MATH 210
Final Exam
May 9, 1997
1.
Which of the following represents the same line as x - 2y = 4?(a)
y=-x+4
(b)
(c)
(d)
(e)
None of the above.
2.
Which of the following equations represents a line that is parallel to the line x + 5y = 1?(a)
-2x - 10y = 2
(b)
x - 5y = 3
(c)
5x + y = 10
(d)
5x - y = 5
(e)
None of the above.
3.
Which of the following equations represents a line passing through ( 1 , 2 )?(a)
3x + y = 3
(b)
x - y = -2
(c)
2x + 2y = 5
(d)
2x + y = 3
(e)
None of the above.
4.
Find all solutions of the following system of linear equations:x + y + z = 6
2x + 2y - z = 3
x - y + z = 4
(a)
The system has no solutions.
(b)
The system has infinitely many solutions.
(c)
The system has a unique solution. In this solution, y = 1.
(d)
The system has a unique solution. In this solution, y = 2.
(e)
None of the above.
5.
Find all solutions of the following system of linear equations:x + y + z = 6
2x + 2y - z = 3
x + y - 2z = -3
(a)
The system has no solutions.
(b)
The system has infinitely many solutions.
(c)
The system has a unique solution. In this solution, y = 1.
(d)
The system has a unique solution. In this solution, y = 2.
(e)
None of the above.
6.
Let
If possible, find the entry in the second row and the first column of the matrix product AB.
(a)
4
(b)
8
(c)
-3
(d)
13
(e)
The product is undefined.
7.
Is the following matrix invertible? If it is, find its inverse.
(a)
The matrix A is not invertible.
(b)
The entry in the first row and third column of
is 1.
(c)
The entry in the first row and third column of
is -4.
(d)
The entry in the first row and third column of
is -1.
(e)
None of the above.
For the next two problems, solve the following linear programming problem.
Maximize the objective function P = 3x - 2y subject to the constraints:
8.
The maximum value is:(a)
2
(b)
4
(c)
6
(d)
8
(e)
None of the above.
9.
The maximum occurs at a point with x equal to(a)
0
(b)
1
(c)
2
(d)
3
(e)
None of the above.
10.
Let A = {1,2,3,4,5}, let B = {2,4,6,8,10}, and let U = {1,2,3,4,5,6,7,8,9,10}.(U is the universal set for this problem.) Find
(a)
{1,2,3,4,5,7,9}
(b)
{2,4,6,7,8,9,10}
(c)
{1,3,5}
(d)
{6,8,10}
(e)
None of the above.
For the next two problems consider the following information:
100 people were surveyed about which magazines they read.
40 read Time.
30 read Newsweek.
25 read Cosmopolitan.
15 read Time and Newsweek.
12 read Time and Cosmopolitan.
10 read Newsweek and Cosmopolitan.
4 read all three.
11.
How many read Time, but not Newsweek?(a)
25
(b)
17
(c)
40
(d)
0
(e)
None of the above.
12.
How many do not read any of these magazines?(a)
38
(b)
0
(c)
55
(d)
5
(e)
None of the above.
13.
How many different subcommittees of 4 can be selected from a committee with 10 members?(a)
164
(b)
5,005
(c)
5,040
(d)
210
(e)
None of the above.
14.
A restaurant offers 6 different toppings for pizza. The pizzas come in 3 sizes and 2 different types of crust. How many different pizzas can be ordered? (From 0 to 6 toppings can be selected for any pizza.)(a)
384
(b)
4,320
(c)
120
(d)
36
(e)
None of the above.
15.
State employees must choose a health care package consisting of a medical plan and a dental plan. The state offers 4 different medical plans and 5 different dental plans. How many different health care packages can be chosen?(a)
9
(b)
20
(c)
(d)
C(5,4)
(e)
None of the above.
16.
The Smith family wants their 6 children to stand in line for a photograph. However, Mary and Tom are fighting and refuse to stand next to each other. In how many different ways can the children line up with Mary not next to Tom?(a)
720
(b)
120
(c)
480
(d)
600
(e)
None of the above.
