A 45 minute test Sample Exam 1, Math 141
1. For the line 3𝑥 + 5𝑦 = 45, if 𝑥 decreases by 2 then the change in 𝑦 is ____________.
For problems 2, 3 and 4 the demand and supply equations for the same item are
20𝑝 + 4𝑥 = 2800 𝑎𝑛𝑑 30𝑝 − 2𝑥 = 1200 .
2. At what price will no-one buy any?
3. At what price will suppliers provide 120 units?
4. The price at equilibrium is __________________.
5. Find the vertex of the quadratic 𝑦 = 3𝑥 2 − 6𝑥 − 72 is
6. The demand price at quantity 𝑥units is given by 𝑝 = −.02𝑥 + 24. At price is revenue a maximum?
7. A person wants to make a meal with twice as many grams of protein as of carbohydrates.
The total number of grams is to be 100. Let 𝑝 = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑎𝑚𝑠 𝑜𝑓 𝑝𝑟𝑜𝑡𝑒𝑖𝑛
and 𝑐 = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑟𝑎𝑚𝑠 𝑜𝑓 𝑐𝑎𝑟𝑏𝑜ℎ𝑦𝑑𝑟𝑎𝑡𝑒. Write a set of equations for this
information.
8. The system of equations
6𝑥 − 10𝑦 = 𝑘
a) Has no solution for any 𝑘.
9𝑥 − 15𝑦 = 12
b) has a unique solution if 𝑘 = 8
c) has infinitely many solutions if 𝑘 = 8
d) has a unique solution for 𝑘 = 16
e) none of these
2
9. For the matrix [
1
1
𝑎) [
0
3 5
], the result of performing the row operation R22R2-R1 is:
−2 7
3 5
]
−7 9
2
𝑏) [
0
3 5
]
−7 9
2
𝑐) [
0
e) none of these
3 −18 9 −3
10. Pivot the matrix [ 4
−2 3] 1 ] on row1 column1.
−2
5
1 0
2 𝑦
For problems 11 and 12, 𝐴 = [ 𝑥 4]
−2 5
11.
Find the matrix 2𝐴 + 3𝐵𝑇 .
12. Find 𝐴𝐵.
𝐵=[
1 5
2 7
−1
]
3
3 5
]
−7 2
𝑑) [
−2 −3 −5
]
2 −4 14
The table shows the g/oz of protein, carbohydrate and fat in three foods.
Almonds Bread
Cheese
protein
6
3
8
carb
6
14
1
fat
13
2
5
Use this for problems 13 and 14.
13. What matrix product shows the calories per ounce in almonds, bread and cheese?
14.What is the total calories in a meal of 1 ounce of almonds, 3 ounces of bread and 2 ounces of
cheese?
15. One package of dogfood made from chicken, rice and soy products costs $2.50 and contains 28
grams of protein. The costs per ounce of chicken rice and the soy products are $0.60, $0.20 and $0.30
respectively. The protein contents in grams per ounce for chicken, rice and soy are 7, 2, and 4
respectively. How many ounces of each ingredient does the package contain?
a) Define variables.
b) Write equations showing all the information.
c) Solve the system in the calculator. Give the parametric solution and two realistic, particular solutions.
Key:
1.
8. c
1.2
9. b
2. $140 3. $48
4. $65
1  6 3  1


10. 0 22  9 5


0  7 7  2
5. (1, -75)
6. $12
7. p + c = 100
p – 2c = 0
2 y  6
 7

11. 2 x  15
29 

19 
  7
2  2 y 10  7 y  2  3 y 
 x  8 5 x  28  x  12 
12.


 8
25
17 
 6 3 8


13. 4 4 9 6 14 1


13 2 5
 6 3 8 1

 
14. 4 4 9 6 14 1 3

 
13 2 5 2
15. a) c = # of oz. of chicken r = # of oz. of rice s = #oz. soy
15. b) 0.6c+0.2r+0.3s = 2.5 7c+2r+4s=28
15. c) (3-s,3.5+1.5s,s) 0  s  3
To get particular solutions, choose values for s and substitute into the parametric solution
For example: s=0 gives (3, 3.5, 0) s=2 gives (1, 6.5, 2)