Exam 1 Review with key    Math 166

 

  1. A police department employs two grades of personnel, rookies and sergeants. A rookie spends 20 hours training and 20 hours on patrol each week. A sergeant spends 5 hours training and 30 hours on patrol each week. The training center can handle 240 person hours each week. The police department needs 440 person hours for patrol duty each week. How many rookies and sergeants should they have?

 

  1. A chemistry department wants to make 2 liters of an 18% acid solution by mixing a 21% acid solution with a 14% solution. How many liters of each type of acid should they use?

 

  1. A kitchen appliance manufacturer makes can openers and dough cutters using two machines, a press and a riveter. Each can opener requires 0.2 minutes in the press and 0.4 minutes in the riveter. A dough cutter requires 0.5 minutes in the press and 0.3 minutes in the riveter. The press can be operated only 3 hours per day and the riveter can be operated only 2.5 hours per day. If these two machines are fully used, how many can openers and dough cutters can be produced each day?

 

  1. A dietitian is planning a meal which should contain 444 calories. She knows that carbohydrate and protein both contain 4 calories per gram and fat contains 9 calories per gram. She wants the weight of protein to be twice as much as the weights of carbohydrate and fat combined. The weight of fat should be 1/8 of the total weight. How many grams of carbohydrate, protein and fat should the meal contain?

 

 

  1. Solve each system of equations using Gauss-Jordan elimination on an augmented matrix or state no solution. If the solution is infinite, give the parametric equation and two particular solutions.

 

a)        b)         c)        d)

 

 

 

 

 

 

 

 

 

  1. Each augmented matrix represents a system of equations in x, y, and z. You will have to supply the vertical bar between the last two columns. Find each solution or state none exists. If the solution is infinite, give the parametric solution and two particular solutions.

 

a)       b)     c)      d)

 

 

 

 

 

 

 

  1. Use your rref in your calculator to solve each system of equations. If infinite, give the parametric solution and two particular solutions.

a)      b)      c)      

d)

 

  1. Each of the matrices is not fully row reduced. Perform the next pivot of the row reduction. Show all work and row operations.

 

a)     b)     c)

 

9. A=    B=      C=

Perform the indicated operations for A, B, C as above.

a) A + B    b)     c) BC

 

  1. A pollution control officer is monitoring two waste treatment plants. They measure the amount of nitrates, lead and oxygen in the plant discharge in parts per million. The measurements at the beginning of the monitoring period are given by matrix E. The measurements at the end of the period are given by matrix F. What matrix represents the change in the amounts of pollutants in the discharge?

E=       F=

 

  1. The Black’s are each investing in two stocks. Matrix A shows the amounts each of them had in stocks 1 and 2 at the beginning of the year. Matrix B shows the amounts each had at the end of the year. If they owe 25% in taxes on their gains, what matrix shows the after tax gain in each stock for each person?

A=       B=

 

  1. Write a product of 3 matrices to answer the following: A company produces two nut mixtures, regular and premium. The regular mix contains 60% peanuts, 25% cashews and 15% pecans. The premium mix contains 30% peanuts, 40% cashews and 30% pecans. Peanuts cost them $.80 per pound, cashews cost $1.15 per pound and pecans cost $1.50 per pound. How much will it cost them to fill an order for 4 pounds of regular mix and 12 pounds of premium mix?

 

  1. Solve for x, y and z.

 

 

14. Which matrix operations can be performed for A=   B=   C=  If not possible, why not and if possible, what is the size of the resulting matrix?

 

a)  A+B    b) BC   c) AC    d)  CA       d) AB+C  

 

 

 

 

 

  1. Which matrix has an inverse? Find the inverse when it exists.

 

A=      B=     C=

 

  1. Write the system of equations as a matrix equation and solve it using an inverse matrix.

 

     

 

  1. Show that A=  and B=  are inverses and use this fact to solve

A= .   Your solution will contain the unknown, a.

 

18. Solve the matrix equation AX+B=X  for X if  A=  and  B=.

 

 

19.  An economy is based on three sectors, electricity, natural gas, and oil. Production of a dollar’s worth of electricity requires inputs of $0.30 from electricity, $0.10 from natural gas and $0.20 from oil. Production of a dollars worth of natural gas requires inputs of $0.20 from electricity, $0.15 from natural gas, and $0.25 from oil. Production of a dollar’s worth of oil requires inputs of $0.10 from each sector. Find the output from each sector needed to satisfy a demand for $25 billion for electricity, $15 for natural gas, and $20 for oil. Round to 4 decimal places and express as billions.

 

20. p is the proposition, “I will go to the beach.” .  q is “I will go to the park.”  r is “I will study.”  Make a truth table for each statement.

a) I will not go to both the beach and the park.

b) I will neither go to the beach  nor the park.

c) I will not go to the beach or the park but I will study.

