Processing math: 100%
MCQExams
0:0:2
CBSE
JEE
NTSE
NEET
Practice
Homework
×
CBSE Questions for Class 12 Commerce Maths Inverse Trigonometric Functions Quiz 7 - MCQExams.com
CBSE
Class 12 Commerce Maths
Inverse Trigonometric Functions
Quiz 7
If
tan
−
1
x
=
π
10
for some
x
∈
R
, then the value of
cot
−
1
x
is
Report Question
0%
π
5
0%
2
π
5
0%
3
π
5
0%
4
π
5
Explanation
(B) is the correct answer.
We know that
tan
−
1
x
+
cot
−
1
x
=
π
2
.
⇒
cot
−
1
x
=
π
2
−
tan
−
1
x
⇒
cot
−
1
x
=
π
2
−
π
10
=
4
π
10
=
2
π
5
The value of
sin
−
1
{
cos
(
43
π
5
)
}
is
Report Question
0%
3
π
5
0%
−
7
π
5
0%
π
10
0%
−
π
10
Explanation
(D) is the correct answer.
We have
sin
−
1
(
cos
40
π
+
3
π
5
)
=
sin
−
1
{
cos
(
8
π
+
3
π
5
}
}
=
sin
−
1
(
cos
3
π
5
)
=
sin
−
1
{
sin
(
π
2
−
3
π
5
)
}
=
sin
−
1
(
sin
(
−
π
10
)
)
=
−
π
10
The value of
cot
(
sin
−
1
x
)
is
Report Question
0%
√
1
+
x
2
x
0%
x
√
1
+
x
2
0%
1
x
0%
√
1
−
x
2
x
Explanation
(D) is the correct answer.
Let
sin
−
1
x
=
θ
,
Then
sin
θ
=
x
⇒
csc
θ
=
1
x
⇒
csc
2
θ
=
1
x
2
⇒
1
+
cot
2
θ
=
1
x
2
⇒
cot
θ
=
√
1
−
x
2
x
The domain of
y
=
cos
−
1
(
x
2
−
4
)
is
Report Question
0%
[
3
,
5
]
0%
[
0
,
π
]
0%
[
−
√
5
,
−
√
3
]
∩
[
−
√
5
,
√
3
]
0%
[
−
√
5
,
−
√
3
]
∪
[
√
3
,
5
]
Explanation
(D) is the correct answer.
Given
y
=
cos
−
1
(
x
2
−
4
)
⇒
cos
y
=
x
2
−
4
∴
−
1
≤
x
2
−
4
≤
1
( since
−
1
≤
cos
y
≤
1
)
⇒
3
≤
x
2
≤
5
⇒
√
3
≤
|
x
|
≤
√
5
⇒
x
∈
[
−
√
5
,
−
√
3
]
∪
[
√
3
,
√
5
]
Hence the domain of
y
is
[
−
√
5
,
−
√
3
]
∪
[
√
3
,
5
]
The domain of the function
y
=
sin
−
1
(
−
x
2
)
is
Report Question
0%
[
0
,
1
]
0%
(
0
,
1
)
0%
[
−
1
,
1
]
0%
ϕ
Explanation
(C) is the correct answer,
Given
y
=
sin
−
1
(
−
x
2
)
⇒
sin
y
=
−
x
2
∴
−
1
≤
−
x
2
≤
1
( since
−
1
≤
sin
y
≤
1
)
⇒
1
≥
x
2
≥
−
1
⇒
0
≤
x
2
≤
1
⇒
|
x
|
≤
1
⇒
−
1
≤
x
≤
1
Hence the domain is
[
−
1
,
1
]
The value of
sin
(
2
sin
−
1
(
0.6
)
)
is
Report Question
0%
0.48
0%
0.96
0%
1.2
0%
sin
1.2
Explanation
(B) is the correct answer.
Let
sin
−
1
(
0.6
)
=
θ
⇒
sin
θ
=
0.6
.
∴
sin
(
2
sin
−
1
(
0.6
)
)
=
sin
(
2
θ
)
=
2
sin
θ
cos
θ
=
2
sin
θ
√
1
−
sin
2
θ
=
2
(
0.6
)
(
√
1
−
0.6
2
)
=
2
(
0.6
)
(
0.8
)
=
0.96
The value of the expression
sin
[
cot
−
1
{
cos
(
tan
−
1
1
)
}
]
is
Report Question
0%
0
0%
1
0%
1
√
3
0%
√
2
3
Explanation
(D) is the correct answer.
Let
P
=
sin
[
cot
−
1
{
cos
(
tan
−
1
1
)
}
]
=
sin
[
cot
−
1
(
cos
π
4
)
]
[
∵
tan
−
1
1
=
π
4
]
=
sin
[
cot
−
1
1
√
2
]
[
∵
cos
π
4
=
1
√
2
]
Let,
cot
−
1
1
√
2
=
y
⇒
cot
y
=
1
√
2
⇒
sin
y
=
√
2
3
⇒
y
=
sin
−
1
√
2
3
∴
P
=
sin
[
sin
−
1
√
2
3
]
=
√
2
3
The value of
tan
2
(
sec
−
1
2
)
+
cot
2
(
csc
−
1
3
)
is
Report Question
0%
5
0%
11
0%
13
0%
15
Explanation
(B) is the correct answer.
