A positive charge q is fixed at the point x = 0, y = 0, and a negative charge -2q is fixed at the point x = a, y = 0.

a) Derive an expression for the potential V at points on the x -axis as a function of the coordinate x . Take V to be zero at an infinite distance from the charges. (Express your answer in terms of the given quantities and appropriate constants).

V(x) =

b) At which positions on the x -axis is V = 0? (Enter your answers numerically separated by commas).

x/a =

c) What does the answer to part A become when x>>a ?

V =

d) Explain why this result is obtained.

Assume the input graph is directed but may or may not have cycle in it, that is, may or may not be a directed acyclic graph. Consider the recursive topological sorting algorithm shown below. TopologicalSort(G) { If G has only one node v then return v. Find a node v with no incoming edge and remove v from G. Return v followed by the order returned by TopologicalSort(G). }

(a) Extend the recursive algorithm shown below so it can detect and output a cycle if the input graph G is not a directed acyclic graph. Clearly mark the extended part of the algorithm and state your reasoning behind the extension made.

A professor is studying to see if there is a relationship between the grades and gender of his students. He picks a random sample of his students and notes their grades (A, B, or C) and their gender (male or female). He finds that:

1. 28% are A students.
2. 58% are B students.
3. 62% of his students are female.
4. Of those who are female, one-third are A students.
5. Of those who are B students, three-fifths are female students.

What percentage of the students are male and have a C grade?

A professor is studying to see if there is a relationship between the grades and gender of his students. He picks a random sample of his students and notes their grades (A, B, or C) and their gender (male or female). He finds that:

1. 28% are A students.
2. 58% are B students.
3. 62% of his students are female.
4. Of those who are female, one-third are A students.
5. Of those who are B students, three-fifths are female students.

What percentage of the students are male and have a C grade?

Outline the Constitutional rights of public employees. What restrictions can be placed upon public employees? Refer to Supreme Court cases where appropriate.

            Pickering v Board of Education (1968)

            Norton v Macy (1969)

            Kelly v Johnson (1976)

In the figure below, the switch is left in position a for a long time interval and is then quickly thrown to position b. Rank the magnitudes of the voltages across the four circuit elements a short time thereafter from the largest to the smallest.
a.ΔV1200 Ω > ΔVL > 12.0 V > ΔV12.0 Ω
b.ΔVL > ΔV1200 Ω > 12.0 V > ΔV12.0 Ω
c.ΔV1200 Ω > ΔVL = 12.0 V > ΔV12.0 Ω
d.ΔV1200 Ω = ΔVL > 12.0 V > ΔV12.0 Ω

1.let {v=(1,2,3,5,9),v2=(3,1,2,8,9),v3=(2,-5,5,9,4)} and {u1=(0,1,1,1,2),u2=(0,2,-2,-2,0)} be basis of subspaces V and U of R5 respectively.find a basis and the dimension of V+U and V intersection U.

2.does a matrix have a right inverse ?if so find one A=[2,-3,-7,11;3,-1,-7,13;1,2,0,2]

3.find the interpolating polynomial that passes through the point (1,2),)(-1,-8) and (2,1)

Calculate E o , E, and ΔG for the following cell reactions

(a) Mg(s) + Sn2+(aq) ⇌ Mg2+(aq) + Sn(s)

where [Mg2+] = 0.035 M and [Sn2+] = 0.040 M

E o = V

E = V

ΔG = kJ

(b) 3Zn(s) + 2Cr3+(aq) ⇌ 3Zn2+(aq) + 2Cr(s)

where [Cr3+] = 0.090 M and [Zn2+] = 0.0085 M

E o = V

E =    V

ΔG = kJ

(From Hardcover Book, Marsden/Tromba, Vector Calculus, 6th ed., Review Exercises for Chapter 2, # 7)

Use the chain rule to find D ( f ∘ g ) ( − 1 , 2 ) for f ( u , v , w ) = ( v 2 + w 2 , u 3 − v w , u 2 v + w ) and g ( x , y ) = ( 3 x + 2 y , x 3 y , y 2 − x 2 ).

There is a Java program that is missing one recursive function:

public class BinarySearch {

  /*                          / -1                    when min > max
   *                          | mid                   when A[mid] = v
   * search(A, v, min, max) = | search(A,v,mid+1,max) when A[mid] < v
   *                          \ search(A,v,min,mid-1) otherwise
   * where mid = (min+max)/2
   */
  public static int search_rec(int[] A, int v, int min, int max) {
    return 0;
  }

  public static int search(int[] A, int v) {
    return search_rec(A, v, 0, A.length-1);
  }

  /* Binary Search Test Framework
   *
   */
  public static void main(String[] args) {
    int[] inputA  = { 0, 1,    3, 4, 5, 6, 7, 8, 9, 10};
    int[] inputV  = { 3, 8,  2};
    int[] expect  = { 2, 7, -1};

    boolean error = false;

    for(int i = 0 ; i < inputV.length; i++) {
      int answer = search(inputA, inputV[i]);
      if(answer != expect[i]) {
        System.out.printf("ERROR: search(A,%d) returned %d not %d.\n",
                           inputV[i], answer, expect[i]);
        error = true;
      }
    }

    if(error)
      System.exit(1);
    else
      System.out.println("Good Job!");
  }
}