ECAT Mathematics
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ECAT Mathematics
Chapter 13: Inverse Trigonometric Functions
87 Questions
90 Minutes
Pass: 60%+
ECAT Prep
0
sin⁻¹x is defined for x in:
A
(−∞,∞)
B
[−1,1]
C
(0,1)
D
[0,π]
1
Range of sin⁻¹x:
A
[0,π]
B
(−π/2,π/2)
C
[−π/2,π/2]
D
[−1,1]
2
cos⁻¹x is defined for x in:
A
(−∞,∞)
B
(−1,1)
C
[−1,1]
D
[0,π]
3
Range of cos⁻¹x:
A
[−π/2,π/2]
B
[0,π]
C
(0,π)
D
[−1,1]
4
tan⁻¹x is defined for x in:
A
[−1,1]
B
[0,π]
C
(−∞,∞)
D
[−π/2,π/2]
5
Range of tan⁻¹x:
A
[0,π]
B
[−π/2,π/2]
C
(−π/2,π/2)
D
(−∞,∞)
6
sin(sin⁻¹x) for x∈[−1,1] =
A
sin⁻¹(sinx)
B
x
C
sinx
D
1
7
sin⁻¹(sinx) = x for x in:
A
[−1,1]
B
[−π/2,π/2]
C
[0,π]
D
(−∞,∞)
8
cos(cos⁻¹x) =
A
x
B
cosx
C
cos⁻¹(cosx)
D
1
9
tan(tan⁻¹x) =
A
x
B
tanx
C
1/x
D
π/4
10
sin⁻¹(1/2) =
A
π/3
B
π/4
C
π/6
D
π/2
11
cos⁻¹(1/2) =
A
π/6
B
π/3
C
π/4
D
2π/3
12
tan⁻¹(1) =
A
π/4
B
π/3
C
π/6
D
π/2
13
sin⁻¹(0) =
A
0
B
π/2
C
π
D
−π/2
14
cos⁻¹(0) =
A
0
B
π/6
C
π/2
D
π
15
tan⁻¹(0) =
A
0
B
π/4
C
π/2
D
π
16
sin⁻¹(−1/2) =
A
π/6
B
−π/6
C
5π/6
D
2π/3
17
cos⁻¹(−1/2) =
A
π/3
B
2π/3
C
π/6
D
5π/6
18
tan⁻¹(−1) =
A
π/4
B
−π/4
C
3π/4
D
−3π/4
19
sin⁻¹x+cos⁻¹x = ? for x∈[−1,1]
A
π
B
π/2
C
0
D
2π
20
tan⁻¹x+cot⁻¹x = ?
A
π
B
π/2
C
0
D
2π
21
sin⁻¹(−x) =
A
sin⁻¹x
B
−sin⁻¹x
C
cos⁻¹x
D
π−sin⁻¹x
22
cos⁻¹(−x) =
A
−cos⁻¹x
B
cos⁻¹x
C
π+cos⁻¹x
D
π−cos⁻¹x
23
tan⁻¹(−x) =
A
tan⁻¹x
B
−tan⁻¹x
C
π−tan⁻¹x
D
π+tan⁻¹x
24
sin⁻¹(1) =
A
0
B
π/3
C
π/2
D
π
25
cos⁻¹(1) =
A
0
B
π/3
C
π/2
D
π
26
sin⁻¹(−1) =
A
0
B
−π/2
C
π/2
D
π
27
cos⁻¹(−1) =
A
0
B
π/2
C
π
D
2π
28
sin⁻¹(√3/2) =
A
π/6
B
π/3
C
π/4
D
π/2
29
cos⁻¹(√3/2) =
A
π/6
B
π/3
C
π/4
D
π/2
30
tan⁻¹(√3) =
A
π/6
B
π/3
C
π/4
D
π/2
31
tan⁻¹(1/√3) =
A
π/6
B
π/3
C
π/4
D
π/2
32
sin(cos⁻¹(3/5)) (right triangle with hyp 5, adj 3, opp 4):
A
3/5
B
4/5
C
4/3
D
5/4
33
cos(sin⁻¹(5/13)) (adj=12):
A
5/13
B
12/13
C
13/12
D
5/12
34
tan(sin⁻¹(x)) =
A
x/√(1−x²)
B
√(1−x²)/x
C
x
D
√(1+x²)/x
35
tan(cos⁻¹(x)) =
A
x/√(1−x²)
B
√(1−x²)/x
C
x
D
√(1+x²)
36
sin(2sin⁻¹(x)) =
A
2x√(1−x²)
B
x√(1−x²)
C
2x/(1−x²)
D
2x²−1
37
cos(2sin⁻¹(x)) =
A
1−2x²
B
2x²−1
C
x²
D
√(1−x²)
38
tan⁻¹(1)+tan⁻¹(2)+tan⁻¹(3) =
A
π
B
π/2
C
2π
D
0
39
2tan⁻¹(x) =
A
tan⁻¹(2x/(1−x²))
B
tan⁻¹(x/2)
C
2arctan(x)
D
sin⁻¹(2x/(1+x²))
40
sin⁻¹x+sin⁻¹y = sin⁻¹(x√(1−y²)+y√(1−x²)) when:
A
x²+y²≤1
B
x²+y²=1
C
x=y
D
xy=1
41
The function arcsin is:
A
Even
B
Odd
C
Neither
D
Constant
42
The function arccos is:
A
Even
B
Odd
C
Neither
D
Constant
43
The function arctan is:
A
Even
B
Odd
C
Neither
D
Constant
44
lim_{x→∞} tan⁻¹x =
A
0
B
π/2
C
π
D
−π/2
45
lim_{x→−∞} tan⁻¹x =
A
0
B
π/2
C
π
D
−π/2
46
cot⁻¹x range:
A
(−π/2,π/2)
B
(0,π)
C
[0,π]
D
[−π/2,π/2]
47
sec⁻¹x domain (one convention):
A
[−1,1]
B
(−∞,−1]∪[1,∞)