We have
tan
2
(
sec
−
1
2
)
cot
2
(
csc
−
1
3
)
=
sec
2
(
sec
−
1
2
)
−
1
+
csc
2
(
csc
−
1
3
)
−
1
[
∵
tan
2
θ
=
sec
2
θ
−
1
,
cot
2
θ
=
csc
2
θ
−
1
]
=
(
2
)
2
−
1
+
(
3
)
2
−
1
=
11
If
α
≤
2
sin
−
1
x
+
cos
−
1
x
≤
β
, then
Report Question
0%
α
=
−
π
2
,
β
=
π
2
0%
α
=
0
,
β
=
π
0%
α
=
−
π
2
,
β
=
3
π
2
0%
α
=
0
,
β
=
2
π
Explanation
(B) is the correct answer.
We know that
−
π
2
≤
sin
−
1
x
≤
π
2
⇒
−
π
2
+
π
2
≤
sin
−
1
x
+
π
2
≤
π
2
+
π
2
[
Addidng
π
2
]
⇒
0
≤
sin
−
1
x
+
(
sin
−
1
x
+
cos
−
1
x
)
≤
π
[
∵
(
sin
−
1
x
+
cos
−
1
x
)
=
π
2
]
⇒
0
≤
2
sin
−
1
x
+
cos
−
1
x
≤
π
sin
−
1
(
1
−
x
)
−
2
sin
−
1
x
=
π
2
,
then x
Report Question
0%
0
,
1
2
0%
1
,
1
2
0%
0
0%
1
2
Explanation
sin
−
1
(
1
−
x
)
−
2
sin
−
1
x
=
π
2
−
2
sin
−
1
x
=
π
2
−
sin
−
1
(
1
−
x
)
−
2
sin
−
1
x
=
cos
−
1
(
1
−
x
)
cos
(
−
2
sin
−
1
x
)
=
1
−
x
cos
(
2
sin
−
1
x
)
=
1
−
x
cos
(
2
s
i
n
−
1
x
)
=
1
−
x
1
−
2
sin
2
(
s
i
n
−
1
x
)
=
1
−
x
1
−
2
x
2
=
1
−
x
⇒
2
x
2
−
x
=
0
x
(
2
x
−
1
)
=
0
⇒
x
=
0
,
x
=
1
2
1
2
does not satisfy the equation
∴
x
=
0
is the only solution
Choose the correct answer
cos
−
1
(
cos
7
π
6
)
is equal to
Report Question
0%
7
π
6
0%
5
π
6
0%
π
3
0%
π
6
Explanation
cos
−
1
cos
(
7
π
6
)
cos
−
1
(
cos
(
π
+
π
6
)
)
=
cos
−
1
(
−
cos
π
6
)
=
π
−
cos
−
1
(
cos
π
6
)
=
π
−
π
6
=
5
π
6
Choose the correct answer :
sin
(
π
3
−
sin
−
1
(
−
1
2
)
)
is equal to
Report Question
0%
π
0%
−
π
2
0%
1
0%
2
√
3
Explanation
sin
[
(
π
3
−
sin
−
1
(
−
1
2
)
)
]
[
sin
[
(
π
3
−
sin
−
1
(
−
1
2
)
)
]
]
sin
(
π
3
+
sin
−
1
(
sin
π
6
)
)
(
a
s
sin
−
1
(
−
1
2
)
=
sin
−
1
1
2
a
n
d
sin
π
6
=
1
2
)
=
sin
(
π
3
+
π
6
)
=
sin
π
2
=
1
If
sin
−
1
x
=
y
, then
Report Question
0%
0
≤
y
≤
π
0%
−
π
2
≤
y
≤
π
2
0%
0
<
y
<
π
0%
−
π
2
<
y
<
π
2
Explanation
pv of
sin
−
1
x
is
[
−
π
2
,
π
2
]
∴
(
B
)
−
π
2
≤
y
≤
π
2
is correct
tan
−
1
√
3
−
sec
−
1
(
−
2
)
is equal to
Report Question
0%
π
0%
−
π
3
0%
π
3
0%
−
2
π
3
Explanation
tan
−
1
√
3
=
π
3
ϵ
[
−
π
2
,
π
2
]
sec
−
1
(
−
2
)
=
π
−
π
3
=
2
π
3
ϵ
[
0
,
π
]
∴
tan
−
1
√
3
−
s
e
c
−
1
(
−
2
)
=
π
3
−
2
π
3
=
−
π
3
∴
Option (B) is correct .