17.
A red and a green die are cast. What is the probability that the sum of the numbers on the top faces is 8?(a)
5/36
(b)
1/6
(c)
1/9
(d)
2/9
(e)
None of the above.
18.
A red and a green die are cast. What is the probability that the red die is a 3, given that the sum of the numbers on the top faces is 9?(a)
1/12
(b)
1/9
(c)
1/6
(d)
1/4
(e)
None of the above.
19.
A coin is tossed 6 times. Find the probability that at least one of the tosses is a head.(a)
5/6
(b)
31/32
(c)
1/2
(d)
63/64
(e)
None of the above.
20.
A coin is tossed 6 times. Find the probability that exactly two of the tosses are heads, given that the first is a head.(a)
5/32
(b)
1/32
(c)
5/64
(d)
1/64
(e)
None of the above.
21.
A plane has two engines, one on each wing. For each engine, the probability that it will fail during a given flight is 1/1000. The plane can fly with just one engine, but will crash if both engines fail. Assume that failure of the left engine is independent of failure of the right engine. What is the probability that the plane will crash during a given flight?(a)
1/1,000
(b)
1/1,000,000
(c)
1/2,000
(d)
1/500
(e)
None of the above.
22.
A bag contains 9 tomatoes, of which one is rotten. A sample of 3 tomatoes is selected at random. What is the probability that the sample contains the rotten tomato?(a)
1/9
(b)
1/3
(c)
1/506
(d)
1/12
(e)
None of the above.
For the next two problems, consider the following information.
A class of six people took a ten-point quiz. The scores were 9,9,8,6,5,5.
23.
The mean score is:(a)
7
(b)
8
(c)
20/3
(d)
22/3
(e)
None of the above.
24.
The variance is:(a)
21/6
(b)
11/6
(c)
5/3
(d)
3
(e)
None of the above.
For the next two questions, consider the following information.
In a Calculus class, 40% of the students were Math majors, 40% were Physics majors and 20% were Chemistry majors.
The probability that a student passed the course, given that the student was a Math major, is 0.8.
The probability that a student passed the course, given that the student was a Physics major, is 0.9.
The probability that a student passed the course, given that the student was a Chemistry major, is 0.7.
25.
A student from the class is chosen at random. What is the probability that this student passed the course?(a)
0.80
(b)
0.81
(c)
0.82
(d)
0.83
(e)
None of the above.
26.
What is the probability that a student from the class was a Math major, given that the student passed the course?(a)
2/5
(b)
16/41
(c)
15/42
(d)
3/8
(e)
None of the above.
For the next two questions, consider the following information.
Two brands of Cola are available in Kalbville. Right now, 60% of the population buys Brand I and 40% buys Brand II. Every year, 40% of the people buying Brand I switch to Brand II. Every year, 20% of the people buying Brand II switch to Brand I.
27.
What percent of the population will buy Brand I two years from now? (Round off your answer to the nearest whole percent.)(a)
44%
(b)
56%
(c)
38%
(d)
62%
(e)
None of the above.
28.
In the long run, what percent of the population will buy Brand I? (Round off your answer to the nearest whole percent.)(a)
44%
(b)
33%
(c)
67%
(d)
56%
(e)
None of the above.
29.
Which of the following are absorbing stochastic matrices?
(a)
A and C only.
(b)
B and C only.
(c)
C only.
(d)
A only.
(e)
Some other selection.
30.
A company offers life insurance policies for 65-year-old men. If the policy holder dies in the next ten years, the policy pays $10,000. If the policy holder survives the next ten years, the policy pays nothing. The probability that a 65-year-old man will die in the next ten years is 0.2. How much should the company charge for this policy, in order to break even if it sells many of them? (A company breaks even when it takes in exactly as much as it pays out.)(a)
$500
(b)
$1,000
(c)
$2,000
(d)
$5,000
(e)
None of the above.
© Department of Mathematical Sciences, Northern Illinois University, DeKalb IL 60115
Prepared 8/2/97 by Dr. Anders Linnér (
alinner@math.niu.edu)