Test #6 Review Questions.
Topic 1:
Solving Systems
1. 𝑥 + 𝑦 − 2𝑧 = 2
3𝑥 − 𝑦 + 𝑧 = 4
−2𝑥 − 2𝑦 + 4𝑧 = 6
2. – 𝑥 − 5𝑦 + 𝑧 = 17
−5𝑥 − 5𝑦 + 5𝑧 = 5
2𝑥 + 5𝑦 − 3𝑧 = −10
3. 4𝑥 + 4𝑦 + 𝑧 = 24
2𝑥 − 4𝑦 + 𝑧 = 0
5𝑥 − 4𝑦 − 5𝑧 = 12
4. 𝑥 − 𝑦 − 2𝑧 = −6
3𝑥 + 2𝑦
= −25
−4𝑥 + 𝑦 − 𝑧 = 12
Topic 2:
Partial Fractions
Write each fraction as a sum of partial fractions
1.
2𝑥−7
𝑥 2 −3𝑥+2
2.
3𝑥+17
𝑥 2 −2𝑥−15
3.
6𝑥
𝑥 2 +4𝑥−5
4.
7𝑥+6
4𝑥 2 +9𝑥−9
5.
𝑥+6
𝑥 2 −4
6.
12𝑥+4
𝑥 2 −6𝑥−16
7.
2𝑥+1
𝑥 2 −5𝑥+4
8.
4𝑥+1
6𝑥 2 −7𝑥+2
Topic 3:
Graphing Systems of Inequalities
Graph the solution set of each system in a coordinate plane
1. 𝑥 + 𝑦 < 9
2. 𝑦 ≥ −2𝑥 + 6
3. 𝑥 + 𝑦 > 3
𝑦 ≥ 2𝑥
𝑦 ≤ 2𝑥 − 2
2𝑥 + 𝑦 < 1
5. 2𝑥 + 𝑦 < 2
𝑥 − 𝑦 > −2
𝑥 + 𝑦 > −2
6. 𝑥 + 𝑦 ≥ 3
𝑥−𝑦 ≤3
𝑥 + 5𝑦 ≥ 15
7. 2𝑥 + 𝑦 < 6
𝑥 + 2𝑦 < 12
4𝑥 − 𝑦 > 12
4. 3𝑥 − 𝑦 < −6
4𝑥 + 𝑦 < −1
8. 𝑥 > −2
𝑦>0
2𝑦 < 𝑥 + 4
𝑦 < −3𝑥 + 9
Topic 4:
Parabolas
1. Find the vertex, focus, directrix and focal width of the following parabolas
a. 𝑦 2 = 12𝑥
b. 𝑥 2 = −8𝑦
c. (𝑥 + 2)2 = −4(𝑦 − 1)
d. (𝑦 + 2)2 = 16𝑥
2. Find the equation of the parabola in standard form that satisfies the given conditions
a. Vertex (0, 0), Focus (2, 0)
b. Vertex (0, 0), opens downward, focal width = 12
c. Vertex (-3, -3), directrix y = 1
d. Vertex (1, -2), opens left, focal length = 2
e. Focus (4, 5), directrix x = -2
Topic 5:
Ellipses
1. Find the equation of the ellipse that satisfies the given conditions
a. Center (0, 0), foci (±12, 0), vertices (±13, 0)
b. Center (0, 0), foci (0, ±2), vertices (0, ±6)
c. Center (0, 2), semimajor axis is 3, one focus is (2, 2)
d. Center (-3, -4), semimajor axis is 4, one focus is (0, -4)
e. Major axis length is 10 on y – axis, minor axis length is 6
f. Major axis endpoints (4, -13) and (4, 7), minor axis length 12
Topic 6:
Hyberbolas
1. Find the equation of the hyperbola that satisfies the given conditions
a. Center (0, 0), foci (0, ±6), vertices (0, ±5)
b. Center (0, 0), vertices (±2, 0), asymptotes 𝑦 = ±2𝑥
4
c. Center (2, 1), vertices (5, 1) and (-1, 1), one asymptote is 𝑦 = 3 (𝑥 − 2) + 1
d. Center (-5, 0), one focus is (-5, 3), one vertex is (-5, 2)
Topic 7:
Conics
1. Identify the type of conic. Find the center, vertices and foci. Sketch the conic.
a.