Multiple choice Questions :
2
sin
(
cos
−
1
(
−
4
5
)
)
×
cos
cos
−
1
(
−
4
5
)
Report Question
0%
24
25
0%
−
24
25
0%
−
6
25
0%
−
6
25
Explanation
2
sin
(
cos
−
1
(
−
4
5
)
)
×
cos
cos
−
1
(
−
4
5
)
=
2
√
1
−
(
−
4
5
)
2
×
(
−
4
5
)
=
−
8
5
√
9
25
=
−
8
5
×
3
5
=
−
24
25
Multiple choice Questions :
sin
−
1
(
4
5
)
+
cos
−
1
(
4
5
)
=
Report Question
0%
π
2
0%
0
0%
π
0%
−
π
2
Explanation
sin
1
x
+
cos
−
1
x
=
π
2
sin
−
1
(
4
5
)
+
cos
−
1
(
4
5
)
=
π
2
Multiple choice Questions :
cos
−
1
cos
(
4
π
3
)
=
Report Question
0%
4
π
3
0%
2
π
3
0%
−
π
3
0%
−
π
Explanation
cos
−
1
cos
(
4
π
3
)
=
cos
1
cos
(
π
+
π
3
)
=
cos
−
1
[
−
c
o
s
π
3
]
=
π
−
c
o
s
−
1
(
cos
π
3
)
=
π
−
π
3
=
2
π
3
Multiple choice Questions :
The value of
tan
−
1
(
tan
5
)
Report Question
0%
2
π
0%
5
−
2
π
0%
5
0%
2
π
3
Explanation
Let
2
π
−
5
=
θ
⇒
5
=
2
π
−
θ
tan
5
=
tan
(
2
π
−
θ
)
tan
−
1
(
tan
5
)
=
tan
−
1
tan
(
2
π
−
θ
)
=
tan
−
1
[
−
tan
θ
]
=
−
θ
tan
−
1
(
x
y
)
−
tan
−
1
x
−
y
x
+
y
is equal to
Report Question
0%
π
2
0%
π
3
0%
π
4
0%
−
3
π
4
Explanation
tan
−
1
(
x
y
)
−
(
tan
−
1
x
y
−
1
1
+
x
y
)
=
tan
−
1
(
x
y
)
−
tan
−
1
(
x
y
)
+
tan
−
1
(
1
)
tan
−
1
1
=
π
4
Multiple choice Questions :
If a > b > c ,
cot
−
1
(
1
+
a
b
a
−
b
)
+
cot
−
1
(
1
+
b
c
b
−
c
)
+
cot
−
1
(
1
+
a
c
c
−
a
)
Report Question
0%
0
0%
π
0%
2
π
0%
π
2
Explanation
tan
−
1
(
a
−
b
1
+
a
b
)
+
tan
−
1
(
b
−
c
1
+
b
c
)
+
tan
−
1
(
c
−
a
1
+
a
c
)
=
tan
−
1
a
−
tan
−
1
b
+
tan
−
1
b
−
tan
−
1
c
+
π
−
tan
−
1
a
+
tan
−
1
c
=
π
a
s
tan
−
1
(
c
−
a
1
+
a
c
)
=
π
−
tan
−
1
(
a
−
c
1
+
a
c
)
If
sin
−
1
(
1
2
)
=
x
, then general value
x
$ is:
Report Question
0%
2
n
π
±
π
6
0%
π
6
0%
n
π
±
π
6
0%
n
π
+
(
−
1
)
n
π
6
Explanation
Given,
sin
−
1
(
1
2
)
=
x
⇒
sin
x
=
1
2
=
sin
π
6
⇒
x
=
π
6
∴
General value of
x
,
θ
=
n
π
+
(
−
1
)
n
π
6
Hence, option
(
d
)
is correct.
If
tan
−
1
(
1
)
+
cos
−
1
(
1
√
2
)
=
sin
−
1
x
, then value of
x
is
Report Question
0%
−
1
0%
0
0%
1
0%
−
1
2
Explanation
tan
−
1
(
1
)
+
cos
−
1
(
1
√
2
)
=
sin
−
1
x
⇒
sin
−
1
x
=
π
4
+
π
4
⇒
sin
−
1
x
=
π
2
⇒
x
=
sin
π
2
⇒
x
=
1
Hence, option
(
c
)
is correct.
2
tan
(
tan
−
1
x
+
tan
−
1
x
3
)
is:
Report Question
0%
2
x
1
−
x
2
0%
1
+
x
2
0%
2
x
0%
none of these
Explanation
2
tan
(
tan
−
1
x
+
tan
−
1
x
3
)
=
2
tan
{
tan
−
1
(
x
+
x
3
1
−
x
×
x
3
)
}
=
2
tan
{
tan
−
1
(
x
(
1
+
x
2
)
1
−
x
4
)
}
=
2
tan
{
tan
−
1
x
(
1
+
x
2
)
(
1
−
x
2
)
(
1
+
x
2
)
}
=
2
tan
x
1
−
x
2
=
2
x
1
−
x
2
Hence, option
(
a
)
is correct.
Value of
sin
−
1
(
√
3
2
)
+
2
cos
−
1
(
√
3
2
)
is:
Report Question
0%
π
2
0%
π
3
0%
2
π
3
0%
π
Explanation
sin
−
1
(
√
3
2
)
+
2
cos
−
1
(
√
3
2
)
=
[
sin
−
1
(
√
3
2
)
+
cos
−
1
(
√
3
2
)
]
+
cos
−
1
(
√
3
2
)
=
π
2
+
π
6
=
2
π
3
[
∵
sin
−
1
x
+
cos
−
1
x
=
π
2
]
Hence, option
(
c
)
is correct.
