# Department of Mathematics PRELIMINARY EXAMINATION

Work all of the following problems, justifying your answers.
The problems are weighted evenly.\
N denotes the set of positive integers.

1Negate each of the following statements in the most informative way possible (i.e., without using the words "no" or "not").

a. There is an integer x so that $x + y$ is odd for each integer y.

I  is an od integer, then xy is odd for every inteer y.

c. Given $\varepsilon > 0 ,$ , there is $N \in \mathbb { N }$ so that whenever $n > N$ , we have $\left| { \frac { 2 + n } { 1 + n } } - 1 \right| < \varepsilon .$

2. a. State the binomial theorem.

b. Prove that $( x + y ) ^ { 5 } \equiv x ^ { 5 } + y ^ { 5 }$ (mod 5).

c. Prove by mathematical induction that $( x + y ) ^ { 5 ^ { n } } \equiv x ^ { 5 ^ { n } } + y ^ { 5 ^ { n } }$ (mod 5) for every $n \in \mathbb { N }$

3. Suppose $f \colon A \to B$ and $g \colon B \to C$ are functions.
   Prove or give a counterexample:

a. If f and g are injective (one-to-one), then go f is injective.

b. If go f is injective, then f and g are injective.

4. Suppose $f \colon  { \mathbb { R } } \to  { \mathbb { R } }$

a. Give the 8-ε definition of $\operatorname* { l i m } _ { x \to a } f ( x ) = \ell$

b. Determine the limit and use the definition to prove your answer:

$$
\operatorname* { l i m } _ { x  2 } { \frac { 2 x + 1 } { x ^ { 2 } + 1 } } = ?
$$

5.Prove that $\int _ { C } ( x ^ { 3 } - y ^ { 3 } ) d x + ( x ^ { 3 } + y ^ { 3 } ) d y \geq 0$ for every smooth, simple closed curve $C \subset \mathbb { R } ^ { 2 }$ oriented counterclockwise.

6. Find all the cube roots of —2 - 2i.

7. Let $\mathbf { A } { = } \left[ \begin{array} { l l } { 0 } & { 1 } \\ { 6 } & { - 1 } \end{array} \right]$ Prove that A is diagonalizable and find a closed-form expression for $A ^ { k } { \left[ \begin{array} { l } { 1 } \\ { 7 } \end{array} \right] } , k \in \mathbb { N } .$

8. Suppose V and W are vector spaces and $T \colon V \ \to \ W$ is a linear transformation.
   Suppose $v _ { 1 } , \dotsc , v _ { k } \in V$ Prove that if $T ( v _ { 1 } ) , \ldots , T ( v _ { k } )$ form a linearly independent set in W, then $v _ { 1 } , \ldots , v _ { k }$ form a linearly independent set in V.

Prove that $x - { \frac { x ^ { 3 } } { 3 } } \leq \arctan x \leq x$ for all $x \ge 0$