# Preliminary Exam - Fall 1985

Problem 1 Evaluate the integral

$$
\int _ { 0 } ^ { \infty } { \frac { 1 - \cos a x } { x ^ { 2 } } } d x
$$

for $a \in \mathbb { R }$

Problem 2 Prove that for every $\lambda > 1$ , the equation $z e ^ { \lambda - z } = 1$ has exactly one root in the disc $| z | < 1$ and that this root is real.

Problem 3 1. How many different monic irreducible polynomials of degree 2 are there over the field $\mathbb { Z } _ { 5 } { \mathrm { : ~ } } ^ { \ell }$

2. How many different monic irreducible polynomials of degree 3 are there over the field $\mathbb { Z } _ { 5 } { \mathrm { : ~ } } ^ { 2 }$

Problem 4 Let G be a finite subgroup of the group $\mathbb { C } ^ { * }$ of nonzero complex numbers under multiplication. Prove that G is cyclic.

Problem 5 How many roots does the polynomial $z ^ { 4 } + 3 z ^ { 2 } + z + 1$ have in the right half z-plane?

Problem 6 Let k be real, n an integer $\geqslant 2$ , and let $A = \left( a _ { i j } \right)$ be the $n \times n$ matrix such that all diagonal entries $a _ { i i } = k$ , all entries $a _ { i i \pm 1 }$ immediately above or below the diagonal equal 1, and all other entries equal 0. For example, if $n = 5$

$$
A = \left( \begin{array} { l l l l l } { k } & { 1 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { k } & { 1 } & { 0 } & { 0 } \\ { 0 } & { 1 } & { k } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 1 } & { k } & { 1 } \\ { 0 } & { 0 } & { 0 } & { 1 } & { k } \end{array} \right) .
$$

Let $\lambda _ { m i n }$ and $\lambda _ { m a x }$ denote the smallest and largest eigenvalues of A, respectively. Show that $\lambda _ { m i n } \leqslant k - 1$ and $\lambda _ { m a x } \geqslant k + 1$

Problem 7 Let $y ( t )$ be a real valued solution, defined for $) < t < \infty$ , of the differential equation

$$
{ \frac { d y } { d t } } = e ^ { - y } - e ^ { - 3 y } + e ^ { - 5 y } .
$$

Show that $y ( t ) \to + \infty \ u s \ t \to + \infty$

Problem 8 Let $f ( x ) , 0 \leqslant x \leqslant 1$ , be a real valued continuous function. Show that

$$
\operatorname* { l i m } _ { n \to \infty } ( n + 1 ) \int _ { 0 } ^ { 1 } x ^ { n } f ( x ) d x = f ( 1 ) .
$$

Problem 9 Let A be the symmetric matrix

$$
{ \frac { 1 } { 6 } } \left( { \begin{array} { r r r } { 1 3 } & { - 5 } & { - 2 } \\ { - 5 } & { 1 3 } & { - 2 } \\ { - 2 } & { - 2 } & { 1 0 } \end{array} } \right) .
$$

Denote by v the column vector

$$
\binom { x } { y }
$$

in $\mathbb { R } ^ { 3 }$ , and by $x ^ { t }$ its transpose $( x , y , z )$ . Let $\lVert v \rVert$ denote the length of the vector v. As v ranges over the set of vectors for which $v ^ { t } A v = 1$ , show that $\| v \|$ is bounded, and determine its least upper bound.

Problem 10 Let f and $f _ { n } , n = 1 , 2 , . . . ,$ be functions from R to R. Assume that $f _ { n } ( x _ { n } ) \to f ( x )$ as $n \to \infty$ whenever $x _ { n } \to x$ . Show that f is continuous. Note: The functions $f _ { n }$ are not assumed to be continuous.

Problem 11 Let G be a subgroup of the symmetric group on six objects, $\mathrm { S } _ { 6 }$ . Assume that G has an element of order 6. Prove that G has a normal subgroup H of index 2.