If
cot
−
1
x
+
tan
−
1
1
3
=
π
2
then
x
is:
Report Question
0%
1
0%
3
0%
1
3
0%
none of these
Explanation
cot
−
1
(
x
)
+
tan
−
1
(
1
3
)
=
π
2
⇒
cot
−
1
x
=
π
2
−
tan
−
1
(
1
3
)
⇒
cot
−
1
x
=
cot
−
1
1
3
On comparing
x
=
1
3
Hence, option
(
c
)
is correct.
If
tan
−
1
(
3
x
)
+
tan
−
1
2
x
=
π
4
, then
x
is:
Report Question
0%
1
6
0%
1
3
0%
1
10
0%
1
2
Explanation
tan
−
1
(
3
x
)
+
tan
−
1
(
2
x
)
=
π
4
⇒
tan
−
1
{
3
x
+
2
x
1
−
3
x
×
2
x
}
=
π
4
⇒
(
5
x
1
−
6
x
2
)
=
tan
π
4
⇒
5
x
1
−
6
x
2
=
1
⇒
1
−
6
x
2
=
5
x
⇒
6
x
2
+
5
x
−
1
=
0
⇒
6
x
2
+
6
x
−
x
−
1
=
0
⇒
6
x
(
x
+
1
)
−
1
(
x
+
1
)
=
0
⇒
(
x
+
1
)
(
6
x
−
1
)
=
0
⇒
x
=
−
1
,
x
=
1
6
Hence, option
(
a
)
is correct.
lf the equation
sin
−
1
(
x
2
+
x
+
1
)
+
cos
−
1
(
λ
x
+
1
)
=
π
2
has exactly two solutions, then
λ
can not have the integral value(s)
Report Question
0%
−
1
0%
0
0%
1
0%
2
Explanation
sin
−
1
(
x
2
+
x
+
1
)
+
cos
−
1
(
λ
x
+
1
)
=
π
2
cos
−
1
(
λ
x
+
1
)
=
π
2
−
sin
−
1
(
x
2
+
x
+
1
)
Taking
cos
on both sides,
λ
x
+
1
=
x
2
+
x
+
1
x
2
+
(
1
−
λ
)
x
=
0
x
(
x
+
1
−
λ
)
=
0
x
=
0
or
λ
=
x
+
1
sin
−
1
(
x
2
+
x
+
1
)
is defined for
−
1
≤
x
2
+
x
+
1
≤
1
x
2
+
x
+
2
≥
0
is always true.
x
(
x
+
1
)
≤
0
⇒
x
∈
[
−
1
,
0
]
⇒
x
+
1
∈
[
0
,
1
]
λ
∈
[
0
,
1
]
λ
≠
−
1
,
2
Also when
λ
=
1
, there is only one solution to the given equation i.e.,
x
=
0
So,
λ
≠
1
.
Assertion(A):
cos
−
1
x
and
tan
−
1
x
are positive for all positive real values of
x
in their domain.
Reason(R): The domain of
f
(
x
)
=
cos
−
1
x
+
tan
−
1
x
is
[
−
1
,
1
]
.
Report Question
0%
Both A and R are true and R is the correct explanation of A
0%
Both A and R are true but R is not correct explanation of A
0%
A is true but R is false
0%
A is false but R is true
Explanation
Assertion :
cos
−
1
x
range is
[
0
,
π
]
and domain is
[
−
1
,
1
]
∴
cos
−
1
x
>
0
for all
x
>
0
tan
−
1
x
range is
(
−
π
2
,
π
2
)
and range is
R
∀
x
>
0
⟹
tan
−
1
x
>
0
Reason :
The domain of
cos
−
1
x
is
[
−
1
,
1
]
The domain of
tan
−
1
x
is
R
So The domain of
cos
−
1
x
+
tan
−
1
x
is
R
∩
[
−
1
,
1
]
⟹
[
−
1
,
1
]
sin
−
1
|
sin
x
|
=
√
sin
−
1
|
sin
x
|
then
x
=
Report Question
0%
n
π
−
1
0%
n
π
0%
n
π
+
1
0%
n
π
2
+
1
Explanation
Case - 1
|
s
i
n
x
|
=
s
i
n
x
x
ϵ
[
2
n
π
,
(
2
n
+
1
)
π
]
s
i
n
−
1
s
i
n
x
=
√
s
i
n
−
1
s
i
n
x
x
=
√
x
√
x
=
1
o
r
x
=
0
x
=
1
Case - 2
|
s
i
n
x
|
=
−
s
i
n
x
x
ϵ
[
(
2
n
−
1
)
π
,
2
n
π
]
s
i
n
−
1
(
−
s
i
n
x
)
=
√
s
i
n
−
1
(
−
s
i
n
x
)
−
x
=
√
−
x
x
=
−
1
,
x
=
0
So,
x
=
n
π
−
1
f
o
r
n
=
0
x
=
n
π
f
o
r
n
=
0
x
=
n
π
−
1
f
o
r
n
=
0
The number of solutions of:
sin
−
1
(
1
+
b
+
b
2
+
…
.
