pre-calculus math 11 final exam review chapter 6 rational
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Pre-Calculus Math 11 β Final Exam Review
Chapter 6 β Rational Expressions & Equations
1) The undefined value(s) for the rational expression 12
π₯2β4 are
A) π₯ β 2, π₯ β β2
B) π₯ β 2β3
C) π₯ β 2
D) π₯ β 4
2) Simplify the expression and state any undefined value(s): 20π₯2β5π¦2
2π₯2β15π₯π¦β8π¦2
A) 5(2π₯+π¦)
π₯β8π¦, π₯ β β
π¦
2, π₯ β 8π¦
B) 5(2π₯+π¦)
π₯+8π¦, π₯ β β
π¦
2, π₯ β β8π¦
C) 5(2π₯βπ¦)
π₯+8π¦, π₯ β
π¦
2, π₯ β β8π¦
D) 5(2π₯βπ¦)
π₯β8π¦, π₯ β β
π¦
2, π₯ β 8π¦
3) What is the simplified version of β3π₯+12
32β8π₯ ?
A) 3
8(π₯ β 4)
B) π₯ β 4
C) 3
8
D) β3
8
4) Fully simplify, ignoring undefined values: 6π₯9
3π₯3 Γπ₯8
9π₯6
A) 2
9π₯8
B) 9
2π₯4
C) 2
9π₯4
D) 9
2π₯8
5) Simplify using only positive exponents: 4π₯8π¦5
(2π₯π¦)3 Γ·(π₯8π¦5)3
(2π₯π¦8)4
A) 8π₯33π¦3
B) 8π¦19
π₯15
C) π¦19
32π₯33
D) π₯33
32π¦3
6) Simplify π₯2β5π₯β24
π₯2β11π₯+24Γ·
2π₯2+7π₯+3
π₯2+π₯β12
A) 2π₯+1
π₯+4
B) π₯+4
2π₯+1
C) (π₯+3)(2π₯+1)
(π₯β3)(π₯+4)
D) (π₯β3)(π₯+4)
(π₯+3)(2π₯+1)
7) Simplify π₯+8
π₯2+9π₯+20+
π₯+5
π₯2+7π₯+12
A) 2π₯+13
2π₯2+16π₯+32
B) (π₯+8)(π₯+5)
(π₯2+9π₯+20)(π₯2+7π₯+12)
C) 2π₯2β21π₯β49
(π₯+5)(π₯+4)(π₯+3)
D) 2π₯2+21π₯+49
(π₯+5)(π₯+4)(π₯+3)
8) Solve the rational equation π₯
π₯+1=
4βπ₯
π₯2β3π₯β4+
6
π₯β4
A) π₯ = 10
B) π₯ = 4 & π₯ = β1
C) π₯ = β10
D) π₯ = β10 & π₯ = 1
9) Simplify each expression and state any undefined values
a) π₯2β2π₯
π₯+1Γ
π₯2β1
π₯2+π₯β6
b) 4π₯β1
π₯2+7π₯+12Γ·
2π₯β1
π₯2+π₯β12
c) π₯
π₯2β3π₯β4β
4
π₯+1
d)
2
π₯β
2
3π₯1
π₯β
5
6π₯
e) 1+
1
π₯
1β1
π₯2
10) Solve & check: 5
π₯β1+
2
π₯+1= β6
11) If it takes Mike 5 hours to paint a room, and Joe 6 hours to paint the same room, how long
will it take them to paint the room together (answer in hours and minutes)?
12) A jet-ski can travel 104km downstream in the same time that it can travel 74km upstream.
If the current of the river is 9km/h, what is the speed of the jet-ski in still water? Answer to the
nearest tenth.
Chapter 3 β Quadratic Equations
13) Which function is not quadratic?
A) π(π₯) = (6π₯ + 9)(1
9π₯ β 9)
B) π(π₯) = π₯(π₯ β 9)(6π₯ + 8)
C) π(π₯) = 7π₯2 + 8
D) π(π₯) = 6(π₯ β 9)2
14) Solve β8π₯2 + 120π₯ + 432 = 0
15) Determine the roots of the quadratic equation β5π₯2 + 55π₯ = 50
16) A rectangle has dimensions π₯ + 10 and 5π₯ β 4, where π₯ is in centimetres. If the area of the
rectangle is 72ππ2, what is the value of π₯, to the nearest tenth of a centimetre?