C
All reals
D
[0,π]
48
csc⁻¹x domain:
A
[−1,1]
B
(−∞,−1]∪[1,∞)
C
All reals
D
[−π/2,π/2]
49
sin⁻¹(sin(7π/6)) =
A
7π/6
B
−π/6
C
π/6
D
5π/6
50
cos⁻¹(cos(5π/4)) =
A
5π/4
B
3π/4
C
π/4
D
−π/4
51
The principal value of sin⁻¹ lies in:
A
[0,π]
B
[−π/2,π/2]
C
(−π,π)
D
[−π,π]
52
sin⁻¹x can also be written as:
A
arcsin x
B
asin x
C
sin inverse x
D
All of above
53
If sin⁻¹x=π/3, then x=
A
√3/2
B
1/2
C
1
D
√2/2
54
If cos⁻¹x=2π/3, then x=
A
1/2
B
−1/2
C
√3/2
D
−√3/2
55
The derivative of sin⁻¹x is:
A
1/√(1+x²)
B
1/√(1−x²)
C
−1/√(1−x²)
D
x/√(1−x²)
56
The derivative of tan⁻¹x is:
A
1/(1+x²)
B
1/(1−x²)
C
−1/(1+x²)
D
x/(1+x²)
57
tan⁻¹(1/2)+tan⁻¹(1/3) =
A
π/4
B
π/3
C
π/6
D
π/2
58
sin⁻¹(3/5)+sin⁻¹(4/5) =
A
π/2
B
π/3
C
π/4
D
π
59
cos(tan⁻¹(x)) =
A
1/√(1+x²)
B
x/√(1+x²)
C
√(1+x²)
D
1/x
60
sin(tan⁻¹(x)) =
A
1/√(1+x²)
B
x/√(1+x²)
C
√(1+x²)/x
D
x
61
Graph of y=sin⁻¹x passes through:
A
(0,0)
B
(1,π/2)
C
(−1,−π/2)
D
All of the above
62
Graph of y=cos⁻¹x passes through:
A
(0,π/2)
B
(1,0)
C
(−1,π)
D
All of the above
63
sin⁻¹(cos(π/3)) =
A
π/6
B
π/3
C
π/2
D
π/4
64
cos⁻¹(sin(π/6)) =
A
π/3
B
π/6
C
π/4
D
2π/3
65
2sin⁻¹(1/2) =
A
π/6
B
π/3
C
π/2
D
π/4
66
The equation sin⁻¹x=cos⁻¹x is true for x=
A
0
B
1/√2
C
1
D
−1/2
67
3sin⁻¹x = sin⁻¹(3x−4x³) is valid for:
A
x∈[0,1]
B
x∈[−1/2,1/2]
C
All reals
D
x=0
68
3cos⁻¹x = cos⁻¹(4x³−3x) is valid for:
A
x∈[1/2,1]
B
x∈[−1/2,1/2]
C
All reals
D
x=1
69
tan⁻¹x+tan⁻¹y = tan⁻¹((x+y)/(1−xy)) when:
A
xy>1
B
xy=1
C
xy<1
D
Always
70
When xy>1 (and x,y>0): tan⁻¹x+tan⁻¹y =
A
tan⁻¹((x+y)/(1−xy))
B
π+tan⁻¹((x+y)/(1−xy))
C
π−tan⁻¹((x+y)/(1−xy))
D
π/2
71
The equation sin(sin⁻¹x+cos⁻¹x) = sin(π/2) =
A
0
B
x
C
1
D
sinx
72
cot⁻¹(x) = tan⁻¹(1/x) for:
A
x<0
B
x=0
C
x>0
D
All x
73
Principal value of cot⁻¹(−√3):
A
π/6
B
5π/6
C
7π/6
D
−π/6
74
sin⁻¹(2/3)+sin⁻¹(√5/3) = ? (Check: (2/3)²+(√5/3)²=4/9+5/9=1)
A
π/2
B
π/3
C
π
D
2π/3
75
4tan⁻¹(1/5)−tan⁻¹(1/239) = ?
A
π/3
B
π/4
C
π/6
D
π/2
76
The function y=sin⁻¹x is:
A
Decreasing
B
Increasing
C
Constant
D
Periodic
77
The function y=cos⁻¹x is:
A
Decreasing
B
Increasing
C
Constant
D
Periodic
78
The function y=tan⁻¹x is:
A
Decreasing
B
Increasing
C
Constant
D
Periodic
79
sin⁻¹x+sin⁻¹(−x) =
A
π/2
B
0
C
π
D
2sin⁻¹x
80
cos⁻¹x+cos⁻¹(−x) =
A
π/2
B
0
C
π
D
2cos⁻¹x
81
sin(cos⁻¹(1/2)) =
A
1/2
B
√3/2
C
√2/2
D
1
82
cos(sin⁻¹(1/2)) =
A
1/2
B
√3/2
C
√2/2
D
1
83
sin⁻¹(√2/2) =
A
π/6
B
π/4
C
π/3
D
π/2
84
cos⁻¹(√2/2) =
A
π/6
B
π/4
C
π/3
D
3π/4
85
sin⁻¹(sin(4π/3)) =
A
4π/3
B
−π/3
C
π/3
D
2π/3
86
The domain restriction for sin⁻¹ is needed because:
A
sinx is not differentiable
B
sinx is not one-to-one on all reals
C
sinx is bounded
D
sinx is periodic
0
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