∞
)
+
cos
−
1
(
a
−
a
2
3
+
a
3
9
+
…
∞
)
=
π
2
Report Question
0%
1
0%
2
0%
3
0%
∞
Explanation
As
sin
−
1
θ
are defined for
θ
∈
[
−
1
,
1
]
∴
For
sin
−
1
(
1
+
b
+
b
3
+
.
.
.
∞
)
|
b
|
<
1
Hence
1
,
b
,
b
2
,
.
.
.
∞
is
G
.
P
.
of common ratio
b
∴
1
+
b
+
b
2
+
.
.
.
+
∞
=
1
1
−
b
Similarly for
cos
−
1
(
a
−
a
2
3
+
a
3
9
+
.
.
.
∞
)
|
a
3
|
<
1
Hence
1
,
−
a
2
3
,
a
3
9
,
.
.
.
∞
is
G
.
P
.
of common ration
(
−
a
3
)
∴
a
−
a
2
3
+
a
3
9
+
.
.
.
∞
=
a
1
+
a
3
=
3
a
3
+
a
Now, as
sin
−
1
(
x
)
+
cos
−
1
(
x
)
=
π
2
∴
1
1
−
b
=
3
a
3
+
a
⇒
3
+
a
=
3
a
−
3
a
b
⇒
2
a
−
3
=
3
a
b
This has infinite solution.
If
(
tan
−
1
x
)
2
+
(
cot
−
1
x
)
2
=
5
π
2
8
, then
x
=
Report Question
0%
−
1
0%
1
0%
0
0%
2
Explanation
Let
t
a
n
−
1
x
=
y
Therefore
y
2
+
(
π
2
−
y
)
2
=
5
π
2
8
2
y
2
−
2
π
y
2
+
π
2
4
=
5
π
2
8
2
y
2
−
π
y
−
3
π
2
8
=
0
y
=
−
π
4
and
y
=
3
π
4
Hence
t
a
n
−
1
(
x
)
=
−
π
4
x
=
−
1
and
t
a
n
−
1
(
x
)
=
3
π
4
x
=
−
1
However only
x
=
−
1
satisfies the above quadratic equation.
c
o
s
−
1
(
√
a
−
x
a
−
b
)
=
s
i
n
−
1
(
√
x
−
b
a
−
b
)
is possible if
Report Question
0%
a
>
x
>
b
or
a
<
x
<
b
0%
a
=
x
=
b
0%
a
>
b
and x takes any value
0%
a
<
b
and x takes any value
Explanation
In
cos
−
1
x
,
x
∈
[
−
1
,
1
]
In
sin
−
1
x
,
x
∈
[
−
1
,
1
]
−
1
≤
√
a
−
x
a
−
b
≤
1
0
≤
√
a
−
x
a
−
b
≤
1
0
≤
a
−
x
≤
a
−
b
0
≥
x
−
a
≥
b
−
a
[on multiplying by (-ve) sing direction of equality changes]
a
≥
x
≥
b
and
0
≤
√
x
−
b
a
−
b
≤
1
0
≤
x
−
b
≤
a
−
b
b
≤
x
≤
a
x
ϵ
[
b
,
a
]
The solution set of the equation
tan
−
1
x
−
cot
−
1
x
=
cos
−
1
(
2
−
x
)
is
Report Question
0%
(
0
,
1
)
0%
(
−
1
,
1
)
0%
[
1
,
3
)
0%
(
1
,
3
)
Explanation
Given,
tan
−
1
x
−
cot
−
1
x
=
cos
−
1
(
2
−
x
)
x
∈
[
1
,
3
]
2
tan
−
1
x
−
π
2
=
cos
−
1
(
2
−
x
)
[
∵
tan
−
1
x
+
cot
−
1
x
=
π
2
]
2
tan
−
1
x
=
π
2
+
cos
−
1
(
2
−
x
)
Take
sin
on both sides,
⇒
2
x
(
1
+
x
2
)
=
2
−
x
.
.
.
.
.
.
.
.2
tan
−
1
x
=
s
i
n
−
1
2
x
1
+
x
2
⇒
x
3
−
2
x
2
+
3
x
−
2
=
0
⇒
x
=
1
So, set containing solution is
[
1
,
3
)
.
If
sin
−
1
α
+
sin
−
1
β
+
sin
−
1
γ
=
3
π
2
, then
α
β
+
α
γ
+
β
γ
is equal to :
Report Question
0%
1
0%
0
0%
3
0%
−
3
Explanation
The above expression is true for
α
=
1
,
β
=
1
and
γ
=
1
Since
−
π
2
≤
s
i
n
−
1
(
x
)
≤
π
2
Hence
α
β
+
β
γ
+
γ
α
=
(
1
)
+
(
1
)
+
(
1
)
=
3
The number of positive integral solutions of the equation
t
a
n
−
1
x
+
c
o
t
−
1
y
=
t
a
n
−
1
3
is :
Report Question
0%
0
0%
1
0%
2
0%
3
Explanation
c
o
t
−
1
y
=
t
a
n
−
1
3
−
t
a
n
−
1
x
c
o
t
−
1
y
=
t
a
n
−
1
(
3
−
x
1
+
3
x
)
Take tan on both sides
1
y
=
3
−
x
1
+
3
x
y
=
1
+
3
x
3
−
x
⇒
y
>
0
Put
x
=
0
,
y
=
1
3
1
+
3
x
3
−
x
>
0
x
=
1
,
y
=
2
3
x
+
1
x
=
3
<
0
x
=
2
,
y
=
7
x
ϵ
(
−
1
3
,
3
)
The integral values of
x
=
0
,
1
,
2
The domain of
f
(
x
)
=
cot
−
1
(
x
√
x
2
−
[
x
2
]
)
is
(
[
.