17) Solve π₯2 + 2π₯ + 42 = 0 by completing the square.
18) Find the roots of π¦ = β1
2π₯2 β 2π₯ +
7
10
19) Find the roots of the quadratic function π¦ = 5π₯2 + 20π₯ β 6 by completing the square.
20) Solve using the quadratic formula:
a) π₯2 + 4π₯ = 21
b) 2π₯2 = 5π₯ β 8
c) 2π₯2 = 5π₯ + 8
21) Solve 3π₯2 = 8π₯ β 4 by:
a) factoring
b) completing the square
c) quadratic formula
22) Find the x-intercepts of the quadratic function π¦ = 3π₯2 β 10π₯ + 6
23) Solve each of the following by factoring, if possible:
a) π₯2 + 10π₯ = 24
b) 2π₯2 = 8π₯ β 6
c) 0 = βπ₯2 β 15π₯ β 44
d) 3π₯2 β 21π₯ = 0
e) 6π₯2 + 17π₯ β 3 = 0
f) 8π₯2 + π₯ = 9
g) π₯2 β 9 = 0
h) 4π₯2 + 25 = 0
24) A uniform border on a framed photo has an area four times that of the photo. What are the
outside dimensions of the border if the dimensions of the photo are 30 cm by 20 cm?
25) The Parthenon, in Athens, is a temple to the Greek goddess Athena, and was built in about
447 B.C.E. It has a rectangular base with a perimeter of approximately 202 m and an area of
2170 m2. Find the dimensions of the base, to the nearest metre.
26) The sum of the squares of two consecutive odd integers is 1570. Find the integers.
27) The driving distance from Winnipeg, Manitoba to Billings, Montana is 1200 km. A moving
van made the trip in 31 hours, excluding loading/unloading time. The average speed from
Winnipeg to Billings was 5 km/h slower than the average speed of the return trip to Winnipeg.
What was the average speed of each trip?
28) A patrol boat took 2.5 h for a round trip 12 km up river and 12 km back down river. The
speed of the current was 2 km/h. What was the speed of the boat in still water?
Chapter 1/2 β Quadratic Functions
29) Graph the following functions and find the vertex, axis of symmetry equation, domain,
range, max/min, x-intercepts, and y-intercept.
a) π¦ = βπ₯2
b) π¦ = 2π₯2 β 6
c) π¦ = (π₯ β 1)2 + 2
d) π¦ = β1
2(π₯ + 5)2 + 2
e) π¦ = 3(π₯ + 2)2 β 8
30) Given the following information, write the function for the parabola:
a) vertex (-1, 4); passing through point (-2, 2)
b) vertex (-2, 3); y-intercept of 1
c) passes through points (-3, 4), (6, 6), & (5, 4)
31) Write each of the following general form quadratic functions in standard form by
completing the square, then state the vertex.
a) π¦ = π₯2 β 2π₯ + 3
b) π¦ = βπ₯2 + 8π₯ β 12
c) π¦ = 3π₯ β π₯2
d) π¦ = 2π₯2 + 8π₯ + 6
e) π¦ = β1
3π₯2 + 2π₯ + 4
32) Find two numbers whose difference is 10 and whose product is a minimum.
33) Two numbers have a sum of 34. Find the numbers if the sum of their squares is a minimum.
34) A park has an arch over its entrance. The curve of the arch can be graphed on a grid with
the origin on the path directly under the centre of the arch. The arch can be modeled by the
function β(π) = β1.17π2 + 3, where β(π) metres is the height, and π metres is the
horizontal distance from the centre of the arch.
a) What is the maximum height of the arch?
b) If the ends of the arch are level with the path, how wide is the arch to the nearest
tenth of a metre?
c) At a horizontal distance of 0.5m from the centre of the arch, how high is the arch, to
the nearest tenth?
35) The captain of a riverboat cruise charges $36 per person, which includes lunch. The cruise
averages 300 customers per day. The captain is considering increasing the price. A survey of
customers indicates that for every $2 increase in price, there would be 10 fewer customers.
