## Math 3118 Practice Exam 4

1. True or False? Give your reason. There is a largest rational number less than 1.

2. True or False? Give your reason. (-\sqrt(3)+1)/(\sqrt(3)-1) is a rational number.

3. Verify that x=1+i\sqrt(3) satisifes the equation x^2-2x+4=0.

4. Prove that the cube root of 5 is irrational.

5. Suppose that w is complex number such that w^6=-1. Can you find w^15?

6. Let p(x)= x^3+10*x^2+x. Show that there are three distinct real values values of x such that p(x)=0.

## Math 3118 Practice Exam 4-2

1. Let x=(1/2+i)/(2/3-i). Find a rational number r such that x-ir is a real number.

2. True or False? Give your reason. The cube root of a positive integer m is irrational unless m is perfect cube.

3. Prove that the number x=(\sqrt(3)-\sqrt(5))/4 is irrational.

4. Suppose that x^6=1. Find rational numbers a,b,c,d,e,f such that x^1000=a+b*x+c*x^2+d*x^3+e*x^4+f*x^5.

5. Show that there is some number x between 1 and 3 such that x^7+x-3=0.

## Math 3118 Practice Exam 4-3

1. True or False? Give your reason. There is positive real number x such that x^3+3x^2-100*x-4=0

2. Give an example of two irrational numbers x,y such that x/y is rational.

3. True or false: If x and y are any two real numbers, then there is some rational number r between x and y.

4. The infinite decimal which consists of the consecutive integers 0.1234567891011121314... is an irrational number.

5. Let z=(2-i)/i. Find the a+bi form, a and b rational, for z^2.

6. Let z=1-2*\sqrt(5). Find the a+b\sqrt(5) form, a and b rational, for z^2.

## Math 3118 Practice Exam 4-4

1. Prove that the real number (2-\sqrt(2))/\sqrt(2) is irrational.

2. Let z be the complex number z=(1+i)/\sqrt(2). Prove that z^2=i.

3.Suppose that x is a complex number such that x^2+x+1=0. Prove that x^3=1.

4. True or False: The average of two irrational numbers is always irrational.

5. True or False: The square of an irrational number is rational.

6. Prove that the set of algebraic numbers is countable.

## Actual exam spring 1999

1. Find rational numbers a and b such that z=a+bi,

z = {2i-4}/{5+2i}.

2. Find rational numbers a and b such that the number below has the form a+b\sqrt{7},

{\sqrt{7}-5}/{2}+{3}/{\sqrt{7}}.

3. True or False? You must give a reason for your answer in complete sentences.

The product of an irrational number times a non-zero rational number is always an irrational number.

4. Let x=(3-\sqrt{5})/4.

(a) Is the decimal representation for x repeating?

(b) Find a number y such that x+y is rational.

5. (a) Show there is some real number x such that x^4-3x^3+10x^2-11x-4=0.

(b) Suppose that x is the real number you found in part (a). Explain why there exist rational numbers a, b, c, and d such that

x^{12}=a+bx+cx^2+dx^3.

You do NOT need to find explicit formulas for a, b, c, d. Be sure to explain your answer in complete sentences.