# Department of Mathematics, University of California, Berkeley

## GRADUATE PRELIMINARY EXAMINATION, Part A

Spring Semester 2019

1. Please write your 1- or 2-digit exam number on this cover sheet and on all problem sheets (even problems that you do not wish to be graded).

2. Indicate below which six problems you wish to have graded. Cross out solutions you may have begun for the problems that you have not selected.

3. Extra sheets should be stapled to the appropriate problem at the upper right corner. Do not put work for problem p on either side of the page for problem q if p 6= q.

4. No notes, books, calculators or electronic devices may be used during the exam.

## PROBLEM SELECTION

Part A: List the six problems you have chosen:

## GRADE COMPUTATION (for use by grader—do not write below)

1A.

2A.

1B.

Calculus

2B.

Real analysis

3A.

3B.

Real analysis

4A.

4B.

Complex analysis

5A.

5B.

Complex analysis

6A.

6B.

Linear algebra

7A.

7B.

Linear algebra

8A.

8B.

Abstract algebra

9A.

9B.

Abstract algebra

Part A Subtotal:

Part B Subtotal:

Grand Total:

Please cross out this problem if you do not wish it graded

Score:

Let y be a solution of $y ^ { \prime \prime \prime } { - } y = 0$ such that y(t) → 0 as t → ∞. Show that $y ( 0 ) + y ^ { \prime } ( 0 ) + y ^ { \prime \prime } ( 0 ) =$ 0.

Solution:

Please cross out this problem if you do not wish it graded

Problem 2A.

Score:

Let f : R → R be bounded and continuously differentiable. Show that every solution y of $y ^ { \prime } = f ( y )$ is monotone.

Solution:

## Problem 3A.

Score:

Let f be a twice continuously differentiable function on [0, 1] such that $f ( 0 ) = f ( 1 ) = 0$ Prove that

$$
\operatorname* { m a x } _ { x \in [ 0 , 1 ] } | f ( x ) | \leq { \frac { 1 } { 8 } } \operatorname* { m a x } _ { x \in [ 0 , 1 ] } | f ^ { \prime \prime } ( x ) | ,
$$

and find an example where equality holds.

Solution:

YOUR EXAM NUMBER

Please cross out this problem if you do not wish it graded

Problem 4A.

Score:

Evaluate

I = Z ∞−∞ x sin x(1 + x 2 ) 2 d x sin x x.

Solution:

YOUR EXAM NUMBER

Please cross out this problem if you do not wish it graded

## Problem 5A.

Score:

Find the number of complex roots of $e ^ { z } = 3 z ^ { 6 }$ with $| z | < 1$ that have positive imaginary part.

Solution:

Please cross out this problem if you do not wish it graded

## Problem 6A.

Score:

Let n be a positive integer and let a be a complex number. Prove that $a ^ { n } = 1$ if and only if there are invertible n by n complex matrices X, Y such that $Y X = a X Y$

Solution:

YOUR EXAM NUMBER

Please cross out this problem if you do not wish it graded

Problem 7A.

Score:

Let A be a complex $n \times n$ matrix satisfying $A ^ { 3 7 } = I$ . Show that A is diagonalizable.

Solution:

Score:

Show that $F : \mathbb { C } ^ { 3 } \to \mathbb { C } ^ { 3 }$ defined by

$$
F ( u , v , w ) = ( - u - v - w , u v + u w + v w , - u v w )
$$

is surjective but not injective.

Solution:

Please cross out this problem if you do not wish it graded

<table><tr><td>Problem 9A. Score:</td></tr></table>

Let S be a countable set of real numbers. Show that there are function $g _ { n } : \mathbb { N } \to \mathbb { N }$ such that if $f : \mathbb { N } \to \mathbb { N }$ is a function from N to N with $f ( n + 1 ) > g _ { n } ( f ( n ) )$ for all n, then $\textstyle A = \sum _ { n = 1 } ^ { \infty } { \frac { 1 } { f ( n ) } }$ converges to a real that is not in the set S.

## Solution:

Spring Semester 2019

1. Please write your 1- or 2-digit exam number on this cover sheet and on all problem sheets (even problems that you do not wish to be graded).

2. Indicate below which six problems you wish to have graded. Cross out solutions you may have begun for the problems that you have not selected.

3. Extra sheets should be stapled to the appropriate problem at the upper right corner. Do not put work for problem p on either side of the page for problem q if $p \neq q$

4. No notes, books, calculators or electronic devices may be used during the exam.

## PROBLEM SELECTION

Part B: List the six problems you have chosen:

Please cross out this problem if you do not wish it graded

Problem 1B.