]
denotes the greatest integer function)
Report Question
0%
(
0
,
∞
)
0%
R
−
{
0
}
0%
R
−
{
x
:
x
∈
Z
}
0%
(
−
∞
,
0
)
Explanation
f
(
x
)
=
cot
−
1
(
x
√
x
2
−
[
x
2
]
)
For
f
(
x
)
to be defined,
x
2
−
[
x
2
]
>
0
∴
x
2
should not be an integer
Hence,
x
should not be an integer.
Hence, Option C is the answer
If
(
t
a
n
−
1
x
)
2
+
(
c
o
t
−
1
x
)
2
=
5
π
2
8
, then
x
=
Report Question
0%
−
1
0%
0
0%
1
0%
2
Explanation
We have,
t
a
n
−
1
x
+
c
o
t
−
1
x
=
π
2
The given equation can be written as :
(
t
a
n
−
1
x
+
c
o
t
−
1
x
)
2
−
2
t
a
n
−
1
x
(
π
2
−
t
a
n
−
1
x
)
=
5
π
2
8
2
(
t
a
n
−
1
x
)
2
−
2
(
π
2
)
t
a
n
−
1
(
x
)
−
3
π
2
8
=
0
t
a
n
−
1
x
=
−
π
4
o
r
3
π
4
Hence,
t
a
n
−
1
x
=
1
x
=
−
1
The number of integral solutions of
s
i
n
−
1
√
4
x
−
x
2
−
3
+
t
a
n
−
1
√
x
2
−
3
x
+
2
=
π
2
is
Report Question
0%
zero
0%
infinite
0%
four
0%
None of these
Explanation
Given equation is
sin
−
1
√
4
x
−
x
2
−
3
+
tan
−
1
√
x
2
−
3
x
+
2
=
π
2
tan
−
1
√
x
2
−
3
x
+
2
=
π
2
−
sin
−
1
√
4
x
−
x
2
−
3
⇒
tan
−
1
√
x
2
−
3
x
+
2
=
cos
−
1
√
4
x
−
x
2
−
3
Since,
√
x
2
−
3
x
+
2
≥
0
⇒
tan
−
1
√
x
2
−
3
x
+
2
<
π
2
Also,
√
4
x
−
x
2
−
3
≥
0
⇒
0
<
cos
−
1
√
4
x
−
x
2
−
3
≤
π
2
The value of
s
i
n
−
1
(
s
i
n
2010
0
)
+
c
o
s
−
1
(
c
o
s
2010
0
)
+
t
a
n
−
1
(
t
a
n
2010
0
)
is
Report Question
0%
π
6
0%
π
3
0%
2
π
3
0%
5
π
6
Explanation
2010
0
=
11
π
+
π
6
...(i)
Hence
s
i
n
−
1
(
s
i
n
(
2010
0
)
)
=
s
i
n
−
1
(
s
i
n
(
11
π
+
π
6
)
)
...(from(i))
c
o
s
−
1
(
c
o
s
(
2010
0
)
)
=
c
o
s
−
1
(
c
o
s
(
11
π
+
π
6
)
)
t
a
n
−
1
(
t
a
n
(
2010
0
)
)
=
t
a
n
−
1
(
t
a
n
(
11
π
+
π
6
)
)
Therefore
s
i
n
−
1
(
s
i
n
(
11
π
+
π
6
)
)
+
c
o
s
−
1
(
c
o
s
(
11
π
+
π
6
)
)
+
t
a
n
−
1
(
t
a
n
(
11
π
+
π
6
)
)
=
−
π
6
+
π
−
π
6
+
π
6
=
5
π
6
The number of solutions of the equation
1
+
x
2
+
2
x
s
i
n
(
c
o
s
−
1
y
)
=
0
is
Report Question
0%
1
0%
2
0%
3
0%
4
Explanation
1
+
x
2
+
2
x
s
i
n
c
o
s
1
−
y
=
0
⇒
1
+
x
2
+
2
x
√
1
−
y
2
=
0
x
2
+
2
x
√
1
−
y
2
+
1
=
0
x
2
+
2
x
√
1
−
y
2
+
1
−
y
2
=
−
y
2
(
x
+
√
1
−
y
2
)
=
−
y
2
⇒
−
y
2
≥
0
x
+
√
1
−
y
2
=
0
a
n
d
y
=
0
(
x
+
√
1
−
y
2
)
2
≥
0
→
y
=
0
,
x
=
−
1
only one solution
The value of
sec
−
1
(
1
1
−
2
x
2
)
+
4
cos
−
1
√
1
+
x
2
is equal to
Report Question
0%
π
0%
2
π
0%
π
2
0%
None of these
Explanation
Given
sec
−
1
(
1
1
−
2
x
2
)
+
4
cos
−
1
√
1
+
x
2
Let
x
=
cos
θ
⟹
sec
−
1
1
1
−
2
cos
2
x
