whose cosine is ¼ can't be trisected with straightedge and compass.

Test 3 Questions from 1999 Exam

4. Does the equation x⁴ + x – 1 = 0 have a rational root? Justify your answer.

5. State which of the following numbers are constructible and justify your answers:

a) root (3792/1419)

b) 7 to the power of 2/3

c) cos20 degrees

d) cos15 degrees

e) 3.146891

10. Prove that a continuous function mapping R into R must be a constant function if its range is countable.

Test 3 Questions from 2000 Exam

3a) What is the cardinality of the set of all sets of constructible numbers? Justify your answer.

3b) Let S be the set of all functions mapping R into R. Show that the cardinality of S is greater than c.

4 b) Find a polynomial p with integer coefficients such that p(3 + i (root 7)) = 0.

8 a) Can a polyhedron have an odd number of edges, an odd number of vertices, and an odd number of faces? Justify your answer.

b) Can a regular polygon with 20 sides be constructed with straightedge and compass? Justify your answer.

Test 3 Questions from 2001 Exam

3 a) Let N be the set of natural numbers and a and b be distinct numbers. What is the cardinality of the set of all functions with domain {a,b} and range a subset of N? Justify your answer.

5. Prove the following: If S is uncountable and T is a countable subset of S, then the cardinality of S\T is the same as the cardinality of S.

7. Let S denote the collection of all sequences of real numbers. Show that the cardinality of S is c.

10. a) State whether each of the following numbers is constructible and justify.

i) root of (p ²+ 4)

ii) cos10 degrees

iii) root of ( 3 + 4(root 2) + (root5))

iv)sin75 degrees

v) root of (7cos15 degrees)

10b) Prove that the acute angle whose cosine is 2/5 can't be trisected with straightedge and compass.