𝑦2
8
d.
(𝑦−3)2
9
+
𝑥2
5
=1
−
(𝑥−7)2
12
=1
b.
𝑦2
16
e.
(𝑥−2)2
16
−
𝑥2
49
=1
+
(𝑦+1)2
7
=1
c.
(𝑥+3)2
18
e.
𝑦2
36
+
−
(𝑦−5)2
28
(𝑥+6)2
20
2. Write the conic in standard form. Identify the conic. Sketch the graph
a. 𝑥 2 − 6𝑥 − 𝑦 − 3 = 0
b. 𝑥 2 + 4𝑥 + 3𝑦 2 − 5 = 0
c. 𝑥 2 − 𝑦 2 − 2𝑥 + 4𝑦 − 6 = 0
d. 𝑥 2 + 2𝑥 + 4𝑦 − 7 = 0
e. 𝑦 2 − 6𝑥 − 4𝑦 − 13 = 0
f. 3𝑥 2 − 6𝑥 − 4𝑦 − 9 = 0
g. 2𝑥 2 − 3𝑦 2 − 12𝑥 − 24𝑦 + 60 = 0
h. 12𝑥 2 − 4𝑦 2 − 72𝑥 − 16𝑦 + 44 = 0
=1
=1
Topic 8:
Vectors in 3-Dimensions
1. Given point P (-1, 0, 3) and Q (3, -2, -4) and vectors v = ⟨3, −1, −2⟩ and w = ⟨3, −4,0⟩
a. Compute the distance from P to Q
b. Find the midpoint of segment PQ
c. Compute v + w
d. Compute v – w
e. Compute v ∙ w
f. Compute the magnitude of v
g. Write the unit vector in the direction of w
h. Compute (𝐯 ∙ 𝐰)(𝐯 + 𝐰)
2. Repeat #1 given point P (1, 4, -2) and Q (-3, 8, 4) and vectors v = ⟨2, −3, −1⟩ and w = ⟨1, −5, 2⟩
3. Repeat #1 given point P (0, 1, -2) and Q (-1, 2, -5) and vectors v = ⟨0, 2 − 5⟩ and w = ⟨4, −3, 1⟩
Topic 9:
Matrix Applications
1. Marina had $24,500 to invest. She divided the money into three different accounts. At the end
of the year, she had made $1,300 in interest. The annual yield of the three accounts was 4%,
5.5%, and 6%. If the amount of money in the 4% account was four times the amount of money
in the5.5% account, how much had she placed in each account?
2. Find the equation of the parabola 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, that passes through the following three
points: (-2, 40), (1, 7), (3, 15)
3. Billy’s Restaurant ordered 200 flowers for Mother’s Day. They ordered carnations at $1.50 each,
roses at $5.75 each and daisies at $2.60 each. They ordered mostly carnations, and 20 fewer
roses than daisies. The total order came to $589.50. How many of each type of flower was
ordered?
4. Brock’s Discount TV has three types of television sets on sale: a 13-in portable, a 27-in remote
and a 50-in flat screen. They have three types of vehicles to use for delivery: vans, small trucks
and large trucks. The vans can hold 8 portables, 3 remote and 2 flat screen TVs; the small trucks
can hold 15 portables, 10 remote and 6 flat screen TVs; and the large trucks hold 22 portables,
20 remote and 5 flat screens. On a given day, the have 115 portables, 85 remote and 35 flat
screen TVs to deliver. How many vehicles of each type are needed to deliver the TVs?