What increase in price would maximize revenue for the captain?
36) A farmer wants to make a rectangular corral along the side of a large barn, and has enough
materials for 60m of fencing. Only three sides must be fenced, with the barn wall forming the
fourth side. What dimensions should the farmer use to maximize the area?
Chapter 4 β Systems of Equations & Inequalities
37) The line π¦ = 9π₯ β 4 intersects the quadratic function π¦ = π₯2 + 7π₯ β 3 at one point. What
are the coordinates of the intersection?
A) (0, 0)
B) (1, -5)
C) (-1, 5)
D) (1, 5)
38) Find the coordinates of the point(s) of intersection of the line π¦ = 4π₯ + 8 and the quadratic
function π¦ = β4π₯2 β 5π₯ + 8
A) (0, 8) and (9
4, 17)
B) (0, 0)
C) (2, -34)
D) (β9
4, β1) and (0, 8)
39) What are the solutions for the following system?
π¦ = β2π₯2 β 9π₯ β 4 π¦ = 2π₯2 β 5π₯ β 4
A) (-1, 3) & (0, -4)
B) (1, -3) & (0, -4)
C) (1, 3) & (0, -4)
D) (1, -3) & (0, 4)
40) What are the coordinates of the point(s) of intersection of the quadratic functions
π¦ = β2π₯2 β 4π₯ + 5 π¦ = 2π₯2 + 4π₯ + 5
A) (-2, 5) & (0, 5)
B) (2, -5) & (0, -5)
C) (2, 5) & (0, 5)
D) (2, -5) & (0, 5)
41) The solution set to the inequality β2π₯2 + 8π₯ β 6 > 0 is:
A) {π₯|1 < π₯ < 3, π₯ππ }
B) {π₯|β3 < π₯ < β1, π₯ππ }
C) {π₯|π₯ < 1, π₯ > 3, π₯ππ }
D) {π₯|π₯ < β3, π₯ > β1, π₯ππ }
42) If π₯ represents the number of pairs of cross-country skis sold and π¦ represents the number
of pairs of snowshoes sold, what inequality models the combinations of ski and snowshoe sales
that will meet or exceed the daily goal?
A) 50π¦ + 125π₯ β€ 700
B) 50π¦ + 125π₯ > 700
C) 50π₯ + 125π¦ β₯ 700
D) 50π₯ β 125π¦ < 700
43) Graph the solution to the inequality you chose in the previous question.
44) Graph the solution π¦ < β2
3π₯ + 4
45) Graph the solution to the inequality π¦ β€ β5(π₯ + 3)2 + 4
46) Solve the following system by graphing: 1) π¦ = π₯2 β 16
2) π¦ = β(π₯ + 4)2
47) Solve the system using substitution (to the nearest hundredth):
1) π¦ = β3π₯2 β 3π₯ + 2
2) π¦ = β6π₯2 + 4π₯ + 7
48) One or more of the following ordered pairs are a solution to the inequality π¦ β₯ 3π₯ β 5
Circle the solutions: (2, 2) (-1, -9) (1, -2) (0, 0)
49) What is the solution for 2π₯2 β 7π₯ β₯ β3 ?
50) Graph the quadratic inequality π¦ β₯ β2
3(π₯ β 3)2 β 1. Then check with one test point in the
solution region, and one test point outside of the solution region.
51) Solve the system algebraically: 1) π¦ = 4π₯2 + 13
2) π¦ + 7 = 4π₯2
52) Solve the system of inequalities by graphing: π₯ + 2π¦ < 8 β3π₯ + 2π¦ β₯ β4
53) Solve the system of inequalities by graphing: 5π₯ β 4π¦ < 20 π₯ + 2π¦ β€ 4 π₯ β₯ 0 π¦ β€ 0
54) A womenβs clothing store makes an average profit of $125 on each dress sold and $50 on
each blouse. The managerβs target is to make at least $500 per day on dress and blouse sales.
a) What inequality represents the numbers of dresses and blouses that can be sold each day to
reach the target?
b) Graph the inequality.
c) If equal numbers of dresses and blouses are sold, what is the minimum number needed to
reach the target?