Problem 12 Evaluate

$$
\int _ { 0 } ^ { 2 \pi } e ^ { e ^ { i \theta } } d \theta .
$$

Problem 13 Let $f ( z )$ be analytic on the right half-planeH $= \{ z \mid \Re z > 0 \}$ and suppose $| f ( z ) | \leqslant 1 \ f o r \ z \in H$ . Suppose also that $f ( 1 ) = 0$ . What is the largest possible value of $| f ^ { \prime } ( 1 ) | \ell$

Problem 14 Suppose that A and B are endomorphisms of a finite-dimensional vector space V over a field F. Prove or disprove the following statements:

1. Every eigenvector of AB is also an eigenvector of BA.

2. Every eigenvalue of AB is also an eigenvalue of $B A$

Problem 15 Let $0 \leqslant a \leqslant 1$ be given. Determine all nonnegative continuous functions f on $[ 0 , 1 ]$ which satisfy the following three conditions:

$$
\int _ { 0 } ^ { 1 } f ( x ) d x = 1 ,
$$

$$
\int _ { 0 } ^ { 1 } x f ( x ) d x = a ,
$$

$$
\int _ { 0 } ^ { 1 } x ^ { 2 } f ( x ) d x = a ^ { 2 } .
$$

Problem 16 Let $f ( x ) = x ^ { 5 } - 8 x ^ { 3 } + 9 x - 3$ and $g ( x ) = x ^ { 4 } - 5 x ^ { 2 } - 6 x + 3$ Prove that there is an integer d such that the polynomials √ $f ( x )$ and $g ( x )$ have a common root in the field $\mathbb { Q } \left( { \sqrt { d } } \right)$ . What is d?

Problem 17 Let $( M , d )$ be a nonempty complete metric space. Let S map M into M , and write $S ^ { 2 }$ for $S \circ S ;$ that is, $S ^ { 2 } ( x ) = S \left( S ( x ) \right)$ . Suppose that $S ^ { 2 }$ is a strict contraction; that is, there is a constant $\lambda < 1$ such that for all points $x , y \in M , d ( S ^ { 2 } ( x ) , S ^ { 2 } ( y ) ) \leqslant \lambda d ( x , y )$ . Show that S has a unique fixed point in M .

Problem 18 Let G be a group. For any subset X of G, define its centralizer $C ( X )$ to be $\{ y \in G \mid x y = y x$ , for all $x \in X \}$ . Prove the following:

1. $I f X \subset Y$ , then $C ( Y ) \subset C ( X )$

2. $X \subset C \left( C ( X ) \right)$

3. $C ( X ) = C \left( C \left( C ( X ) \right) \right)$

Problem 19 An n×n real matrix T is positive definite if T is symmetric and $\langle T x , x \rangle > 0$ for all nonzero vectors $x \in \mathbb { R } ^ { n }$ , where $\langle u , v \rangle$ is the standard inner product. Suppose that A and B are two positive definite real matrices.

1. Show that there is a basis $\{ v _ { 1 } , v _ { 2 } , \ldots , v _ { n } \}$ of Rn and real numbersλ $\mathbf { \alpha } _ { 1 } , \lambda _ { 2 } , \ldots , \lambda _ { n }$ such that, for $1 \leqslant i , j \leqslant n .$

$$
\langle A v _ { i } , v _ { j } \rangle = { \left\{ \begin{array} { l l } { 1 } & { i = j } \\ { 0 } & { i \neq j } \end{array} \right. }
$$

and

$$
\langle B v _ { i } , v _ { j } \rangle = { \left\{ \begin{array} { l l } { \lambda _ { i } } & { i = j } \\ { 0 } & { i \neq j } \end{array} \right. }
$$

2. Deduce from Part 1 that there is an invertible real matrix U such that $U ^ { t } A U$ is the identity matrix and $U ^ { t } B U$ is diagonal.

Problem 20 Let f be a differentiable function on [0, 1] and let

$$
\operatorname* { s u p } _ { 0 < x < 1 } | f ^ { \prime } ( x ) | = M < \infty .
$$

Let n be a positive integer. Prove that

$$
\left| \sum _ { j = 0 } ^ { n - 1 } { \frac { f ( j / n ) } { n } } - \int _ { 0 } ^ { 1 } f ( x ) d x \right| \leqslant { \frac { M } { 2 n } } \cdot
$$