 

21. a) Make a  truth table for  each of      and    Are they the same?

b) State the other distributive law.

 

 

 

 

 

 

22. A, B, and C are sets in a universal set U. Describe each of the sets in terms of A, B, and C and represent each set with a shaded Venn diagram.

a){x| x is in at least one of A, B, and C} 

b) {x| x is in A and B but not C}

 c) {x| x is in exactly two of A, B, and C}

d) {x| x is in A or B and x is not in C}  

e) {x| x is in exactly one of A, B, and C}  

 

23. A={x| x is an accounting major at TAMU} B={x| x is a junior accounting major at TAMU}  C={x| x is a junior at TAMU}  D={x| x is a psychology major at TAMU}

Which are true?

 a) B   b)    c)    d)    e)   f) B= 

 

 

24. For A, B, C, and D as in 7, describe each of the following sets in words.

 

a)    b)   c)    d)   e)    f)

 

g). All the psychology majors and all the accounting majors are invited to a meeting. What set describes those invited?

 

25. 120 people were asked if they read or listen to music in their leisure time. 85 said they read. 65 said they listen to music. 10 said they do neither.

a) How many do both?     b) How many read but do not listen to music?   c) How many either read or listen to music but not both?

 

 

26. 100 children took a vision test and a hearing test. 50 passed the vision test. 60 passed the hearing test. 15 failed both tests.

 a) How many passed both tests?     b) How many passed the hearing test only?

 

 

27. Sixty students were asked if they belong to the theatre club, the Spanish club, or the art club.  From their responses it was found that:

13 belong to the art club only.

25 belong to the theatre club or Spanish but not the art club.

33 belong to exactly one of the three.

24 belong to the Spanish club only or the art club only.

20 belong to exactly two of the three.

32 belong to the art club.

22 belong to the theatre club but not all three.

 

Fill in a Venn diagram with the number in each region.

a) How many belong to none of the three?

b) How many belong to only one club?

 

Key:

1.      10 rookies, 8 sergeants

 

2.      1.143 liters of 21% solution and 0.857 liters of 14% solution

 

3.      150 can openers and 300 dough cutters

 

4.      20 grams of carb., 64 grams of protein, 12 grams of fat

 

5.      a) x= -2, y=1 or (-2,1)   b) x=12  y=6  or (12,6)   c) no solution 

d) (9/5 + 4/5 t, t) t is any real number   particular solutions vary. If t=0, (9/5,0).  If t=5, (13, 5)

 

6. a) (-3, 2, 5)   b) (-4+t, -3-2t, t) t is any real number   t=0: (-4, -3, 0)

 t=1: (-3, -5, 1)    c) no solution    d) (t, -1, 3)   t=0: (0, -1, 3)   t=1: (1, -1, 3)

 

7. a) (4+t, -3-t, t) t is any real number   t=0: (4, -3, 0)   t=1: (5, -4, 1)

b) no solution    c) (3+5/13 t, 2+14/13 t, t) t is any real number  t=0: (3, 2, 0)

t=13:  (8, 16, 13)       d) (1, -2, 3)

 

8. Different row operations can perform the pivot. Only the completely reduced matrix is unique. You might have a different answer and still be correct.

a)  results in

b)    results in

c)   results in

 

9. a)      b)     

c)

 

10.       

 

11.

 

12.     or       which both equal  $17.77

 

13. x= -6   y=1    z= -2

 

  1. a) not possible to add matrices of different sizes

 

b) 3x2     c) 2x2    d) 2x3

 

  1. A is not square so it cannot have an inverse.  B is square but is singular.  C has an inverse and C=
  2. The matrix equation is   and 

 

  1. A and B are inverses since AB is equal to . ( BA is also). x= -1/a, y=2.

 

  1. Doing the algebra we have

 

  1. $49.6852 billion of electricity, $28.3315 billion of natural gas, and $41.1333 billion of oil.

 

 

 

 

 

 

 

 

 

20 a)

b)  same as ~           

 

 

c) ~    

21 a) yes they are the same   2b)  iff 

 

22.  a)    b)     

 

c)     d)     

e)   

 

23. a) F  b) T  c) T  d) T  e) T  f) T  

 

24. a) All TAMU students who are either accounting majors or psychology majors.

b) All TAMU students who are junior psychology majors.  c) All TAMU students who are juniors or psychology majors   d) All TAMU students who are juniors not majoring in psychology   e) TAMU students who are accounting majors and are not juniors  f)  All TAMU students who are either juniors or not psychology majors   g)

 

25. a) 40   b) 45   c) 70

26.  a) 25    b) 35

27. a) 3   b) 33