55) Jennifer likes to make bagels and cupcakes. In one hour, Jennifer can make 24 bagels or 36
cupcakes. In one week, Jennifer hopes to make at least 252 baked items but spend no more
than 10 hours baking. Solve by graphing. Then describe two possible ways that Jennifer can
meet her requirements.
Chapter 5 β Radicals
56) What does the expression 7β7 β 6β12 β (4β28 + 4β3) simplify to?
A) 15β7 β 16β3
B) 15β7 + 16β3
C) ββ7 β 16β3
D) ββ7 + 16β3
57) Express β64π10π155 in simplified form.
A) 4π2π3 β45
B) 2π2π3 β52
C) 4π2π3 β25
D) 2π2π3 β25
58) Express β7β6(β6β5 β 2β6) in simplest form.
A) 14β6 + 42β30
B) 252
C) 42β30 + 84
D) 1260 + 14β6
59) Express the following in simplest form: 2β21β3β7
β7+
4β3β8
β4
A) 6β3 β 5
B) 6β21 β 14β7
C) 4β3 β 7
D) 2β21 β 3β7 + 4β3 β 2
60) Solve β4π₯ β 5 = 6
A) π₯ =121
16
B) π₯ =11
16
C) π₯ =121
4
D) π₯ =11
4
61) Solve βπ₯ + 3 = β2π₯ + 8
A) β
B) π₯ = β5
C) π₯ =1
25
D) π₯ = β1
5
62) What are the restrictions on π₯ if the solution to the equation β4 β β4 β π₯ = 6 involves
only real numbers?
A) π₯ β€ 10
B) π₯ β₯ 6
C) π₯ β€ 4
D) π₯ β₯ 100
63) Without using a calculator, arrange the following in order from least to greatest:
3β5, 2β11, 4β3, 5β2
64) Simplify each expression:
a) 5β12 β 2β27
b) 24β14
8β2
c) β2(2β2 + 2) β 3(5β2 + 1)
65) Solve each of the following:
a) π₯2 = 36
b) π₯2 = β36
c) π₯3 = 27
d) π₯3 = β27
e) π₯4 = 16
f) π₯4 = β16
66) Simplify each expression. Assume all variables represent positive numbers
a) β20π₯3π¦6
b) β32π₯2π¦π§9
c) β9π₯8π¦2π§3
d) β24π₯2π¦4π§63
e) β16π₯6
π¦12π§
4
67) Simplify each expression. Let the variables be any real numbers.
a) β20π₯3π¦6
b) β32π₯2π¦π§9
c) β9π₯8π¦2π§3
d) β18π₯6
π¦3
e) β32π₯4π¦8π§74
68) Change to an entire radical. Assume all variables are positive
a) 3π2π β2π2π3
b) 5π₯5β6π¦
c) 3π₯2
π¦3 βπ₯34
69) Simplify each expression.
a) 6β5(2β4)
b) 3β2(8β2 β 3β6)
c) 7β2 β β24 + β18 β 5β54
d) 8β18
2β2
e) ββ40
2β5
70) Simplify βπ₯3
(βπ₯5), π₯ β₯ 0
71) Simplify, including rationalizing the denominators.
a) 2β2
β3
b) 4ββ5
3β2
c) 6β5
2β10
d) β4β2
3+β3
e) ββ3β1
2ββ2
f) 2 β5
3
β43
g) β20
β163
72) Solve 4 β β4 + π₯2 = π₯
73) Solve βπ₯ + 10 = π₯ β 2
74) The formula π = 2πβπ
32 represents the swing of a pendulum, where π is the time, in
seconds, to swing back and forth, and π is the length of the pendulum, in feet.
a) Solve the formula for π.
b) What is the length of the pendulum that makes one swing in 1.5π ?
