Name: __________________________ Date: _____________
1. Write the first five terms of the sequence. (Assume that n begins with 1.)
an = ( −1) ( n − 2 )( n − 3)
9, –2, 0, 0, 2
–2, 0, 0, 2, –6
–2, 0, 0, –2, –6
0, 0, –2, 6, –12
–2, 0, 2, –4, 6
n
A)
B)
C)
D)
E)
2. Find the indicated term of the sequence.
an = ( −1)
A)
B)
C)
D)
E)
n
( 2n − 5 )
a32 =
–57
59
69
–5
62
Page 1
3. Match the sequence with its graph of the first 10 terms.
A)
B)
C)
Page 2
D)
E)
4. Write an expression for the apparent nth term of the sequence. (Assume that n begins
with 1.)
3, 5, 7, 9, 11
A) an = n + 2
B)
an = ( −1)
C)
an = 2n
D)
an = 2n + 1
an = 2n + 1
E)
n
( 2n + 1)
Page 3
5. Write an expression for the apparent nth term of the sequence. (Assume that n begins
with 1.)
4
4
4
4
4
6 − ,6 − ,6 − ,6 − ,6 − ,
1
2
3
4
5
n
A) an = 6 −
4
6
B) an = 4 −
n
n
C) an = 4 −
6
4
D) an = 6 −
n
4
E) an = 6 −
n +1
6. Write the first five terms of the sequence defined recursively. Use the pattern to write
the nth term of the sequence as a function of n. (Assume that n begins with 1.)
a1 = 3, ak +1 = ak + 7
A) an = –4 + 7 n
B) an = 3 + 7 n
C) an = –11 + 7 n
D) an = –4
E)
an = 3 ( n − 1)
7. Write the first five terms of the sequence defined recursively. Use the pattern to write
the nth term of the sequence as a function of n. (Assume that n begins with 1.)
a1 = 7, ak +1 = –3ak
A) an = 7 n
B)
an = 7 ( –3)
C)
an = ( –21)
n
D)
an = ( –21)
n−1
E)
an = 7 ( –3)
n
n−1
Page 4
8. Write the first five terms of the sequence. (Assume that n begins with 0.)
7n
an =
( n + 1)!
7 49 343 2401
A) 1, , ,
,
2 6 24 120
49 343 2401 16807 16807
B)
,
,
,
,
6 24 120 720
720
49 343 2401 16807
C) 0, ,
,
,
6 24 120 720
49 343 2401 2401
D) 7, ,
,
,
24 120 720 720
49 49 49 49
E) 7, , ,
,
6 24 120 720
9. Simplify the factorial expression.
16!
13!
A) 43,680
B) 3360
16
C)
13
D) 240
E) 57,120
10. Find the sum.
3
1
∑ k2 + 5
k =1
22
A)
63
1
B)
14
C) 1
73
D)
168
12
E)
11
Page 5
11. Use a calculator to find the sum. Round to four decimal places.
12
–2
∑ k +3
k =3
A)
B)
C)
D)
E)
–2.9698
–3.6365
–0.1333
–2.0698
–1.9365
12. Use sigma notation to write the sum.
1
1
1
+
+ +
3⋅ 2 4 ⋅3
9 ⋅8
7
1
A) ∑
n =1 ( n + 1)( n + 2 )
7
B)
n =1
7
C)
1
∑ ( n + 1)( n + 2 )
n =1
6
E)
n
∑ ( n + 2 )!
n =1
5
D)
1
∑ n ( n + 1)
1
∑ ( n + 1)( n + 2 )
n =0
Page 6
13. Find the indicated partial sum of the series.
∞
A)
B)
C)
D)
E)
i
⎛ 1⎞
∑ 2 ⎜⎝ – 2 ⎟⎠
i =1
third partial sum
1
–
4
5
4
3
–
4
1
4
7
–
12
14. Determine whether the sequence is arithmetic. If so, find the common difference.
6, 11, 16, 21, 26
A) 1
B) 5
C) 6
D) –5
E) not arithmetic
15. Determine whether the sequence is arithmetic. If so, find the common difference.
4, 16, 64, 256, 1024
A) 4
B) 4n
C) 4n – 4n–1
D) –4
E) not arithmetic
Page 7
16. Determine whether the sequence is arithmetic. If so, find the common difference.
(Assume that n begins with 1.)
⎛1⎞
an = 7 ⎜ ⎟
⎝5⎠
n
A) 7
B) 5
1
C)
5
D) –7
E) not arithmetic
17. Find a formula for an for the arithmetic sequence.
a1 = –7, d = 6
A) an = –7 + 5n
B) an = –13 + 6n
n−1
C) an = –7 ⋅ ( 6 )
D) an = 6 – 7(n – 1)
E)
⎛1⎞
an = –7 ⋅ ⎜ ⎟
⎝6⎠
n−1
18. Find a formula for an for the arithmetic sequence.
a5 = 25, a15 = 95
A) an = –3 + 7n
B) an = 10 – 3n
C) an = 7 – 3n
D)
an = –3 ( 7 )
E)
an = –10 + 7n
n
19. Write the first five terms of the arithmetic sequence.
a1 = 4, d = 1
A) 4, 5, 6, 7, 8
B) 4, 3, 2, 1, 0
C) 5, 6, 7, 8, 9
D) 4, 4, 4, 4, 4
E) 4, 5, 9, 13, 17
Page 8
20. Write the first five terms of the arithmetic sequence.
a4 = 11, a10 = 35
A) –1, –5, –9, –13, –17
B) 3, 7, 11, 15, 19
C) –1, 3, 7, 11, 15
D) –1, –4, –16, –64, –256
E) –1, 3, 2, 1, 0
21. The first two terms of the arithmetic sequence are given. Find the indicated term.
a1 = –4, a2 = –10, a8 =
A) –52
B) –34
C) –38
D) –58
E) –46
Page 9
22. Match the arithmetic sequence with its graph from the choices below.
A)
B)
C)
Page 10
D)
E)
23. Find the indicated nth partial sum of the arithmetic sequence.
2.9, 5.4, 7.9, 10.4, ..., n = 35
A) 1813
B) 1676.5
C) 1589
D) 1588.6
E) 1589.4
24. Find the sum of the integers from –11 to 14.
A) 105
B) 25
C) 78
D) 39
E) 210
Page 11
25. Find the partial sum.
300
∑ ( 4n + 1)
n =1
A)
B)
C)
D)
E)
180,901
179,699
180,300
182,105
180,900
26. Determine whether the sequence is geometric. If so, find the common ratio.
–3, 6, –12, 24, ...
A) –2
B) –3
1
C) –
2
D) 2
E) not geometric
27. Determine whether the sequence is geometric. If so, find the common ratio.
–2, 2, 6, 10, ...
A) 4
B) –2
1
C)
4
D) –4
E) not geometric
28. Write the first five terms of the geometric sequence.
1
a1 = 1, r =
4
A) 1, 5, 9, 13, 17
B) 1, 4, 16, 64, 256
1 1 1 1
1
C)
, , ,
,
4 16 64 256 1024
1 1 1
D) 1,1, , ,
4 16 64
1 1 1 1
E) 1, , , ,
4 16 64 256
Page 12
29. Write the nth term of the geometric sequence as a function of n.
a1 = –6, ak +1 = 2ak
A)
an = –6 ( 2 )
n−1
B)
an = 2 ( –6 )
n−1
n−1
D)
⎛1⎞
an = –6 ⎜ ⎟
⎝2⎠
an = –8 + 2n
E)
an = –6 ( 2 )
C)
n
30. Find the indicated term of the geometric sequence. Round to the nearest thousandth.
a1 = 3, r = –0.96, n = 9
A) –4.680
B) –2.078
C) 2.164
D) 1.994
E) –2.880
31. Find the indicated nth term of the geometric sequence.
7th term: 2, –6, 18,...
A) –16
B) 1458
C) –4374
D) –192
E) –384
Page 13
32. Find the indicated nth term of the geometric sequence.
3
3
4th term: a2 = , a8 =
2
128
3
A) –
16
2
B)
27
3
C)
32
3
D)
8
3
E) –
4
Page 14
33. Match the geometric sequence with its graph from the choices below.
A)
B)
C)
D)
Page 15
E)
34. Find the sum of the finite geometric sequence.
5
A)
B)
C)
D)
E)
⎛ 1⎞
∑ ⎜⎝ – 5 ⎟⎠
n=1
2604
–
625
1
5
521
104
125
521
625
n−1
35. Find the sum of the finite geometric sequence. Round to the nearest thousandth.
6
∑ 200 (1.06 )
i
i=0
A)
B)
C)
D)
E)
1678.768
1478.768
1183.465
1395.064
1779.494
Page 16
36. Use summation notation to write the sum.
4 – 12 + 36 – … + 2916
6
A)
∑ 4 ( –3)
n =0
5
B)
∑ 4 ( –3)
n=1
7
C)
n −1
∑ 4 ( –3)
n −1
∑ 4 ( –3)
n +1
n =1
6
E)
n
∑ 4 ( –3)
n =1
5
D)
n −1
n =1
37. Find the sum of the infinite geometric series.
∞
⎛ 1⎞
∑ –2 ⎜⎝ – 2 ⎟⎠
n =0
A)
n
4
3
4
3
2
C) –
3
2
D)
3
E) undefined
B)
–
Page 17
38. Find the rational number representation of the repeating decimal.
0.941
941
A)
9999
94.1
B)
999
941
C)
9
941
D)
999
941
E)
99
39. Find Pk+1 for the given Pk.
Pk =
k2
( 5k + 4 )
2
( k + 1)2
( 5k + 9 )
A)
Pk +1 =
B)
Pk +1 =
k2
( 5k + 4 ) + 1
2
C)
Pk +1 =
k2
( 5k + 8)
2
D)
k 2 +1
Pk +1 =
( 5k + 9 )
2
E)
Pk +1
2
2
k + 1)
(
=
2
( 5k + 4 )
40. Use mathematical induction to prove the formula for every positive integer n. Show all
your work.
n
4 + 7 + 10 + 13 + … + ( 3n + 1) = ( 3n + 5 )
2
41. Use mathematical induction to prove the formula for every positive integer n. Show all
your work.
n
n ( n + 1)( 2n + 7 )
∑ i (i + 2) =
6
i =1
Page 18
42. Find the sum using the formulas for the sums of powers of integers.
7
∑ n3
n=1
A)
B)
C)
D)
E)
441
1568
343
140
784
43. Use mathematical induction to prove the property for all positive integers n.
4
⎡ a n ⎤ = a 4n
⎣ ⎦
44. Find the sum using the formulas for the sums of powers of integers.
∑ ( 6 n – 5n 2 )
11
n =1
A)
B)
C)
D)
E)
–1595
–6402
–539
66
–2134
45. Find a quadratic model for the sequence with the indicated terms.
a0 = 4, a1 = 0, a3 = –2
A)
an = n 2 + 4
B)
an = n 2 – 5n + 4
C)
an = n 2 – 4n + 4
D)
an = n 2 – 4n + 5
E)
an = n 2 – n + 4
46. Calculate the binomial coefficient: 10 C7
A) 604,800
B) 70
C) 120
D) 1
E) 0
Page 19
47. Evaluate using Pascal's triangle.
⎛9⎞
⎜ ⎟
⎝ 3⎠
⎛9⎞
A) ⎜ ⎟ = 120
⎝ 3⎠
⎛9⎞
B) ⎜ ⎟ = 84
⎝ 3⎠
⎛9⎞
C) ⎜ ⎟ = 56
⎝ 3⎠
⎛9⎞
D) ⎜ ⎟ = 165
⎝ 3⎠
⎛9⎞
E) ⎜ ⎟ = 35
⎝ 3⎠
48. Use the Binomial Theorem to expand and simplify the expression.
( p + 6 )5
A)
p5 + 30 p 4 + 360 p3 + 2160 p 2 + 6480 p
B)
p 4 + 24 p3 + 216 p 2 + 864 p + 1296
C)
p5 + 30 p 4 + 540 p3 + 3240 p 2 + 6480 p + 7776
D)
p5 + 24 p 4 + 324 p3 + 1944 p 2 + 5184 p + 7776
E)
p5 + 30 p 4 + 360 p3 + 2160 p 2 + 6480 p + 7776
49. Use the Binomial Theorem to expand and simplify the expression.
( 5 x – 4 y )4
A)
625 x 4 – 2000 x3 y + 2400 x 2 y 2 – 1280 xy3 + 256 y 4
B)
625 x 4 – 500 x3 y + 400 x 2 y 2 – 320 xy3 + 256 y 4
C)
625 x 4 + 4 x3 y + 6 x 2 y 2 + 4 xy3 + y 4
D) 125 x 4 – 400 x3 y + 480 x 2 y 2 – 256 xy3 + 256 y 4
E)
125 x3 – 300 x 2 y + 240 xy 2 – 64 y3
Page 20
50. Find the specified nth term in the expansion of the binomial. (Write the expansion in
descending powers of x.)
( x – 2 y )8 , n = 4
A)
70x 4 y 4
B)
–448x5 y3
C)
70x 4 y 4
D)
256 y8
E)
1680x 4 y 4
51. Find the coefficient a of the term in the expansion of the binomial.
Binomial
Term
( x – 5 y )8
ax3 y 5
A) a = 13
B) a = –175, 000
C) a = 24
D) a = 390, 625
E) a = 6720
52. Use the Binomial Theorem to expand and simplify the expression.
( x3 / 4 + 4 )
A)
B)
C)
D)
E)
4
x3 + 256
x12 + 16 x9 + 96 x 6 + 256 x3 + 256
x3 + 16 x9 / 4 + 96 x3 / 2 + 256 x3 / 4 + 256
x3 – 16 x9 / 4 + 96 x3 / 2 – 256 x3 / 4 + 256
x12 – 16 x9 + 96 x 6 – 256 x3 + 256
53. Use the Binomial Theorem to expand the complex number. Simplify your result.
( 3 + 2i )4
A)
B)
C)
D)
E)
–119 + 120i
–119 – 120i
119 + 120i
119 – 120i
81
Page 21
54. Find the coordinates of the point located five units behind the yz-plane, four units to the
left of the xz-plane, and seven units above the xy-plane.
A) ( –5, 7, –4 )
B)
C)
D)
E)
( –5, –4, 7 )
( –5, 4, 7 )
( –5, 7, 4 )
( –5, –9, –2 )
55. Find the distance between the points.
( 0, 0, 0 ) , ( 4,5, –1)
A) 8
B) 10
C) 2 2
D) 2 42
42
E)
56. Find the lengths of the sides of the right triangle whose vertices are located at the given
points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your
work.
( 3, –5, –5) , (1,3, –6 ) , ( –6, 2, 0 )
57. Find the midpoint of the line segment joining the points.
( 3, –9,5) , ( 2, –4, 0 )
A)
B)
C)
D)
E)
⎛ 1 –5 5 ⎞
⎜ , , ⎟
⎝2 2 2⎠
⎛ 5 –13 5 ⎞
, ⎟
⎜ ,
⎝2 2 2⎠
( 5, –13,5)
(1, –5,5)
( 6,36, 0 )
Page 22
58. Find the standard form of the equation of the sphere with the given characteristics.
Center: ( 3, –3, –5) ; radius: 9
C)
( x + 3)2 + ( y – 3)2 + ( z – 5)2 = 81
( x + 3) 2 + ( y – 3) 2 + ( z – 5 ) 2 = 9
( x – 3)2 + ( y + 3)2 + ( z + 5)2 = 81
D)
x 2 + y 2 + z 2 = 38
E)
( x – 3)2 + ( y – 3)2 + ( z – 5)2 = 81
A)
B)
59. Find the standard form of the equation of the sphere with the given characteristics.
Endpoints of a diameter: ( –7, 2,3) , ( –1, 4,1)
A)
B)
C)
D)
E)
( x + 8)2 + ( x – 6 )2 + ( x – 4 )2 = 11
( x + 4 )2 + ( x – 3)2 + ( x – 2 )2 = 22
( x + 4 )2 + ( x – 3)2 + ( x – 2 )2 = 44
( x + 4 )2 + ( x – 3)2 + ( x – 2 )2 = 11
( x – 4 )2 + ( x + 3)2 + ( x + 2 )2 = 11
60. Find the center and radius of the sphere.
x 2 + y 2 + z 2 + 8 x + 12 y – 2 z + 37 = 0
A) center: ( 4, 6, –1) ; radius: 4
B)
center: ( –4, –6,1) ; radius: 4
C)
center: ( –4, 6, –1) ; radius: 4
D) center: ( 4, 6, –1) ; radius: 16
E)
center: ( –4, –6,1) ; radius: 16
Page 23
61. Write the component form of the vector described below.
Initial point: ( 3,5, –3)
Terminal point: ( 2, 0, –1)
A)
–1, –5, 2
B)
1,5, –2
C)
2, 0, –1
D)
–1, –5, –4
E)
6, 0,3
62. Find the magnitude of the vector described below.
Initial point: ( –5, 6,3)
Terminal point: ( 8,3,1)
A) 8
B) 19
C) 3 2
D) 2 182
182
E)
63. Find a unit vector in the direction of the vector described below.
Initial point: ( –2,5,5)
Terminal point: ( 5, –5,1)
A)
21 7, –10, –4
B)
165 7, –10, –4
C)
D)
E)
1
7, –10, –4
21
7, –10, –4
1
7, –10, –4
165
Page 24
64. Find the vector z, given u = 7, –3,8 and v = 6, –7,5 .
z = 4u + 5v
A) 34, –19,37
B)
58, –47,57
C)
13, –10,13
D)
59, –43, 60
E)
–2, 23, 7
65. Find the vector z, given u = –7, –2, 2 , v = –3, –6, –2 , and w = –2, –4, –33 .
–5u + 4 v – 5z = w
A) –25,10, –15
B)
–2,5,3
C)
–5, 2,3
D)
5, –2,3
E)
6, –2, 4
66. Find the magnitude of the vector v described below.
Initial point: (1,5, –7 )
Terminal point: ( –4, 7, –2 )
A) 0
B) 12
C) 2 3
D) 6 6
E) 3 6
Page 25
67. Find a unit vector in the direction of u.
u = 5,8, –1
A)
B)
C)
D)
E)
14 5,8, –1
1
–5, –8,1
3 10
1
5,8, –1
14
5,8, –1
1
5,8, –1
3 10
68. Find a unit vector in the opposite direction of u.
u = 4, –15,8
A)
B)
C)
D)
E)
3 3 4, –15,8
1
–4,15, –8
305
1
–4,15, –8
3 3
–4,15, –8
1
4, –15,8
305
69. Find the dot product of u and v.
u = 3, –1,5 , v = 3,1,8
A) 48
B) 19
C) 9, –1, 40
D) –32
6, 0,13
E)
Page 26
70. Find the angle between the vectors u and v. Express your answer in degrees and round
to the nearest tenth of a degree.
u = 6i – j – 9k , v = –9i – 2 j + 9k
A) 43.5°
B) 71.9°
C) 46.5°
D) 90°
E) 161.9°
71. Determine whether u and v are parallel, orthogonal, or neither.
u = –5, –9, 7 , v = –25, –45,35
A) parallel
B) orthogonal
C) neither
72. Determine whether u and v are parallel, orthogonal, or neither.
u = 3, –1,10 , v = –4, –2,1
A) parallel
B) orthogonal
C) neither
73. Use vectors to determine whether the points are collinear.
( –9, 7, –2 ) , ( –13,10, –7 ) , ( 7, –5,18)
A) collinear
B) not collinear
74. Use vectors to determine whether the points are collinear.
( 7, –3, 0 ) , ( 2, –8,5) , (11, –5, –9 )
A) collinear
B) not collinear
Page 27
75. The vector v and its initial point are given. Find the terminal point.
v = 4, –1, 2
Initial point: ( –4, –5, 0 )
A)
B)
C)
D)
E)
( 0, –6, 2 )
(8, 4, –2 )
( 4, –1, 0 )
( –16,5, 0 )
( –8, –4, 2 )
76. Determine the values of c such that cu = 4 , where u = 6i + 4 j – 4k .
A) c = ±4
2 17
B) c = ±
17
17
C) c = ±
2
1
D) c = ±
4
17
E) c = ±
34
Page 28
77. The weight of a crate is 200 newtons. Find the tension in each of the supporting cables
shown in the figure. The coordinates of the points A, B, C, and D are given below the
figure. Round to the nearest newton.
z
D
C
B
y
x
A
[Figure not necessarily to scale.]
point A = ( 0, 0, –120 ) , point B = (180, 0, 0 ) , point C = ( –30,50, 0 ) , point D = ( 0, –170, 0 )
A) cable AB = 152; cable AC = 41; cable AD = 70
B) cable AB = 41; cable AC = 70; cable AD = 152
C) cable AB = 41; cable AC = 152; cable AD = 70
D) cable AB = 70; cable AC = 152; cable AD = 41
E) cable AB = 152; cable AC = 70; cable AD = 41
78. Find u × v .
u = 0, –6,1 , v = –2, 6, –4
A) 18, –2, –12
B) –40
C) 32
D) 18, 2, –12
E)
0, –36, –4
79. Find u × v .
u = –9i – 3 j + 5k , v = –6i + j + k
A) –8i – 21j – 27k
B) 56
C) 62
D) –8i + 21j – 27k
E) 54i – 3 j + 5k
Page 29
80. Find the area of the parallelogram that has the vectors as adjacent sides.
u = 3i – j – 3k , v = –5i + 3 j – 2k
A) 18
2
B)
C) 2 3
D) 4
E) 17 2
81. Find the area of the triangle with the given vertices.
( –5, 4, 0 ) , ( –5, 2,3) , ( –8, 7,3)
A) 0
B) 3
3 38
C)
2
3 38
D)
4
E) 3 38
82. Find the triple scalar product u ⋅ ( v × w ) for the vectors
A)
B)
C)
D)
E)
u = –2, 7,8 , v = –4,8,5 , w = 1, –4, –4
–376
11
–11
165
0
Page 30
83. Find the torque on the crankshaft V using the data shown in the figure. Round to the
nearest tenth of a foot-pound.
θ
V
F
V = 2.2 ft
F = 40 lb
θ = 40°
A)
B)
C)
D)
E)
56.6 ft-lb
67.4 ft-lb
73.8 ft-lb
88.0 ft-lb
0
84. Find a set of parametric equations for the line through the point and parallel to the
specified vector. Show all your work.
( –9, –9, –6 ) , parallel to 6, –3,1
85. Find a set of parametric equations for the line through the point and parallel to the
specified line. Show all your work.
x = 7 + 3t
( 4, 2,1) , parallel to
y = –1 + 4t
z = 3+t
86. Find symmetric equations for the line through the point and parallel to the specified
vector. Show all your work.
( 5, –5, –5) , parallel to –3, –2, –2
87. Find symmetric equations for the line through the point and parallel to the specified
line. Show all your work.
x = –5 + 5t
( 4, –3,9 ) , parallel to
y = 2 – 3t
z = 2 – 5t
Page 31
88. Find a set of parametric equations for the line that passes through the given points.
Show all your work.
( 5,3,1) , ( –5, –9,5)
89. Find a set of parametric equations for the line that passes through the given points.
Show all your work.
⎛ –7 –1 ⎞ ⎛ 9 –5 ⎞
⎜ 4, , ⎟ , ⎜ , , 6 ⎟
⎝ 2 2 ⎠ ⎝2 2 ⎠
90. Find the general form of the equation of the plane passing through the point and
perpendicular to the specified vector. [Be sure to reduce the coefficients in your answer
to lowest terms by dividing out any common factor.]
( 6,8, –5) , n = – i + 6 j + k
A) x – 6 y – z = 0
B) x – 6 y – z – 37 = 0
C) x – 6 y – z + 37 = 0
D) 6 x + 8 y – 5 z + 37 = 0
E) 6 x + 8 y – 5 z – 37 = 0
91. Find the general form of the equation of the plane passing through the point and
perpendicular to the specified line. [Be sure to reduce the coefficients in your answer to
lowest terms by dividing out any common factor.]
x = –2 – t
( –7, –6, –2 ) , y = 4 + 6t
A)
B)
C)
D)
E)
z = 1 – 5t
x – 6 y + 5z = 0
x – 6 y + 5 z + 19 = 0
x – 6 y + 5 z – 19 = 0
7 x + 6 y + 2 z + 19 = 0
7 x + 6 y + 2 z – 19 = 0
Page 32
92. Find the general form of the equation of the plane passing through the three points. [Be
sure to reduce the coefficients in your answer to lowest terms by dividing out any
common factor.]
( –6, 6,3) , ( –1,5, –2 ) , ( 6, 6, 6 )
A) x + 25 y – 4 z – 132 = 0
B) x + 25 y – 4 z = 0
C) 2 x – 2 y – z = 0
D) x + 25 y – 4 z + 132 = 0
E) 2 x – 2 y – z + 132 = 0
93. Find the angle of intersection of the planes in degrees. Round to a tenth of a degree.
4x + y – z = 5
– x + y + 4 z = –3
A) 8.4°
B) 22.9°
C) 112.9°
D) 21.3°
E) 2.0°
94. Find parametric equations for the line of intersection of the two planes.
5 x + 4 y + z = –21
–4 x + 3 y + z = 30
95. Find the distance between the point and the plane.
( –4,5, –3)
A)
B)
C)
D)
E)
4 x – 2 y – 5 z = –5
6
2
15
16
3 5
0
2
5
Page 33
Answer Key
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
B
B
C
E
D
A
E
A
B
A
D
A
C
B
E
E
B
E
A
C
E
C
C
D
E
A
E
E
A
C
B
D
E
E
A
C
B
D
A
Page 34
40. 1) When n = 1, S1 = 4 =
1
( 3 + 5) . The formula is valid for n = 1.
2
2) Assume that 4 + 7 + 10 + 13 + … + ( 3k + 1) =
Sk +1 = 4 + 7 + 10 + 13 + … + ( 3 ( k + 1) + 1) =
(
(
)
k
( 3k + 5) is true.
2
k
( 3k + 5) + 3k + 4
2
1
3k 2 + 5k + 3k + 4
2
1
= 3k 2 + 11k + 8
2
k +1
=
( 3 ( k + 1) + 5)
2
=
)
By mathematical induction, the formula is true for all positive integers n.
41. 1) When n = 1,
(1)(1 + 1) ( 2 (1) + 7 )
18
S1 = (1)(1 + 2 ) =
or 3 = = 3 .
6
6
The formula is valid for n = 1.
k
2) Assume that
∑ i (i + 2) =
i =1
k +1
k
i =1
i =1
k ( k + 1)( 2k + 7 )
is true.
6
∑ i ( i + 2 ) = ∑ i ( i + 2 ) + ( k + 1)( k + 3) =
(
(
k ( k + 1)( 2k + 7 )
+ ( k + 1)( k + 3)
6
)
k +1
2k 2 + 7 k + 6k + 18
6
k +1
=
2k 2 + 13k + 18
6
k +1
=
( k + 2 )( 2k + 9 )
6
( k + 1)( k + 2 ) ( 2 ( k + 1) + 7 )
=
6
=
)
By mathematical induction, the formula is true for all positive integers n.
42. E
Page 35
43. 1) n = 1:
4?
⋅
⎡ a1 ⎤ = a 41
⎣ ⎦
?
[ a ]4 = a 4
a4 = a4
The statement is true for n = 1.
4
2) Assume ⎡ a k ⎤ = a 4k . Then,
⎣ ⎦
( a )4 ⎡⎣ a k ⎤⎦
4
= (a)
4
( a 4k )
4
⎡ a ⋅ a k ⎤ = a 4+ 4 k
⎣
⎦
⎡ a k +1 ⎤ = a 4( k +1)
⎣
⎦
By mathematical induction, the property is true for all positive values of n.
E
B
C
B
E
A
B
B
C
A
B
E
Let point P = ( 3, –5, –5) , point Q = (1,3, –6 ) , and point R = ( –6, 2, 0 ) .
4
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
2
PQ = (1 – 3) + ( 3 + 5 ) + ( –6 + 5 ) = 69 ; ⇒ PQ = 69
2
2
2
2
2
2
2
QR = ( –6 – 1) + ( 2 – 3) + ( 0 + 6 ) = 86 ; ⇒ QR = 86
2
2
PR = ( –6 – 3) + ( 2 + 5 ) + ( 0 + 5 ) = 155 ; ⇒ PR = 155
57.
58.
59.
60.
61.
62.
63.
64.
2
2
Note that 69 + 86 = 155 in accordance with the Pythagorean Theorem.
B
C
D
B
A
E
E
B
Page 36
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
D
E
E
B
A
E
A
B
A
B
A
B
C
A
A
E
C
B
A
Answers may vary. One possible answer is shown below.
x = –9 + 6t , y = –9 – 3t , z = –6 + t
Answers may vary. One possible answer is shown below.
x = 4 + 3t , y = 2 + 4t , z = 1 + t
Answers may vary. One possible answer is shown below.
x –5 y +5 z +5
=
=
–3
–2
–2
Answers may vary. One possible answer is shown below.
x – 4 y +3 z –9
=
=
5
–3
–5
Answers may vary. One possible answer is shown below.
x = 5 – 10t , y = 3 – 12t , z = 1 + 4t
Answers may vary. One possible answer is shown below.
1
7
1 13
x = 4 + t, y = – + t, z = – + t
2
2
2 2
C
C
A
C
Answers may vary. One possible answer is shown below.
x = –6 + t
y = 3 – 9t
z = –3 + 31t
95. E
Page 37