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Mathematics
The sum of all the solutions of the equation cos x⋅ cos ( (π /3)+x )⋅ cos ( (π /3)-x )=(1/4) , x∈ [0, 6π ] is
Q. The sum of all the solutions of the equation
cos
x
⋅
cos
(
3
π
+
x
)
⋅
cos
(
3
π
−
x
)
=
4
1
,
x
∈
[
0
,
6
π
]
is
2231
212
Manipal
Manipal 2010
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A
15
π
B
30
π
C
3
110
π
D
None of these
Solution:
We have,
cos
x
⋅
cos
(
3
π
+
x
)
cos
(
3
π
−
x
)
=
4
1
⇒
cos
x
(
4
1
cos
2
x
−
4
3
sin
2
x
)
=
4
1
or
4
c
o
s
x
(
4
cos
2
x
−
3
)
=
4
1
⇒
4
cos
3
x
−
3
cos
x
=
1
(
∵
cos
3
x
=
4
cos
3
x
−
3
cos
x
)
or
cos
3
x
=
1
⇒
3
x
=
2
nπ
⇒
x
=
3
2
nπ
, where
n
=
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
(
∵
x
∈
[
0
,
6
π
])
∴
The required sum
=
3
2
π
n
=
0
∑
9
n
=
30
π