# Graduate Preliminary Exam, Spring 2006 (9 problems counted equally, 3 hours)

# 1. i) State what it means for a sequence $\left\{ a _ { n } \right\}$ of real numbers to be Cauchy.

ii) Prove that every convergent sequence is Cauchy.

# 2. Prove that for every positive integer $n , n ^ { 3 } +$ 2n is divisible by 3.

# 3. Find all complex numbers $z = x + i y , ( x , y \in \mathbb { R } )$ , such that $e ^ { z } = 2 i$

# 4. Let $L \subset \mathbb { R } ^ { 2 }$ be the line spanned by the vector $\binom { 1 } { 2 }$ . Let $P \subset \mathbb { R } ^ { 2 }$ be the line defined by the equation $x + 2 y = 0$ Find the standard matrix for the linear transformation $T : \mathbb { R } ^ { 2 } \to \mathbb { R } ^ { 2 }$ such that $T ( v ) = 0$ for $v \in L$ and $T ( v ) = v$ for $v \in P$

# 5. Find an invertible matrix Q so that $Q ^ { - 1 } \left( \begin{array} { c c } { { - 1 } } & { { 1 } } \\ { { 0 } } & { { 2 } } \end{array} \right)$ Q is diagonal.

# 6. Let f be a Riemann integrable function on a closed interval $[ a , b ]$ For $x \in [ a , b ]$ , let $\textstyle g ( x ) = \int _ { a } ^ { x } f ( t ) d t$ Prove that g is continuous.

# 7. Let 7 be the arc of the unit circle in the first quadrant, from (1,0) to $( \mathbf { 3 } / 5 , \mathbf { 4 } / 5 )$ . Compute

$$
\int _ { \tau } x d y + y d x \ .
$$

# 8. Let $\textstyle f ( x , y ) = { \frac { x ^ { 2 } y } { x ^ { 2 } + y ^ { 2 } } }$ for $( x , y ) \neq ( 0 , 0 )$ and let $f ( 0 , 0 ) = 0$

i) Prove that f is continuous at (0, 0).

ii) Prove that f is not differentiable at (0, 0).

# 9. For each positive integer n, let $\textstyle f _ { n } ( x ) = { \frac { x } { x + n } }$ for $x \in [ 0 , \infty )$ . Show that the sequence of functions $\{ f _ { n } \}$ converges pointwise on $\lbrack 0 , \infty )$ to the O-function but does not converge uniformly.