+
4
cos
−
1
(
√
1
+
cos
θ
2
)
=
sec
−
1
(
−
1
cos
2
θ
)
+
4
cos
−
1
(
√
cos
2
θ
/
2
)
=
π
−
sec
−
1
(
sec
2
θ
)
+
4
cos
−
1
(
cos
θ
/
2
)
=
π
−
2
θ
+
4
(
θ
2
)
=
π
−
2
θ
+
2
θ
=
π
The largest interval lying in
(
−
π
2
,
π
2
)
for which the function
[
f
(
x
)
=
4
−
x
2
+
cos
−
1
(
x
2
−
1
)
+
log
(
cos
x
)
]
is defined, is-
Report Question
0%
[
0
,
π
]
0%
(
−
π
2
,
π
2
)
0%
[
−
π
4
,
π
2
)
0%
[
0
,
π
2
)
Explanation
For given function to be defined
−
1
≤
x
2
−
1
≤
1
and
cos
x
>
0
⇒
0
≤
x
2
≤
2
and
−
π
2
≤
x
<
π
2
⇒
0
≤
x
≤
4
and
−
π
2
≤
x
<
π
2
Taking common of both we get
x
∈
[
0
,
π
2
)
The number of real solutions of
t
a
n
−
1
(
√
x
(
x
+
1
)
+
s
i
n
−
1
√
(
x
2
+
x
+
1
)
=
π
2
is
Report Question
0%
0
0%
1
0%
2
0%
infinite
Explanation
From the above expression
−
1
≤
√
x
2
+
x
+
1
≤
1
and
x
(
x
+
1
)
≥
0
⇒
0
≤
x
2
+
x
+
1
≤
1
a
n
d
x
2
+
x
+
1
≥
1
⇒
x
2
+
x
+
1
=
1
⇒
x
2
+
x
=
0
⇒
x
(
x
+
1
)
=
0
Hence there will be two solutions. One at
x
=
−
1
and another at
x
=
0
.
If
x
>
0
and
c
o
s
−
1
(
12
x
)
+
c
o
s
−
1
(
35
x
)
=
π
2
,
then x is
Report Question
0%
7
0%
39
0%
37
0%
−
37
Explanation
c
o
s
−
1
(
a
)
+
c
o
s
−
1
(
b
)
=
π
2
⇒
c
o
s
−
1
(
a
)
=
π
2
−
c
o
s
−
1
(
b
)
⇒
c
o
s
−
1
(
a
)
=
s
i
n
−
1
(
b
)
Therefore,
a
2
+
b
2
=
1
Substituting the value of a and b we get
12
x
2
+
35
x
2
=
1
x
2
=
1369
x
=
37
,
−
37
However , since
x
>
0
x
=
37
The value of
sin
−
1
{
tan
(
cos
−
1
√
2
+
√
3
4
+
cos
−
1
√
12
4
−
cosec
−
1
√
2
)
}
, is
Report Question
0%
0
0%
π
2
0%
−
π
2
0%
π
Explanation
Given:
E
=
sin
−
1
(
tan
(
cos
−
1
√
2
+
√
3
2
+
cos
−
1
√
3
2
−
cosec
−
1
√
2
)
)
We know that,
cos
15
0
=
√
2
+
√
3
4
Therefore,
⇒
E
=
sin
−
1
(
tan
(
15
0
+
30
0
−
45
0
)
)
⇒
E
=
sin
−
1
(
tan
0
0
)
⇒
E
=
sin
−
1
0
=
0
If
c
o
s
e
c
−
1
(
c
o
s
e
c
(
x
)
)
and
c
o
s
e
c
(
c
o
s
e
c
−
1
(
x
)
)
are equal functions, then the maximum range of value of
x
is
Report Question
0%
[
−
π
2
,
−
1
]
∪
[
1
,
π
2
]
0%
(
−
π
2
,
−
1
)
∪
(
1
,
π
2
)
0%
(
−
∞
,
−
1
]
∪
[
1
,
∞
)
0%
(
−
∞
,
−
1
)
∪
(
1
,
∞
)
Explanation
The range of
c
o
s
e
c
−
1
(
c
o
s
e
c
(
x
)
)
=
[
−
π
2
,
0
)
∪
(
0
,
π
2
]
The range of
c
o
s
e
c
(
c
o
s
e
c
−
1
(
x
)
)
=
(
−
∞
,
−
1
]
∪
[
1
,
∞
)
Since it is given that both the functions are equal, then the range of value x common to both will be
=
[
−
π
2
,
−
1
]
∪
[
1
,
π
2
]
If
[
sin
−
1
cos
−
1
sin
−
1
tan
−
1
θ
]
=
1
, where
[
.
