Ryerson University
Ted Rogers School of Management
QMS 110 – APPLIED MATHEMATICS FOR
BUSINESS
Practice Questions for
Test 2
Dear students,
These “Practice Questions for Test 2” were created to help you prepare for
Test 2. This does not mean that your actual Test 2 will look like these
questions. Your test may be completely different. The practice questions
were selected to reflect the material from the lecture notes, the textbook
and the assignments. The final answers are also included to help you verify
your practice answers.
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1
1. Which of the following equations define  as a function of x
2x + 3y −7 = 0
1 + y2 =x
4x2 + 5y2 = 19
a) 1 + y2 = x and 2x + 3y −7 = 0
b) 4x2 + 5y2 = 19 and 2x + 3y −7 = 0
c) 1 + y2 = x and 4x2 + 5y2 = 19
d) 2x + 3y −7 = 0
e) None of the above
2. The slope of the line that passes through the points (a −1, b) and (a + 1, b + 1) is

a) 
b)
c)
d)






e) None of the above
3. The equation of the line passing through (5, 2) and perpendicular to  = −3 is

a)  − 2 =  ( − 5)

b)  − 2 = −  ( − 5)
c) 2 − 5 + 3 = 0
d)  = 5
e) None of the above
4. The lines ax + 3y + 1 = 0 and 2x + ay −1 = 0 are parallel if a equals
a)


b) ±2
c) ±√6
d) −4
e) None of the above
5. The lines a2x −3y −1 = 0 and 6x + ay −1 = 0 are perpendicular if a equals
a)


b) ±2
c) ±√6
d) −4
e) None of the above
2
6. The number of corporate fraud cases pending stood at 545 at the beginning of
2008 (t = 0), and was 767 cases at the beginning of 2012. If the growth was
approximately linear, the estimated number of corporate fraud cases pending at
the beginning of 2014 was?
a) 898
b) 1001
c) 780
d) 878
e) None of the above
7. Given that the point P (−2, 3) lies on the line −2x + ky + 10 = 0, find k.
8. Find value(s) of a so that the line passing through the points (1, a) and (4, −2) is