Score:

Evaluate

$$
I = \int _ { 0 } ^ { \pi / 2 } { \frac { \sin { x } \cos { x } } { \sin ^ { 4 } { x } + \cos ^ { 4 } { x } } } d x
$$

Solution:

Problem 2B.

Score:

For $t \geq 0$ let

$$
F ( t ) = \int _ { 0 } ^ { t } \exp ( - x ^ { 2 } ) d x
$$

and

$$
G ( t ) = \int _ { 0 } ^ { 1 } \frac { \exp ( - t ^ { 2 } ( 1 + x ^ { 2 } ) ) } { 1 + x ^ { 2 } } d x .
$$

Show that $F ( t ) ^ { 2 } + G ( t )$ is constant and deduce the value of $F ( \infty )$

Solution:

Please cross out this problem if you do not wish it graded

Problem 3B.

Score:

Show that $x _ { n + 1 } = ( 1 + x _ { n } ) ^ { - 1 }$ converges and find its limit for any $x _ { 0 } > 0$

Solution:

Score:

Let $c _ { 0 } , c _ { 1 } , . . . , c _ { n - 1 }$ be complex numbers. Prove that all the zeroes of the polynomial

$$
z ^ { n } + c _ { n - 1 } z ^ { n - 1 } + \cdot \cdot \cdot + c _ { 1 } z + c _ { 0 }
$$

lie in the open disc with center 0 and radius

$$
1 + \left| c _ { n - 1 } \right| + \cdot \cdot \cdot + \left| c _ { 1 } \right| + \left| c _ { 0 } \right| .
$$

Solution:

## Problem 5B.

Score:

If $f ( z )$ is analytic in the open disc $\mathbb { D } = \{ z : | z | < 1 \}$ , and $\mathrm { i f } \ | f ( z ) | < 1 / ( 1 - | z | )$ for $\mathrm { a l l } ~ z \in \mathbb { D }$ show that

$$
\left| \frac { f ^ { ( n ) } ( 0 ) } { n ! } \right| \le ( n + 1 ) \left( 1 + \frac { 1 } { n } \right) ^ { n } < e ( n + 1 ) .
$$

Solution:

Problem 6B.

Score:

Let $\mathbb { Z } _ { 2 }$ be the ring of integers mod 2. Prove the following identity in $\mathbb { Z } _ { 2 } [ x _ { 1 } , \ldots , x _ { n } ]$

$$
\operatorname* { d e t } { \left[ \begin{array} { l l l } { x _ { 1 } } & { \ldots } & { x _ { n } } \\ { x _ { 1 } ^ { 2 } } & { \ldots } & { x _ { n } ^ { 2 } } \\ & { \ldots } & \\ { x _ { 1 } ^ { 2 n - 1 } } & { \ldots } & { x _ { n } ^ { 2 n - 1 } } \end{array} \right] } = \prod _ { ( a _ { 1 } , \ldots , a _ { n } ) \neq ( 0 , \ldots , 0 ) } ( a _ { 1 } x _ { 1 } + \cdot \cdot \cdot + a _ { n } x _ { n } ) ,
$$

where $\left( a _ { 1 } \ldots a _ { n } \right)$ run all non-zero values in $\mathbb { Z } _ { 2 } ^ { n }$

Solution:

YOUR EXAM NUMBER

Please cross out this problem if you do not wish it graded

## Problem 7B.

Score:

Is it true that elements of the group $G L _ { 2 } ^ { + } ( \mathbb { R } )$ of real 2 × 2-matrices with positive determinant are conjugate in $G L _ { 2 } ^ { + } ( \mathbb { R } )$ if and only if the matrices are similar (conjugate in $G L _ { 2 } ( \mathbb { R } ) ) \colon$ Either prove this or give a counterexample.

Solution:

Please cross out this problem if you do not wish it graded

Problem 8B.

Score:

Let R be a ring (possibly non-commutative, possibly without an identity 1) in which every element is idempotent (this means that for all $a \in R , a ^ { 2 } = a )$ . Show that R has characteristic 2 (2a = 0 for all a) and is commutative.

Solution:

Please cross out this problem if you do not wish it graded

## Problem 9B.

Score:

Recall that $S _ { 6 }$ and $A _ { 6 }$ are the symmetric group and alternating group on 6 letters, respectively.

Prove or give a counterexample (with explanation): For every $\sigma \in A _ { 6 }$ there is ${ \mathrm { ~ a ~ } } \tau \in S _ { 6 }$ such that $\tau ^ { 2 } = \sigma$

## Solution: