Name: ________________________ Rec/Sec: ____________ 1 Review Test4 Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Find the equivalent expression. tan 4 x sec 4 x a. cot 2 x csc 2 x b. csc 2 x cot 2 x c. csc 2 x tan 2 x d. tan 2 x sec 2 x e. sec 2 x tan 2 x ____ 2. Simplify the following trigonometric expression. sec 2 x 1 sec 2 x a. 1 b. sin x c. sin 2 x d. sec 2 x ____ 3. Simplify the following trigonometric expression. sin(z) + cos(–z) + sin(–z) a. sin z b. cos z c. 2sin z – cos z d. 2sin z ____ 4. Simplify the following trigonometric expression as much as possible. csc x sin x csc x a. sin x b. cos 2 x c. sin 2 x d. csc x ____ 5. Simplify the following trigonometric expression as much as possible. sin 2 t + cos 2 t + tan 2 t a. tan x b. sec 2 x c. sec x d. tan 2 x Instructor: Koshal Dahal Test 4 date: Fri, May 1 (Math1650:500)
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Review Test4 (Math1650:500) Instructor: Koshal Dahalmath.unt.edu/~koshal/SYLLABUS/Math1650_Spring_015/Test4 Review.pdf1 cosx sinx sinx 1 cosx a. 2 sin x b. cos x c. sin x d. 2 csc
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____ 23. Write the following expression in terms of sine only.
sin z + cos z
a. 3 sin6 z
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
b. 2 sin4 z
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
c. 3 sin6 z
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
d. 2 sin z 4
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
____ 24. Write the following expression in terms of sine only.
5sinx 5 3 cosx
a. 5sin x 3
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
b. 5sin x 3
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
c. 10sin x 3
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
d. 10sin x 3
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
Name: ________________________ ID: A
7
____ 25. Rewrite the expression as an algebraic expression in x.
tan (sin – 1 x)
a.1
x2 1
b.x
1 x2
c. 1 x2
d. x2 1
____ 26. Rewrite the expression as an algebraic expression in x.
sin (cos – 1 x)
a. x2 1b. x – 1
c. 1 x2
d. 1 – x
e. x
____ 27. Find the exact value of the expression.
cos sin1 32
Ê
Ë
ÁÁÁÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃˜̃̃
a.2
b.12
c.2
2
d.3
2
Name: ________________________ ID: A
8
____ 28. Simplify the expression.
sin14xsin13x sinx
a.sin6xsin7x
b.cos 6xcos 7x
c.sin13xsin6x
d.sin7xsin6x
e.cos 7xcos 6x
____ 29. Find all solutions of the equation.
2cos x 2 0
Select the correct answer, where k is any integer:
a.5 2k,
45
2k
b.4 2k,
74
2k
c.4 k,
74
k
d.5 2k,
95
2k
____ 30. Find all solutions of the equation.
2sinx 1 0
Select the correct answer, where k is any integer:
a.6 k,
116
k
b.6 k,
56
k
c.6 2k,
56
2k
d.6 2k,
116
2k
Name: ________________________ ID: A
9
____ 31. Find all solutions of the following equation.
4cos2x 3 0
Select the correct answer, where k is any integer:
a.6 2k,
56
2k,76
2k,11
6 2k
b.6 k,
116
k
c.6 k,
56
k,76
k,11
6 k
d.6 2k,
56
2k
____ 32. Find all solutions of the following equation.
4 cos 2 x – 4 cos x + 1 = 0
Select the correct answer, where k is any integer:
a.4 2k,
74
2k
b.3 2k,
53
2k
c.6 2k,
56
2k
d.4 2k,
34
2k
____ 33. Use an addition or subtraction formula to simplify the following equation. Then find all the solutions in the
interval 0,4
È
Î
ÍÍÍÍÍÍÍÍ
ˆ
¯
˜̃̃˜̃̃ .
cos x cos 7 x – sin x sin 7 x = 0
a.16
b.8
,38
c.8
d.16
,316
Name: ________________________ ID: A
10
____ 34. Plot the point that has the polar coordinates 5,4
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃ .
a. d.
b. e.
c.
Name: ________________________ ID: A
11
____ 35. Plot the point that has the polar coordinates 3,76
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃ .
a. d.
b. e.
c.
____ 36. Find the third term of the sequence.
an = 2n + 1
a. a 3 = 7b. a 3 = 6c. a 3 = 5d. a 3 = 1e. a 3 = 2
Name: ________________________ ID: A
12
____ 37. Find the fourth term of the sequence.
an =1
n + 1
a. a4 = 5
b. a4 =15
c. a4 =45
d. a4 =14
e. a4 =11
____ 38. Find the 200th term of the sequence.
an = 10a. a 200 = 1b. a 200 = 10c. a 200 = 200d. a 200 = 210e. a 200 = 2000
____ 39. Find the nth term of the sequence.
2, 4, 8, 16, ...a. an = 2 n – 1
b. an = 2nc. an = 2 n
d. an = 2 n + 1
e. an = 2 + 2n
____ 40. Find the partial sum S 7 of the sequence.
5, 10, 15, 20, ...a. S 7 = 140b. S 7 = 50c. S 7 = 280d. S 7 = 141e. S 7 = 20
Name: ________________________ ID: A
13
____ 41. Find the partial sum S 5 of the sequence.
1, –1, 1, –1, ...a. S 5 = 0b. S 5 = 2c. S 5 = –2d. S 5 = –1e. S 5 = 1
____ 42. Find the sum.
4i 4
18
a. 4 13i 4
18
b. 4 60i 4
18
c. 4 72i 4
18
d. 4 56i 4
18
e. 4 4i 4
18
____ 43. Find the sum.
k2k
k 1
4
a. k2k 99k 1
4
b. k2k 64k 1
4
c. k2k 91k 1
4
d. k2k 10k 1
4
e. k2k 98k 1
4
Name: ________________________ ID: A
14
____ 44. Write the following sum.
kk 5
7
(k 9)
a. k(k 9)k 5
7
5(5 9) 7(7 9)
b. k(k 9)k 5
7
6(6 9) 7(7 9)
c. k(k 9)k 5
7
6(6 9) 7(7 9) 8(8 9)
d. k(k 9)k 5
7
5(5 9) 6(6 9) 7(7 9)
e. k(k 9)k 5
7
5(5 9) 6(6 9) 8(8 9)
____ 45. Write the following sum using sigma notation.
5 + 10 + 15 + 20 + ... + 50
a. 5k 0
50
b. k 5
k 0
10
c. 5kk 1
10
d. 5k
k 0
10
e. kk 5
50
____ 46. The first term of the arithmetic sequence a is 4 and common difference d is 6. Find the nth term and the 10th term.a. a n 1 6(n 4), a 10 62
b. a n 4 6(n 1), a 10 58
c. a n 4 6(n 4), a 10 56
d. a n 6 4(n 1), a 10 59
e. a n 6 4(n 6), a 10 55
Name: ________________________ ID: A
15
____ 47. Find the common difference d of the arithmetic sequence.
5, 7, 9, 11, ...
a. 2b. 2nc. nd. 7e. 5
____ 48. Find the first five terms and determine if the sequence is arithmetic.
a n 2 6n
a. a 1 8, a 2 14, a 3 20, a 4 23, a 5 35 The sequence is not arithmetic.
b. a 1 8, a 2 14, a 3 20, a 4 30, a 5 28 The sequence is not arithmetic.
c. a 1 8, a 2 14, a 3 20, a 4 24, a 5 34 The sequence is arithmetic.
d. a 1 8, a 2 14, a 3 20, a 4 26, a 5 32 The sequence is arithmetic.
e. a 1 8, a 2 14, a 3 20, a 4 27, a 5 33 The sequence is arithmetic.
____ 49. If it is arithmetic, express the nth term of the sequence in the standard form a n a d(n 1) and find the
common difference.
a n 8n 1
a. a n 3 7(n 1), d 7
b. a n 3 6(n 1), d 6
c. Not an arithmetic sequence.d. a n 3 5(n 1), d 5
e. a n 3 8(n 1), d 8
____ 50. Find the fifth term of the arithmetic sequence.
2, 10, 18, 26, ...a. 26b. 27c. 34d. 44e. 5
____ 51. Find the fifth term of the arithmetic sequence.
5, 9, 13, 17, ...
a. 21b. 17c. 30d. 5e. 18
Name: ________________________ ID: A
16
____ 52. Find the nth term of the arithmetic sequence.
2, 2 + s, 2 + 2s, 2 + 3s, ...a. s 2nb. 2 snc. s 2 n 1( )d. 2 s n 1( )
e. 2n 12
sn n 1( )
____ 53. The 12th term of an arithmetic sequence is 13 and the 5th term is 6. Find the 22th term.a. 24b. 43c. 21d. 10e. 23
____ 54. The 20th term of an arithmetic sequence is 97, and the common difference is 5. Find a formula for the nth term.a. 20 + 5(n – 1)b. 2 + 5(n)c. 5 + 2(n – 1)d. 2 + 5(n – 1)e. 5 + 2(n + 1)