perpendicular to the line passing through (3, −1) and (, ).
9. Find an equation of the line that passes through the point (−2, −4) and is
parallel to the line x −3y −24 = 0. Express your answer in the slope-intercept
form.
10. Find the equation of the line that passes through (3, 1) and is parallel to the line
joining points (4, −1) and (−3, 2). Express your answer in the general form.
11. As cell phones proliferate, the number of pay phones continues to drop. The
number of pay phones from 2004 through 2009 (in millions) are shown in the
following table (x = 0 corresponds to 2004):
Table 1: Number of Pay Phones from 2004 through 2009 (in millions)
Year, x
0
1
2
3
4
5
Number of Pay Phones, y 1.3 1.15 1.00 0.84 0.69 0.55
a) Plot the number of pay phones (y) versus the year (x).
5
2
►
0
0
5
b) Draw the line L through the points (0, 1.30) and (5, 0.55).
c) Find an equation of the line L.
d)Assume that this trend continues, estimate the number of pay phones in 2012.
12. If x1 , x2 are the solutions of the equation x2 −x = 0, compute (x1 )3 + (x2 )3 .
13. Jesse has $3.02 worth of pennies and nickels in his piggy bank.
The number of nickels is three more than eight times the number
of pennies. How many nickels and how many pennies does Jesse
have?
3
14. Nataly is considering two job offers. The first job would pay her $83,000 per year.
The second would pay her $66,500 plus 15% of her total sales. What would her
total sales need to be for her salary on the second offer be higher than the first?
15. Solve the inequality
1
( − 3) + 0.03 ≤ −0.3(−2 + )
2
16. Solve the inequality
1 + 2|3 − 1| < 6
17. Solve the equation
|5 − 1| = |2 + 3|
18. A total of $40,000 was invested in two accounts. The first part was invested in CD
at a rate 4% annual interest, and the second part was invested in a money market
fund at a rate 9% annual interest rate. If the total simple interest for one year was
5%, then how much was invested in each account?
19. A science center sold 1363 tickets on a busy weekend. The receipts totaled
$12,146. How many $12 adult tickets and how many $7 child tickets were sold?
20. Determine the value of k for which the following system of equations
3x −2y = 3
6x + ky = 4
does not have a solution?
21. Determine the value of r for which the following system of equations
−2x + 6y = 11
−rx + 18y = 33
has infinitely many solutions?
22. Determine the value of a for which the following system of equations
a2 x −3y = 1
9x + ay = 1
has a unique solution?
23. Marcy inherited $25,000 and invested part of it in a money market account, part in
municipal bonds, and part in a mutual fund. After one year, she received a total of
$1,620 in simple interest from the three investments. The money market paid 6%
annually, the bonds paid 7% annually, and the mutual fund paid 8% annually. There
was $6,000 more invested in the bonds than the mutual funds. Find the amount
Marcy invested in each category.
4
24. Solve the following system of equations
3x −4z = 0
3y + 2z = −3
2x + 3y = −5
25. Solve the following system of equations
x + 2y −3z = −1
x − 3y + z = 1
2x −y −2z = 2
26. Maximize the objective function Z = 4x −6y subject to the constraints
y ≤7
3x −y ≤3
x + y ≥5
x ≥0
y ≥0
27. A company produces two types of can openers: manual and electric. Each requires
in its manufacture the use of three machines: A, B, and C. Table 2 gives data
relating to the manufacture of these can openers. Each manual can opener requires
the use of machine A for 2 hours, machine B for 1 hour, and machine C for 1 hour.
An electric can opener requires 1 hour on A, 2 hours on B, and 1 hour on C.
Furthermore, suppose the maximum numbers of hours available per month for the
use of machines A, B, and C are 180, 160, and 100, respectively. The profit on a
manual can opener is $4, and on an electric can opener it is $6. If the company can
sell all the can openers it can produce, how many of each type should it make in
order to maximize the monthly profit?
Manual
Electric
Hours Available
A
2 hr
1 hr
180
B
1 hr
2 hr
160
C
1 hr
1 hr
100
Profit/Unit
$4
$6
Table 2: Can Opener Production Summary
28. A diet is to contain at least 16 units of carbohydrates and 20 units of protein. Food
A contains 2 units of carbohydrates and 4 units of protein; food B contains 2 units
of carbohydrates and 1 unit of protein. If food A costs $1.20 per unit and food B
costs $0.80 per unit, how many units of each food should be purchased in order to
minimize cost? What is the minimum cost?
5
29. Find a such that the equation 3x2 + 2x + a = 0
a) Has two distinct real solutions
b) Has exactly one real solution
c) Has no real solutions
d) Factor polynomial p(x) = 3x2 + 2x −1
30. If  is the smaller and  the larger solution of the equation 2x2 + 7x − 15 = 0,
then
!"
""
equals
a) 63
b)


c) 34
d)


e) None of the above
31. Which of the following is true for −x2 + x −2
a) It has two real roots
b) It factors as (x + 1)2
c) It has one real root
d) It is negative for every real number x
e) None of the above
32. Simplify the expression
−2 − 2
−1
−
  + 2 − 8 2 − 
33. The table represents a linear function.
a) What is (2)?
b) What is the slope of the line?
c) What is the y-intercept?
d) Using answers from part b and c, write an equation for ().
e) If () = 2.1, what is the value of ?

 = ()
0
3.5
1
2.3
2
1.1
3
-0.1
4
-1.3
34. A satellite telephone is leased at a cost  of $200 plus $10 per minutes  the phone is
used. a) Solve for t as a function of  and b) find (500).
35. Find the distance from (4,1) to the line 4 − 3 + 12 = 0.
6
36. An oil-storage tank is emptied at a constant rate. At 10:00 a.m., 1800 barrels remain,
and at 2:00 p.m., 600 barrels remain. If pumping started at 8:00 a.m., find the equation
relating the number of barrels  at time  ( ℎ) from 8:00 a.m. when will the tank be
empty?
37. Suppose that the supply and demand equations for printed T-shirts for a particular week
are
 = 0.7 + 3
 = −1.7 + 15
 −  
 −  
Where  is the price in dollars and  is the quantity in hundreds.
a) Find the supply and demand (to the nearest unit) if T-shirts are $4 each.
b) Find the supply and demand (to the nearest unit) if T-shirts are $9 each.
c) Find the equilibrium price and quantity.
d) Graph the two equations in the same coordinate system and identify the equilibrium point., supply
curve, and demand curve.
38. A company market exercise DVDs that sell for $19.95, including shipping and
handling. The monthly fixed costs (advertising, rent, etc.) are $24,000 and variable
costs (materials and shipping, etc.) are $7.45 per DVD.
a) Find the cost equation and revenue equation.
b) How many DVDs must be sold each month for the company to break even?
39. Elizabeth has score of 74 and 82 on her first two algebra tests. What must she score on
her third test so that het average is at least 80?
40. Solve the inequality
2|6 − | + 1 ≥ 5
41. Solve the quadratic equation
(x −2)2 + 7(x −2) + 12 = 0
42. An international phone call costs $2 plus $0.3 per minute or fractional part of a minute.
If Jorge has $5.60 to spend on a call, what is the maximum total time he can use the
phone?
43. Solve the inequality and write the solution set in interval notation and graph it.
−4 < 2( + 1) ≤ 6
44. Solve the following equation using quadratic equation (Round answers to 2 decimal places)
 +
 1
=
3 6
45. Simplify the expression
  − 3 + 2   − 4 
  − 4
. 
÷ 

3 − 3  − 6 + 8  + 3 + 2
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Answers
1. d
2. a
3. d
4. c
5. a
6. d
7. −14/3
8.  = 7
9.  =


−


10. 3 + 7 − 16 = 0
11. c)  = −0.15 + 1.3 d) 100,000
12. 1
13. 7 pennies and 59 nickels
14. $110,000
15.  ≤




16. −  <  < 


17.  = −  or  = 
18. $32,000 was invested at a rate of 4% annual interest and $8,000 was invested at a rate of 9%
annual interest
19. Adult ticket=521, and child ticket=842
20.  = −4
21.  = 6
22. The system has a unique solution if and only if  ≠ −3.
23. Marcy has invested $15,000 in money market, and $ 8,000 in municipal bond, and $2,000 in
mutual fund.
24.  = −4,  = 1,  = −3
25. The system is inconsistent and has no solution.
26. The maximum value of Z, subject to the constraints, is -10, and it occurs when x = 2 and y = 3.
27. The maximum profit subject to the constraints is $520, which is obtained by producing
40 manual can openers and 60 can openers per month.
28. 4 units of food A, 4 units of food B; $8




29. a)  < , )  = , )  > , ) () = 3( + 1)( − )




30. b
31. d
32.


for  ≠ 2,  ≠ −4
,
33. a)1.1, ) − 1.2, c) 3.5, d) () = −1.2 + 3.5, e) 1.16
34. a) () =


, b) t(500)=30 min
35. d=5
36.  = −300 + 2400, the tank will be empty when  = 0. Therefore, the tank will be empty 8h
past 8 a.m., or at 4 p.m.
37. a) supply: 143 T-shirts; demand: 647 T-shirts
b) supply: 857 T-shirts; Demand: 353 T-shirts
c) Equilibrium price =$6.5; Equilibrium quantity=500 T-shirts
38. a)  = 24,000 + 7.45,  = 19.95
b) 1,920
c) Break-even point (1,920, 38,304)
39. Her score on third test must be at least 84 (ℎ  ≥ 84)
40. x ≤4 or x ≥8.
41. x = −2 or x = −1
42. 12 minutes
43. −3 <  ≤ 2, (−3,2]
44.  = 0.27,  = −0.61

45. ()   ≠ 0,  ≠ 4,  ≠ ±1,  ≠ ±2
8