Exam
Name
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Match the point in polar coordinates with either A, B, C, or D on the graph.
"H
1)
/' AA 3 " 1 "
/ ,,-£•••
1- •
-»
1
I
1
h-
- 5 - 4 - 3 - 2 -1,
-\
A
,2 3
4 5
-*C
--4- •
B)B
A) A
C)D
D)C
2, -3,-f
<-»
2)
1
1
I
f-
-5 -4 -3 -2 -lx
~" '
•D
H
A
1
1
1
h
, 2 3 4 5
-3--
=5- -
A)D
B)B
C)C
D)A
The polar coordinates of a point are given. Find the rectangular coordinates of the point.
3>|3,
A)
3)
4.
4) (-3, -135°)
» -f
2
c)rfM)
4)
17) y = 5
A) r sin 6 = 5
17)
B)r = 5
C) sin 6 cos 0 = 5
D) r cos 6 = 5
Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation.
18) r = 6sin6
18)
6- • ........
..3
.2--
/ ; / / / V
< I I I I I I
I I I I I I
-24-- i/2 3 4 5 6
/-2
-3
'.//'/ ''
-' ,/
/ ,
20) r sine =4
19) r = 2 cos9
5- •
.3- ••
< i i i i ir
i i i Ii i >
C I I I I I I
-2'-3
x
\
X \•
"-*5"••-=6"
22) r sec 6 = -6
21) .5--
1
I I I I
-5--
5
M i l l !
-6 -5 -4 -3 -2 -it- .-'I Ji ,3 4 5 6 r
r
'-3-" -4' '
' -5- •
26) r = 4 sin(29)
27) r2 = 4cos(29)
-54'
.,.-'•••••0" "•
-' ^ , •
.4--
X
,2- •-.
/ / / /"',2---,s '\
M i l l
I I I I I I
-6-4
- 2 ..
. '• '..
-2-
- 5 - 4 - 3 - 2 -ill i 5 3 4 5 r
2
4
^4. .
Graph the polar equation.
28) r = 2-2cos9
5-4-. ...3- •
,,2->1- •
<—I
I I I
-5 -4 -3 -2 -lj. . 1 2 3 4 5 r
-3--" ' ^ x - /
"-4- •
'' ,,' '
Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary.
29)l+<x/3i
29)
A)2(cos60° + isin60°)
B) 4(cos 60° + i sin 60°)
C) 2(cos 30° + i sin 30°)
D) 4(cos 30° + i sin 30°)
30) -5i
30)
A) 5(cos 0° + i sin 0°)
B)5(cos90° + isin90°)
C)5(cos270° + isin270°)
D) 5(cos 180° + i sin 180°)
17
6 r
Use the vectors in the figure below to graph the following vector.
..... !
! 1
l j_...
1...
,— i—
*'"'T
—;.... i "*J —
..».
—
—
I
-k „.„*r
\.
T""
!
NJ
!
y<
-4-4- .....L. i
• y • • *• -<.
i
i
+
-"! f™ ™ t— »"—t—••
......
TV- &H-H
•ys-p
.....
~T-t 1
».....».....»
*...««.....<
1 1 ]
42) u + z
.......—
~-y»-
=l+:
,....,
"•••t
1
7T
"/
...
.„..
I
....
—•'••••• .....
42)
„... .....
*
...1..U
!..„„
pi
i
"-}--
~H
L..1
....1
;
4.4...]
T-T~T
JLJLJ.
i
._..i.....
.....^ .....«»...
.»..«..«•.
""]
I., i
J
43) 3w
43)
TI
*•"••?
II
=F
"T
.....fa
-f-f-
I {.
^L...
,
,.-..
i
i
i
:
..44.4..,,.4.4...
t
t
;
:
:
;
:
:
^"
n:l:LLil:LiTi:ri
49) Ifv = 6i + 8j / findH|.
A) 14
49)
C)100
D)10
50) If v = 9i + 5j, what is || 4v|| ?
A) -4-V/106
B) Siv/14
50)
D) 4^/106
Find the quantity if v = 5i - 7j and w = 3i + 2j.
51)
51) || v + HI
A)2<x/6+<v/13
Find the unit vector having the same direction as v.
52) v = -4j
A) u = - ^-j
52)
C)u = -4j
B) u = -j
D) u = 16j
53) v = -3i-4j
5.
53)
5.
3.4.
C)u = -15i-20j
Solve the problem.
54) If P = (-4, -1) and Q = (\, -49), find all numbers x such that the vector represented by PQ has
length -80.
A) (68,60}
B){60,-68}
C){60,-72}
D){68,-68}
54)
Write the vector v in the form ai + bj, given its magnitude || v)| and the angle a it makes with the positive x-axis.
55) |H| =13, a = 120°
55)
13. 13-S/3.1
13. 13^/3.'
A)v = 13 " o
B)v = 13
o J
2 'J
1+
C)v=13
^
56) || vf =7, a = 210°
A)v = 7
C)v = 7
13.
D)v = 13
56)
7.
B)v = 7
"I" 2
7.
1.
D)v = 7
——i —-—i
2
2J
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
57) A truck pushes a load of 45 tons up a hill with an inclination of 35°. Express the force
vector F in terms of i and j. Round the components of F to two decimal places.
58) Two forces, Fj of magnitude 35 newtons (N) and f2 °l magnitude 55 newtons, act on an
object at angles of 45° and -60° (respectively) with the positive x-axis. Find the direction
and magnitude of the resultant force; that is, find Fj + ^2- Round the direction and
magnitude to two decimal places.
24
57)
58)
67) v = 4i + 2j,
w = 2i-4j
67)
A) Orthogonal
B) Parallel
C) Neither
Solve the problem.
68) Find a so that the vectors v = i + aj and w = 2i - 4j are orthogonal.
A)-|
B)2
C)-2
68)
D)|
Decompose v into two vectors v\d V2/ where vj is parallel to w and \ is orthogonal to w.
69) v = i + 2j,
w = -3i + j
.
3 . 1 .
11. 21.
1. 1.
'
2. 19.
69)
Solve the problem. Round your answer to the nearest tenth.
70) Find the work done by a force of 4 pounds acting in the direction of 36° to the horizontal in moving
an object 5 feet from (0, 0) to (5, 0).
A)16.2ft-lb
B)32.4ft-lb
C) 17.1 ft-lb
D)11.8ft-lb
70)
71) A wagon is pulled horizontally by exerting a force of 60 pounds on the handle at an angle of 25° to
the horizontal. How much work is done in moving the wagon 50 feet?" .
A) 2718.9 ft-lb
B) 1617.4 ft-lb
C) 1267.9
ft-lb
D) 2110.8 ft-lb
71)
72) Find the work done by a force of 200 pounds acting in the direction -i + 2j in moving an object 75
feet from (0,0) to (-75,0).
A) 13,416.1
ft-lb
6)6708.2
ft-lb
C) 15,000.0 ft-lb
D) 8944.9 ft-lb
72)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
73) An airplane has an air speed of 550 miles per hour bearing N30°W. The wind velocity is 50
kilometers per hour in the direction N30°E. Find the resultant vector (with exact
components) representing the path of the plane relative to the ground. To the nearest tenth,
what is the ground speed of the plane? What is its direction?
73)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
74) A plane is headed due south with an airspeed of 210 miles per hour. A wind from a direction of S30 74)
°W is blowing at 20 miles per hour. Find the groundspeed and resulting direction of the plane,
rounded to the nearest whole number.
A) 193 mph; S3°E
B) 193 mph; S3°W
C) 201 mph; S5°E
D) 211 mph; S15°W
75) An SUV weighing 4600 pounds is parked on a street which has an incline of 8°. Find the force
required to keep the SUV from rolling down the hill and the force of the SUV perpendicular to the
hill. Round the forces to the nearest hundredth.
A) 320.10 Ib and 2277.62 Ib
B) 1034.77 Ib and 4482.10 Ib
C) 640.20 Ib and 4555.23 Ib
D) 240.75 Ib and 4593.70 Ib
26
75)
Use the given vectors to find the requested expression.
86) v = -3i + 5j + k, w = -4i-2j-k
Find (-5v) x w.
A) -15i - 35j + 130k
86)
B)-15i + 17j - 94k
C) 27i + llj - 10k
D) 15i + 35j - 130k
87) v = 3i-2j-3k, w = 5i-4j-2k, u = -4i-5j+ 2k
Find u. (v x w).
A) 32
87)
B)36
C)-35
Find the requested vector.
88) v = -3i + 2j + k, w = 5i + 5j-k
Find a vector orthogonal to both v and w.
A)-7i + 2j-25k
B)-15i + 13j - 10k
D) 73
88)
C)7i-2j + 2k
D)-9i-9j + 21k
Find the area of the parallelogram.
89) One corner at PI and adjacent sides PjP2 and PiPs
Pi (1,2,0), P2 (-2,2,4), P3 (0, -2,3)
A)<v/26
B)Sx/17
89)
C)5
90) One corner at PI and adjacent sides PjP2 and Pi?3
Pi (-1,0,1), P2 (2,1, -1), P3 (3, -1,2)
28
D) 14
90)
Find the distance between the two points.
76) PI = (-1, 4, -3) and P2 = (3, 0, -4)
A)-v/33
76)
B)<v/17
Find the position vector for the vector having initial point P and terminal point Q.
77) P = (-4, -3, 0) and Q = (3, 4, -2)
A)v = 2i + 7j + 3k
B)v = 7i + 7j-2k
C) v = -7i-7j + 2k
D)v = 3i + 7j + 6k
Perform the operation.
78) v = 5i - 6j - 5k Find || v|| .
A) 2-7l4
'
D)<N/86
B)^
77)
C)6
79) v = -6i-2j-3k and w = 5i-4j-3k Find|v-wfl.
B)5<s/5
78)
79)
C)7
Find the dot product v • w.
80) v = 3i + j + 2kandw = i + 3j-2k
A) -4
B)-2
80)
C)5
Find the angle between v and w. Round to one decimal place, if necessary.
81) v = 3i + j + 2kandw = i + 2j-2k
A) 5.1°
B)87.9°
C)84.9°
D)2
81)
D) 86.6°
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
82) A sail on a sailboat is supported by a mast and a boom. The length and direction of the
mast and boom can be described by the vectors m = i - 4j + 5k and b = -2i + 3j + k,
respectively, where the components are in meters. Find the angle (to the nearest tenth of a
degree) between the mast and boom.
82)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the direction angles of each vector. Round to the nearest degree, if necessary.
83) v = -6i + 12j + 4k
A) a = 118°, p = 19°, Y = 72°
B) a = 115°, p = 31°, y = 73°
C)a = 920,P = 860,Y = 89°
D) a = 117°,P = 23°,Y = 720
Find the indicated cross product.
84) v = 6i - 3j + k, w = -2i - 6j - k
Find v x w.
A)-42i-9j-4k
B)9i + 4j-42k
85) v = -3i + 2j + 4k,
Find w x v.
A)39i + 18j + 6k
83)
84)
C)-3i + 8j-30k
D)-9i-4j-30k
w = 6i-2j + 5k
85)
B) -18i - 42j - 6k
C) -18i - 39j + 6k
27
D)6i + 18j-18k
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
59) Two forces of magnitude 25 pounds and 40 pounds act on an object. The force of 40 Ib acts along
the positive x-axis, and the force of 25 Ib acts at an angle of 80° with the positive x-axis. Find the
direction and magnitude of the resultant force. Round the direction and magnitude to the nearest
whole number.
A) Direction: 40°; magnitude: 47 Ib
B) Direction: 4°; magnitude: 65 Ib
C) Direction: 51°; magnitude: 51 Ib
D) Direction: 29°; magnitude: 51 Ib
59)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
60) A box of supplies that weighs 1500 kilograms is suspended by two cables as shown in the
figure. To two decimal places, what is the tension in the two cables?
60)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the dot product v • w.
61) v = 7i + 9j,
w = -5i - 6j
A) 19
B)-89
62) v = -3i,
w = -3j
A) 3^
61)
Q-35
D) -54
62)
B)-6
C)0
D)9
Find the angle between v and w. Round your answer to one decimal place, if necessary.
63) v = -5i + 7j,
w = -6i - 4j
A) 110.8°
B)20.7°
C)90.9°
D) 88.2°
64) v = -3i,
w =j
A) 270°
63)
64)
B)0°
C)90°
Solve Jhe problem.
65) Which of the following vectors is parallel to v = -lOi - 8j?
A)w = 20i + 16j
B)w = -20i + 25j
C)w = 4i + 4j
State whether the vectors are parallel, orthogonal, or neither.
66) v = 4i + j,
w = i-4j
A) Orthogonal
B) Parallel
25
D) 180°
65)
D)w = 3i-5j
66)
C) Neither
44) 2u - z - w
44)
i ! i I<
.I...! I...1 1...... j !
i ! 1 ] i |
M i l
[ ! f 1 "—
> * t t <
t t t t • r! iI
i ! !1
!i 1i ii ii «< ; j
i i j.....j
i i ! ! !
MM
1 j
> * * ••••••» * *•••••* * * « » <
ii
i i i i i
! i ! i i I i
* * t t ? f t ? ? * t * 1
j t t * *
* t * *•••••« * <
i i.....i i i
i j i i i i •
I i i i i
i i ! i i
f-4 * i ! * 1
i
i
! ! i
i !
rKn i n i ! r
* r\ i * i ! r in
1 1...1 ! !.....! 1.....I I
i i i I i I i ! i j
I
"*
.i i i u i i
!"
1 1 !.....! 1 1 1
! ! 14 1 1 ! \;
j..
! I 1
f n ri I
tttttttd
1 1 1 I.../ i i i i i i
i\/\ ! ] ] ] I
....i... .1U U
••f t t t f
4 i i ii
., ! ! I I
U !I
4 ! !!
i i ii
! i I..J.....M
i |....|3--.-|44 j I--! !•| |—*""•*—?—f f—i—f""i—f~ i
..... 1.....1...........i..........
1.....1 .....
I
The vector v has initial position P and terminal point Q. Write v in the form ai + bj; that is, find its position vector.
45) P = (6,6);Q = (-4,-l)
45)
A)v = -7i-10j
B)v = 7i + 10j
C)v = 10i + 7j
D)v = -10i46) P = (2,l);Q = (4,-2)
A) v = -4i + 3j
46)
B) v = 3i - 4j
C)v = -3i + 2j
D)v = 2i-3j
Solve the problem.
47) If u = -4i - 2j and v = 8i + 7j, find u - v.
A) -12i - 9j
47)
B) -14i + 5j
C)4i + 5j
D) -13i + 5j
48) Ifv = 3i-5j andw = -7i + 4j,find3v-4w.
48)
C) -19i + j
23
D) 37i - 31j
Plot the complex number in the complex plane.
31) -2-3i
31)
6-4..
2-
I I I I I I
-6
-4
-2
2
4
6R
-2-.4. .
-6--
32) 3i
32)
- 6 - 4 - 2 1
I I I I I I >
2
4
6R
-2- •
-4'
-6
Solve the problem. Leave your answer in polar form.
- 33) z = 10(cos 30° + i sin 30°)
w = 5(cos 10° + i sin Iff")
Find zw.
33)
A) 15(cos 40° + i sin 40°)
C) 50(cos 300° + i sin 300°)
B)15(cos300° + isin300°)
D)50(cos400 + isin40°)
19
Use a graphing utility to graph the polar equation.
23) r = 3 +sin 9
-6
-4
-2
2
1-1
4
6
x
-2--4..
-6--
Identify and graph the polar equation.
24) r = 6 + 5sine
12
10
.&•
6+
4+
<
I I I I I (
-6-4-^1 2 4 f> 8 10 |2 r
-4-f
-104-124-
25) r = 4 + 5 sin 6
5)
5) Uf
A)
2 ' 2
B)
2 '
2
The letters r and 6 represent polar coordinates. Write the equation using rectangular coordinates (x, y).
6) r = 1 + 2sin0
A) ^x2 + y2 = x2 + y2 + 2y
B) A/x2 + y2 = x2 + y2 + 2x
C) x2 + y2 = Vx2 + y2 + 2y
D) x2 + y2 = Vx2 + y2 + 2x
7) r = 10 sin 9
A) X2 + y2 = 10x
8) i = 5
A) X2 + y2 = 25
9) r sin 0 = 10
A ) x = lOy
7)
B) x2 + y2 = lOy
C) A/x2 + y2 = 10x
D) Vx2 + y2 = 10y
8)
B) x + y = 5
C) x + y = 25
D) x2 - y2 = 25
9)
B) x = 10
C) y = lOx
D) y = 10
The rectangular coordinates of a point are given. Find polar coordinates for the point.
10) (-8,0)
•HP]
A) 18,^-1
1 1 )(
,
B)(-8,n)
10)
0|8,f
11)
-
C)|2,f
D) 2,-.
12) (-4,4)
A)-*>/2,~
6)
I2)
"4
3n]
C)L/2,
D)k/2,-21
The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, 6).
13) x2 + 4y2 = 4
A) cos2 0 + 4 sin2 0 = 4r
B) 4 cos2 0 + sin2 0 = 4r
C) r2(cos2 0 + 4 sin2 0) = 4
D) r2(4 cos2 0 + sin2 0) = 4
14)
13)
14)
A)rcos 2 0 = 4sin0
B) rsin2 0 = 4cos0
C)r = 4cos0
D)r = 4sin0
15)x2 = 4y
A)4cos 2 0 = rsin0
16) xy = 1
A) r sin 20 = 2
C) r2 sin 20 = 2
15)
B) r cos2 0 = 4sin0
C)rsin 2 0 = 4cos0
D)4sin 2 0 = rcos6
16)
B) 2r2 sin 0 cos 0 = 1
D) 2r sin 0 cos 0 = 1
Answer Key
Testname: PRE CALC CHAPTER 8 PRACTICE
1) C
2) D
3) A
4) A
5) B
6) C
7) B
8) A
51) B
52) B
53) D
54) B
55) A
56) C
57) 36.861 + 25.81J
58) Direction: -23.65°; magnitude: 57.04 N
9) D
59) D
60) Tension in right cable: 1098.08 kg; tension in left cable: 776.46 kg
61) B
62) C
63) D
64) C
65) A
66) A
67) A
68) D
69) C
70) A
71) A
72) B
73) -250i + 300^j; 576.6 mph; N25.7°W
74) A
75) C
76) A
77) B
78) D
79) B
80) D
81) C
82) 6 « 111.8°
83) B
84) B
85) C
86) D
87) D
88) A
89) B
90) A
10) D
11) A
12) C
13) C
14) C
15) B
16) C
17) A
18) A
19) B
20) D
21) A
22) B
23) C
24) A
25) B
26) D
27) B
28) D
29) A
30) C
31) C
32) D
33) D
34) A
35) A
36) B
37) C
38) B '
39) C
40) D
41) A
42) A
43) A
44) D
45) D
46) D
47) A
48) D
49) D
50) D
30

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