Graduate Preliminary Exam, August 12, 2005

Three hours.
problems counted equally.

1. Let Z be a set and let $X _ { 1 } , X _ { 2 } , . . .$ be subsets of Z. Prove the formula

$\left( \bigcap _ { i } X _ { i } \right) ^ { c } = \bigcup _ { i } X _ { i } ^ { c }$ , where $( \cdot ) ^ { c }$ denotes complement.

2. i) Let f be a function from R to R and let $a \in \mathbb { R }$ From the definition of the derivativ, provethat i irentiableatathen ontinuous

ii)Prove the product rule, $( f g ) ^ { \prime } = f ^ { \prime } g + g ^ { \prime } f$

3. For $n = 5 , 6 ,$ either give an example of a nonabelian group of order n, or prove that none exists.

4Find al cube roots of 8 in the complex plane.
Write your answers in the form $a + b i$ , where a and b are real numbers, and justify your answer.

5. Let C be a circle in the xy-plane, oriented counterclockwise, and not passing through the origin.
   Prove that $\oint _ { C } { \frac { - y } { x ^ { 2 } + y ^ { 2 } } } d x + { \frac { x } { x ^ { 2 } + y ^ { 2 } } } d y$ equals 0 if the origin is outside the circle, and 2π if the origin is inside the circle.

Let $f ( x , y )$ be a differentiable function from $\mathbb { R } ^ { 2 }$ to R and let $g ( t ) = ( x ( t ) , y ( t ) )$ be a differentiable function from R to $\mathbb { R } ^ { 2 }$ . State the chain rule for the derivative of the function $f ( g ( t ) )$ .

i Let $f ( x , y )$ be a continuously differentiable function on $\mathbb { R } ^ { 2 }$ ,and let $\boldsymbol { P } \in \mathbb { R } ^ { 2 }$ be a point.
Assume that in a neighborhood of P, the equation $f ( x , y ) = f ( P )$ implicitly defines y as a differentiable function of $x ,$ say $y = h ( x )$ . Show that the tangent line to the graph of $y = h ( x )$ at P is perpendicular to the gradient of f at P.

7. Let $a _ { 0 } , a _ { 1 } , a _ { 2 } , \ldots$ be a decreasing sequence of positive real numbers.
   Prove that for every nonnegative integer m, $0 \leq \sum _ { i = 0 } ^ { m } ( - 1 ) ^ { i } a _ { i } \leq a _ { 0 }$ .(Hint: $\sum _ { i = 0 } ^ { m + 1 } ( - 1 ) ^ { i } a _ { i } = a _ { 0 } - \sum _ { i = 0 } ^ { m } ( - 1 ) ^ { i } a _ { i + 1 } . )$

8.Find an invertible matrix A and a diagonal matrix B such that $\left( { \begin{array} { c c } { 8 } & { 9 } \\ { - 6 } & { - 7 } \end{array} } \right) = A B A ^ { - 1 }$

9. i) State the binomial theorem.

ii) Prove that if p is a prime number, then $( x + y ) ^ { p } \equiv x ^ { p } + y ^ { p }$ mod $p .$