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Byju's Answer
Standard XI
Mathematics
Trigonometric Equations
The inequalit...
Question
The inequality
2
cos
x
≤
∣
∣
√
1
+
sin
2
x
−
√
1
−
sin
2
x
∣
∣
≤
√
2
where
x
∈
[
0
,
2
π
]
holds true in the interval
[
a
π
2
,
b
π
2
]
. Then the value of
a
+
b
is
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Solution
∣
∣
√
1
+
sin
2
x
−
√
1
−
sin
2
x
∣
∣
=
∣
∣
∣
|
cos
x
+
sin
x
|
−
|
cos
x
−
sin
x
|
∣
∣
∣
=
2
min
{
|
cos
x
|
,
|
sin
x
|
}
Now,
2
cos
x
≤
2
min
{
|
cos
x
|
,
|
sin
x
|
}
≤
√
2
⇒
cos
x
≤
min
{
|
cos
x
|
,
|
sin
x
|
}
≤
1
√
2
Clearly, from the graph,
x
∈
[
π
4
,
7
π
4
]
a
+
b
=
1
2
+
7
2
=
4
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0
Similar questions
Q.
Determine all the values of
x
in the interval
x
ϵ
[
0
,
2
π
]
which satisfy the inequality
2
cos
x
≤
∣
∣
√
1
+
sin
2
x
−
√
1
−
sin
2
x
∣
∣
≤
2
Q.
If the inequality
sin
2
x
+
a
cos
x
+
a
2
>
1
+
cos
x
holds for any
x
∈
R
,
then the largest negative integral value of
a
is
Q.
The inequality
2
cos
x
≤
∣
∣
√
1
+
sin
2
x
−
√
1
−
sin
2
x
∣
∣
≤
√
2
where
x
∈
[
0
,
2
π
]
holds true in the interval
[
a
π
2
,
b
π
2
]
. Then the value of
a
+
b
is
Q.
If the inequality
sin
2
x
+
a
cos
x
+
a
2
>
1
+
cos
x
holds for any
x
∈
R
, then the largest negative integral value of
a
is
Q.
The number of solution(s) of
sin
2
x
+
cos
4
x
=
2
in the interval
(
0
,
2
π
)
is
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