Deck 8: Systems of Equations and Inequalities

Question 1

Find the determinant of the matrix after introducing zeros.

A) - 9
B) - 8
C) - 10
D) 7
E) 0

Answer 1

A

Question 2

A manufacturer of tennis rackets makes a profit of $15 on each oversized racket and $10 on each standard racket. To meet dealer demand, daily production of standard rackets should be between 25 and 75, and production of oversized rackets should be between 8 and 27. To maintain high quality, the total number of rackets produced should not exceed 75 per day. How many of each type should be manufactured daily to maximize the profit?

A) 25 standard and 8 oversized
B) 25 standard and 27 oversized
C) 27 standard and 48 oversized
D) 48 standard and 27 oversized
E) 48 standard and 8 oversized

Answer 2

D

Question 3

Sketch the graph of the inequality.

A)
B)
C)
D)
E)

Answer 3

A

Question 4

Use matrices to solve the system.

A) ( 0, 0, 0)
B) ( 0, z, z )
C) ( 0, - z, z )
D) ( 2, 8, 4 )
E) The system is inconsistent

Question 5

Find, if possible, .

A)
B)
C)
D)
E)

Question 6

Find the determinant of the matrix.

A) - 89,090
B) - 1,425,432
C) - 356,359
D) 356,358
E) - 356,358

Question 7

Find the determinant of the matrix.

A) -1083.6
B) -173
C) -433.44
D) -443.44

Question 8

Find the inverse of the matrix if it exists. $$\left[ \begin{array} { l l } 5 & 2 \\4 & 9\end{array} \right]$$

A) $$9 \left[ \begin{array} { l l } 5 & 4 \\2 & 1\end{array} \right]$$
B) $$\frac { 1 } { 5 } \left[ \begin{array} { c c } 9 & 4 \\- 2 & 37\end{array} \right]$$
C) $$\frac { 1 } { 37 } \left[ \begin{array} { c c } 9 & - 2 \\- 4 & 5\end{array} \right]$$
D) $$\frac { 1 } { 37 } \left[ \begin{array} { c c } 63 & - 14 \\- 28 & 35\end{array} \right]$$
E) Does not exist

Question 9

Find the values of b such that the system has no solution.

A)
B)
C)
D)
E) no solution

Question 10

Find the maximum and minimum values of the objective function on the region in the figure.

A) The maximum is at ,
The minimum is at .
B) The maximum is at ,
The minimum is at .
C) The maximum is at ,
The minimum is at .
D) The maximum is at ,
The minimum is at .
E) The maximum is at ,
The minimum is at .

Question 11

Solve the system using the inverse method.

A)
B)
C)
D) The system is inconsistent
E) The equations are dependent

Question 12

Use matrices to solve the system.

A)
B)
C)
D) The system is inconsistent
E) The equations are dependent

Question 13

Three solutions contain a certain acid. The first contains 10% acid, the second 30%, and the third 50%. A chemist wishes to use all three solutions to obtain a 100-liter mixture containing 28% acid. If the chemist wants to use twice as much of the 50% solution as of the 30% solution, how many liters of each solution should be used?

A) 49 of 10% , 17 of 30% , 34 of 50%
B) 46 of 10% , 36 of 30% , 18 of 50%
C) 45 of 10% , 19 of 30% , 36 of 50%
D) 46 of 10% , 18 of 30% , 36 of 50%
E) 43 of 10% , 19 of 30% , 38 of 50%

Question 14

A shop specializes in preparing blends of gourmet coffees. From Colombian, Costa Rican, and Kenyan coffees, the owner wishes to prepare 3-pounds bags that will sell for $8.50. The cost per pound of these coffees is $10, $6, and $8, respectively. The amount of Colombian is to be three times the amount of Costa Rican. Find the amount of each type of coffee in the blend.

A) 1.275 lb Colombian , 0.425 lb Costa Rican , 1.3 lb Kenyan
B) 1.575 lb Colombian , 0.525 lb Costa Rican , 0.9 lb Kenyan
C) 1.125 lb Colombian , 0.375 lb Costa Rican , 1.5 lb Kenyan
D) 1.125 lb Colombian , 0.375 lb Costa Rican , 3 lb Kenyan
E) 1.425 lb Colombian , 0.475 lb Costa Rican , 1.1 lb Kenyan

Question 15

Find, if possible, .

A)
B)
C)
D)
E)

Question 16

The data in the table are generated by the function $f ( x ) = a e ^ { - b x }$ . Approximate the unknown constants a and b to four decimal places. x
1 2 3 4
F ( x )
0)71939 0.41687 0.24157 0.13998

A) $a = 1.2414 , a = 0.5456$
B) $a = 0.0000 , a = 0.5456$
C) $$a = 0.5456 , a = 0.5456$$
D) $a = 1.2414 , a = 0.0000$
E) $a = 0.5456 , a = 1.2414$

Question 17

Find the inverse of the matrix if it exists. $$\left[ \begin{array} { l l l } 2 & 0 & 0 \\0 & 9 & 0 \\0 & 0 & 8\end{array} \right]$$

A) $$\left[ \begin{array} { c c c } 1 & \frac { 1 } { 9 } & 1 \\\frac { 1 } { 8 } & 0 & - 2 \\0 & 1 & \frac { 1 } { 2 }\end{array} \right]$$
B) $$\left[ \begin{array} { c c c } \frac { 1 } { 2 } & 0 & 0 \\0 & \frac { 1 } { 9 } & 0 \\0 & 0 & \frac { 1 } { 9 }\end{array} \right]$$
C) $$\left[ \begin{array} { c c c } 0 & 0 & \frac { 1 } { 2 } \\0 & \frac { 1 } { 8 } & 0 \\\frac { 1 } { 9 } & 0 & 0\end{array} \right]$$
D) $$\left[ \begin{array} { c c c } \frac { 1 } { 2 } & 0 & 0 \\0 & \frac { 1 } { 9 } & 0 \\0 & 0 & \frac { 1 } { 8 }\end{array} \right]$$
E) Does not exist

Question 18

A hospital dietician wishes to prepare a corn-squash vegetable dish that will provide at least grams of protein and cost no more than cents per serving. An ounce of creamed corn provides gram of protein and costs cents. An ounce of squash supplies gram of protein and costs cents. For taste, there must be at least ounces of corn and at least as much squash as corn. It is important to keep the total number of ounces in a serving as small as possible. Find the combination of corn and squash that will minimize the amount of ingredients used per serving.

A) ounces of corn and ounces of squash
B) ounces of corn and ounces of squash
C) ounces of corn and ounces of squash
D) ounces of corn and ounces of squash
E) ounces of corn and ounces of squash

Question 19

Let be the identity matrix of order 2, and let . Find the polynomial for the given matrix A in order to find the zeros of .
(In the study of matrices, is the characteristic polynomial of A, and the zeros of are the characteristic values (eigenvalues) of A.)

A)
B)
C)
D)
E)

Question 20

Use the method of substitution to solve the system.

A) ( - 3, -16 )
B) ( - 3, -16 ), ( 6, 11 )
C) ( 6, 11 )
D) ( - 3, -16 ), ( 5, 11 )
E) no solution

Question 21

Find the partial fraction decomposition.

A)
B)
C)
D)
E)

Question 22

Find the partial fraction decomposition.

A)
B)
C)
D)
E)

Question 23

Use Cramer's rule, whenever possible, to solve the system.

A)
B)
C)
D) The equations are dependent.
E) The system is inconsistent.

Question 24

Find the partial fraction decomposition.

A)
B)
C)
D)
E)

Question 25

Find the partial fraction decomposition.

A)
B)
C)
D)
E)