____ 55. Which term of the arithmetic sequence 3, 8, 13,... is 73?a. 15b. 17c. 14d. 16e. 13
____ 56. Find the partial sum S n of the arithmetic sequence that satisfies the following conditions.
a = 1, d = 4, n = 15
a. 57b. 435c. 465d. 870e. 61
Name: ________________________ ID: A
17
____ 57. Find the product of the numbers.
10
1
10,10
2
10,10
3
10,10
4
10, ... , 10
19
10
a. 380
b. 10380
c. 10
19
10
d. 190
e. 1019
____ 58. Find the nth term of the geometric sequence with given first term a and common ratio r. What is the fifth term?
a 73
, r 13
a. a n 73
13
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
n
, a 5 1
243
b. an 73
13
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
n 1
, a 5 781
c. an 13
13
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
n 1
, a 5 781
d. an 73
13
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
n 1
, a 5 781
e. a n 73
13
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
n 1
, a 5 7
243
____ 59. Determine whether the sequence
6, 24, 96, 384...
is geometric. If it is geometric, find the common ratio.a. Geometric sequence, r = 6
b. Geometric sequence, r 14
c. Not a geometric sequence.d. Geometric sequence, r = 4
e. Geometric sequence, r 15
Name: ________________________ ID: A
18
____ 60. Determine whether the sequence is geometric.
8, –4, 2, –1,...
If it is geometric, find the common ratio.
a. Geometric, 12
b. Not geometric.
c. Geometric, 12
d. Geometric, 2e. Geometric, –2
____ 61. Determine whether the sequence is geometric. If it is geometric, find the common ratio.
e4 , e7 , e10 , e13 , ...a. Geometric, r = e 3
b. Not geometric.c. Geometric, r = 3
d. Geometric, r = 1
e3
e. Geometric, r = e 4
____ 62. Find the first five terms of the sequence and determine if it is geometric. If it is geometric express the nth term
of the sequence in the standard form a n arn 1 .
an 1( ) n3n
a. –3, 9, –27, 81, –243; ; it is not geometric.
b. –3, 9, –27, 81, –243; a n 3(4)n 1
c. –3, 9, –27, 84, –243; a n 3(4)n 1
d. –3, 9, –27, 81, –243; a n 3(3)n 1
e. –3, 9, –27, 84, –243; a n 3(3)n 1
____ 63. Determine the common ratio, the 6th term, and the nth term of the geometric sequence.
5, 20, 80, 320, ...
a. Common ratio 5, the 6th term 5,120, and the nth term 4n 1
b. Common ratio 4, the 6th term 5,120, and the nth term 5 4n 1
c. Common ratio 4, the 6th term 12,500, and the nth term 5n 1
d. Common ratio 5, the 6th term 12,500, and the nth term 5 4n 1
e. Common ratio 4, the 6th term 5,120, and the nth term 4n 1
Name: ________________________ ID: A
19
____ 64. Determine the nth term of the geometric sequence.
1, 11,11,11 11, ...
a. 11n 1
b.1
11
Ê
Ë
ÁÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃˜
n 1
c.111
Ê
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃̃˜̃̃
n 1
d. Not a geometric series.
e. 11ÊËÁÁÁ
ˆ¯˜̃̃
n 1
____ 65. Determine the nth term of the geometric sequence.
x,x2
5,x3
25,
x4
125, ...
a.xn
5
b.xn
5n
c. xn 1
d.xn 1
5n 1
e.xn
5n 1
____ 66. The first term of a geometric sequence is 6, and the second term is 3. Find the fifth term.
a.32
b.38
c.68
Name: ________________________ ID: A
20
____ 67. The common ratio in a geometric sequence is 43
, and the fourth term is 73
. Find the third term.
a.34
b.47
c.63
d.144
e.74
____ 68. Which term of the geometric sequence 5, 20, 80, . . . is 20480?a. 7thb. 13thc. 6thd. 8the. 9th
____ 69. Find the partial sum Sn of the geometric sequence that satisfies the given conditions.
a = 3, r = 4, n = 6a. Sn = 4,092b. Sn = 16,383c. Sn = 8,190d. Sn = 4,094e. Sn = 4,095
____ 70. Find the partial sum Sn of the geometric sequence that satisfies the given conditions.
a 4 16, a 6 64, n 4
a. Sn = 60b. Sn = 62c. Sn = 28d. Sn = 29e. Sn = 30
____ 71. Find the sum.
1 + 4 + 16 + ... + 4096a. Sn = 1,365b. Sn = 10,922c. Sn = 21,845d. Sn = 5,460e. Sn = 5,461
Name: ________________________ ID: A
21
____ 72. Find the sum of the infinite geometric series.
1 13 1
9 1
27 ...
a.42
b.54
c. 1
d.23
e.32
____ 73. Find the sum of the infinite geometric series.