75) Solve each of the following:
a) βπ₯ β 2 = 0
b) βπ₯ β 3 + 6 = 2
c) β4(π₯ + 3) = 6
d) β2 β π₯ = βπ₯ β 2
e) βπ₯
2+ 8 = β4π₯ + 1
f) β4(π₯ + 1) = β2π₯ + 3
g) βπ₯ β 4 β π₯ = β4
Functions Unit
76) Graph each of the following and state the domain, range, x-int, & y-int.
a) π(π₯) = βπ₯
b) π¦ = ββπ₯
c) π¦ = ββπ₯
d) π¦ = βββπ₯
e) π(π₯) = 2βπ₯ + 3 β 1
f) π¦ = β1
2βπ₯ β 2 + 4
g) π¦ = β2β4 β π₯ + 1
h) π(π₯) = βπ₯3
i) π¦ = ββπ₯3
j) π¦ = 2ββπ₯3
β 4
77) Graph each and state the domain and range.
a) π¦ = β1
2π₯ β 2
b) π(π₯) = β(π₯ + 2)2 β 4
c) π¦ = ββπ₯2 + 9
78) Solve each equation.
a) |6π₯ + 9| + 2 = 8
b) |4π₯ + 8| = β8π₯ + 3
c) |1
2π₯ + 1| = π₯ + 1
d) |π₯ + 1| = π₯ + 1
e) |4 β π₯| + 3 = 3
f) β2|π₯ + 5| β 3 = 5
g) |π₯2 β π₯ β 9| = 3
79) Simplify: 5 β 3|1 + 2(β3 β 4)|
80) Graph each and state the domain and range. Write each of b, c, & d as a piecewise function.
a) π¦ = 2|π₯ β 3| β 6
b) π¦ = β|π₯ + 2| + 4
c) π¦ = |4 β π₯| β 1
d) π¦ = |(π₯ β 1)2 β 4|
e) π¦ = |β(π₯ + 3)2 β 2|
81) Graph and state the equations of the asymptotes. Find the y-intercept for each.
a) π¦ =1
π₯β2
b) π(π₯) = β1
π₯+3β 4
82) For each, state the equations of the asymptotes.
a) π¦ =3π₯
π₯+2
b) π¦ =π₯2βπ₯β6
π₯
c) π¦ =π₯+1
(π₯β4)(π₯β3)
d) π¦ =1
π₯+ 5
e) π¦ =π₯β1
π₯2β6π₯+5
f) π¦ =π₯
π₯2+4
83) Graph each function:
a) π¦ =1
π₯2+2π₯β8
b) π¦ =4π₯2
π₯2+2
c) π(π₯) =2π₯
(π₯β3)(π₯+3)
84) Graph each function, and give the coordinates of any holes.
a) π¦ = βπ₯+2
π₯2βπ₯β6
b) π¦ =π₯2β1
π₯β1
c) π¦ =π₯+4
π₯2+4π₯
Chapter 7 - Trigonometry
85) Draw each in standard position, state the reference angle, and give one angle coterminal.
a) 120Β°
b) 50Β°
c) 225Β°
d) 350Β°
e) 271Β°
86) Find the exact length of the line from (0, 0) to (-6, 2), then state all three trigonometric
ratios in exact form.
87) Find the exact length of the line from (0, 0) to (-3, -4), then state all three trigonometric
ratios in exact form.
88) Find each ratio in exact form.
a) sin 45Β°
b) cos 135Β°
c) tan 150Β°
d) sin 300Β°
e) cos 210Β°
f) tan 225Β°
g) cos 270Β°
h) sin 90Β°
i) tan 270Β°
89) Find any solutions 0 β€ π β€ 360Β°
a) cos π =1
β2
b) tan π = ββ3
c) sin π = β1
2
d) tan π = 1
e) cos π = ββ3
2
f) sin π = β1
β2
g) cos π = β1
90) Solve to the nearest tenth for 0 β€ π β€ 360Β°
a) cos π = β12
13
b) tan π =4
7
91) In βπ΄π΅πΆ, π = 7ππ, π = 9ππ, πππ < π΅ = 50Β°. Solve the triangle to the nearest tenth.
92) In βπ·πΈπΉ, π = 7ππ, π = 9ππ, πππ π = 10ππ. Solve the triangle to the nearest tenth.
93) In βπΊπ»πΌ, π = 4 ππ, β = 6 ππ, πππ < πΌ = 85Β°. Solve the triangle to the nearest tenth.