]
denotes the greatest integer function, the
θ
lies in the interval
Report Question
0%
[
tan
sin
cos
1
,
sin
tan
cos
sin
1
]
0%
[
sin
tan
cos
1
,
tan
sin
cos
sin
1
]
0%
[
tan
sin
cos
1
,
tan
sin
cos
sin
1
]
0%
None of these
Explanation
We have
[
sin
−
1
cos
−
1
sin
−
1
tan
−
1
θ
]
=
1
⇒
1
≤
sin
−
1
cos
−
1
sin
−
1
tan
−
1
θ
≤
π
2
⇒
sin
1
≤
cos
−
1
sin
−
1
tan
−
1
θ
≤
1
⇒
cos
sin
1
≥
sin
−
1
tan
−
1
θ
≥
cos
1
⇒
sin
cos
sin
1
≥
tan
−
1
θ
≥
sin
cos
1
⇒
tan
sin
cos
sin
1
≥
θ
≥
tan
sin
cos
1
cos
−
1
x
+
cos
−
1
(
x
2
+
1
2
√
3
−
3
x
2
)
is equal to
Report Question
0%
π
3
for
x
ϵ
[
1
2
,
1
]
0%
π
3
for
x
ϵ
[
0
,
1
2
]
0%
2
cos
−
1
x
−
cos
−
1
1
2
for
x
ϵ
[
1
2
,
1
]
0%
2
cos
−
1
x
−
cos
−
1
1
2
for
x
ϵ
[
0
,
1
2
]
Explanation
c
o
s
−
1
x
+
c
o
s
−
1
(
x
.
1
2
+
√
3
2
√
1
−
x
2
)
=
c
o
s
−
1
x
+
c
o
s
−
1
1
3
−
c
o
s
−
1
x
For
1
2
≤
x
≤
1
c
o
s
−
1
x
+
c
o
s
−
1
(
x
.
1
2
+
√
3
2
√
1
−
x
2
)
=
c
o
s
−
1
x
+
π
3
−
c
o
s
−
1
x
=
π
3
And for
0
≤
x
≤
1
2
o
s
−
1
x
+
c
o
s
−
1
(
x
.
1
2
+
√
3
2
√
1
−
x
2
)
=
c
o
s
−
1
x
+
c
o
s
−
1
x
−
c
o
s
−
1
1
2
=
2
c
o
s
−
1
x
−
c
o
s
−
1
1
2
The range of values of
p
for which the equation
sin
cos
−
1
(
cos
(
tan
−
1
x
)
)
=
p
has a solution is
Report Question
0%
(
−
1
√
2
,
2
√
2
)
0%
[
0
,
1
)
0%
(
1
√
2
1
)
0%
(
−
1
,
1
)
Explanation
Here The range of
c
o
s
(
t
a
n
−
1
(
x
)
)
will be [-1,1] since the range of
c
o
s
θ
is always
[
−
1
,
1
]
Hence for
c
o
s
(
t
a
n
−
1
(
x
)
)
=
1
s
i
n
(
c
o
s
−
1
(
1
)
)
=
s
i
n
(
0
)
=
0
For
c
o
s
(
t
a
n
−
1
(
x
)
)
=
0
s
i
n
(
c
o
s
−
1
(
0
)
)
=
s
i
n
(
π
2
)
=
1
For
c
o
s
(
t
a
n
−
1
(
x
)
)
=
−
1
s
i
n
(
c
o
s
−
1
(
−
1
)
)
=
s
i
n
(
π
)
=
0
Howere p=1 for
x
→
∞
Hence range of p=[0,1).
The solution set of the equation
sin
−
1
√
1
−
x
+
cos
−
1
x
=
cot
−
1
(
√
1
−
x
2
x
)
−
sin
−
1
x
Report Question
0%
[
−
1
,
1
]
−
{
0
}
0%
(
0
,
1
]
∪
{
−
1
}
0%
[
−
1
,
0
)
∪
{
1
}
0%
{
1
}
Explanation
sin
−
1
√
1
−
x
+
cos
−
1
x
=
cot
−
1
(
√
1
−
x
2
x
)
−
sin
−
1
x
⇒
sin
−
1
√
1
−
x
+
π
2
=
cot
−
1
(
√
1
−
x
2
x
)
⇒
π
2
−
cot
−
1
(
√
1
−
x
2
x
)
=
−
sin
−
1
√
1
−
x
⇒
tan
−
1
(
√
1
−
x
2
x
)
=
−
sin
−
1
√
1
−
x
Thus is only true when,
x
=
{
1
}
0:0:2
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
40
41
42
43
44
45
46
47
48
49
50
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
40
41
42
43
44
45
46
47
48
49
50
0
Answered
1
Not Answered
49
Not Visited
Correct : 0
Incorrect : 0
Report Question
×
What's an issue?
Question is wrong
Answer is wrong
Other Reason
Want to elaborate a bit more? (optional)
Practice Class 12 Commerce Maths Quiz Questions and Answers
<
>
Support mcqexams.com by disabling your adblocker.
×
Please disable the adBlock and continue.
Thank you.
Reload page