5. The sum of three numbers is 42. The third number is twice the sum of the first two and is 4
more than three times their difference. What are the three numbers?
6. The proceeds from the school car wash in one, five and ten dollar bills come to a total of $355.
If there are 120 bills altogether and twice as many one dollar bills as five and ten combined, how
many bills of each denomination were there?
Topic 10:
Linear Programming Applications
1. Wally’s woodworking shop produces 2 sizes of bookcases: large and small. Each week, his staff
must produce at least 23 large bookcases and 40 small bookcases to meet demand. The
combined total is limited to 75 bookcases, because of a shortage of materials. The cost to
produce a small bookcase is $72, while the cost to produce a large bookcase is $104. If the
selling price for the small size is $125 and the selling price for the large size is $159, how many of
each type should be produced each week to maximize his profit? What is his maximum profit?
2. Mrs. Tritz raises only pigs and goats and this year intends to raise 16 animals. There is plenty of
room in the pigpen, but a lack of space limits the number of goats to 12. One other limitation is
money: it costs $5 per day to raise a pig and $2 per day to raise a goat, and Mrs. Tritz can only
spend $50 per day on animals. If Mrs. Tritz can make a profit of $17.50 per goat and $14.00 per
pig, how many of each should she raise to maximize her profit? What is her maximum profit?
3. ABC makes 2 models of door handles, plain and decorative, each of which must be assembled
and packaged. The plain handles require 12 minutes to assemble and 2 minutes to package,
while the decorative handles require 18 minutes to assemble but only one minute to package.
Each week there are 240 available hours of assembly time and 20 available hours for packaging.
If the plain handles sell for $2.20 and the decorative handles sell for $3.50, how many of each
type should be made to obtain the maximum weekly income? What is the maximum weekly
income?
4. It takes a tailoring company 2 hours of cutting and 4 hours of sewing to make a casual suit. To
make a formal suit, it takes 4 hours of cutting and 2 hours of sewing. At most 20 hours per day
are available for cutting and at most 16 hours per day are available for sewing. If the profit on a
casual suit is $47 and the profit on a formal suit is $61, how many of each type of suit should be
made each day to maximize profit? What is the maximum profit?
5. The Balfour Tool Company uses 3 machines to manufacture 2 models of power drill, standard
and super. The standard drill requires 1 hour on machine A, 2 hours on machine B and 1.6 hours
on machine C. The super drill requires 2 hours on machine A, 1 hour on machine B and 1.6
hours on machine C. Each machine can be used for at most 40 hours per week. If the profit on
each standard drill is $7.45 and the profit on each super drill is $8.95, how many of each type
should be made to maximize profit? What is the maximum profit?
6. In Santa Fe, the Indian Cultural Center makes woven blankets and shirts. Each blanket requires
24 hours of spinning the yarn, 4 hours of dyeing and 15 hours for weaving. Shirts require 12
hours for spinning, 3 hours for dyeing and 9 hours for weaving. There are 216, 44 and 147 hours
available for spinning, dyeing and weaving respectively. The profit on each blanket is $32 and
$18 on each shirt. How many of each item should be made in order to maximize profit? What is
the maximum profit?
7. A manufacturer of ice skates can produce as many as 60 pairs of hockey skates and as many as
45 pairs of figure skates per day. It takes 3 hours of labor to produce a pair of hockey skates and
4 hours of labor to produce a pair of figure skates. The company has up to 240 hours of labor
available per day. The profit on hockey skates is $12 and $18 on figure skates. Find the number
of pairs of each type of skate the firm should produce to earn the maximum profit each day.
8. A vendor, selling peaches and apples, has space for 180 pieces of fruit and sells all of the fruit
during one day. Customers buy at least twice as many apples as peaches. If the vendor makes a
profit of 8 cents on each peach and 10 cents on each apple, how many pieces of each fruit
should the vendor sell each day to earn maximum profit?