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JEE Advanced 2022 Paper 1
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© examsnet.com
Question : 15
Total: 54
Consider the following lists:
List-I
List-II
(I)
{
x
∈
[
−
2
π
3
,
2
π
3
]
:
cos
x
+
sin
x
=
1
}
(P)
has two elements
(II)
{
x
∈
[
−
5
π
18
,
5
π
18
]
:
√
3
tan
3
x
=
1
}
(Q)
has three elements
(III)
{
x
∈
[
−
6
π
5
,
6
π
5
]
:
2
cos
(
2
x
)
=
√
3
}
(R)
has four elements
(IV)
{
x
∈
[
−
7
π
4
,
7
π
4
]
:
sin
x
−
cos
x
=
1
}
(S)
has five elements
(T)
has six elements
The correct option is:
(
I
)
→
(
P
)
;
(
II
)
→
(
S
)
;
(
III
)
→
(
P
)
;
(
IV
)
→
(
S
)
(
I
)
⟶
(
P
)
;
(
II
)
⟶
(
P
)
;
(
III
)
⟶
(
T
)
;
(
IV
)
⟶
(
R
)
(
I
)
⟶
(
Q
)
;
(
II
)
→
(
P
)
;
(
III
)
→
(
T
)
;
(
IV
)
→
(
S
)
(
I
)
→
(
Q
)
;
(
II
)
⟶
(
S
)
;
(
III
)
→
(
P
)
;
(
IV
)
→
(
R
)
Validate
Solution:
(I)
{
x
∈
[
−
2
π
3
,
2
π
3
]
:
cos
x
+
sin
x
=
1
}
cos
x
+
sin
x
=
1
⇒
1
√
2
cos
x
+
1
√
2
sin
x
=
1
√
2
⇒
cos
(
x
−
π
4
)
=
cos
π
4
⇒
x
−
π
4
=
2
n
π
±
π
4
;
n
∈
Z
⇒
x
=
2
n
π
;
x
=
2
n
π
+
π
2
;
n
∈
Z
⇒
x
∈
{
0
,
π
2
}
in given range has two solutions
(II)
{
x
∈
[
−
5
π
18
,
5
π
18
]
:
√
3
tan
3
x
=
1
}
√
3
tan
3
x
=
1
⇒
tan
3
x
=
1
√
3
⇒
3
x
=
n
π
+
π
6
⇒
x
=
(
6
n
+
1
)
π
18
;
n
∈
Z
⇒
x
∈
{
π
18
,
−
5
π
18
}
in given range has two solutions
(III)
{
x
∈
[
−
6
π
5
,
6
π
5
]
:
2
cos
(
2
x
)
=
√
3
}
2
cos
2
x
=
√
3
⇒
cos
2
x
=
√
3
2
=
cos
π
6
⇒
2
x
=
2
n
π
±
π
6
;
n
∈
Z
⇒
x
=
n
π
±
π
12
;
n
∈
Z
x
∈
{
±
π
12
,
π
±
π
12
,
−
π
±
π
12
}
Six solutions in given range
(IV)
{
x
∈
[
−
7
π
4
,
7
π
4
]
:
sin
x
−
cos
x
=
1
}
cos
x
−
sin
x
=
−
1
⇒
cos
(
x
+
π
4
)
=
−
1
√
2
=
cos
3
π
4
⇒
x
+
π
4
=
2
n
π
±
3
π
4
;
n
∈
Z
⇒
x
=
2
n
π
+
π
2
or
x
=
2
n
π
−
π
;
n
∈
Z
⇒
x
∈
{
π
2
,
−
3
π
2
,
π
,
−
π
}
four solutions in given range
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