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Question

Let S={x(π,π):x0,±π2}. The sum of all distinct solutions 3secx+cscx+2(tanxcotx)=0 in the set S is equal to

A
7π9
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B
2π9
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C
0
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D
5π9
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Solution

The correct option is C 0
3secx+cscx=2(cotxtanx)
3cosx+1sinx=2[cosxsinxsinxcosx]
3sinx+cosx=2(cos2xsin2x)
Dividing both sides by 2,
32sinx+12cosx=cos2x
cos60sinx+cos60cosx=cos2x
cos(xπ3)=cos2x
2x=2nπ±(xπ3)
When 2x=2nπ+xπ3
x=2nππ3

When 2x=2nπx+π3
3x=2nπ+π3
x=2nπ3+π9

For n=0,x=π3x=π9
For n=1,x=7π3x=2π3+π9=5π9
For n=1, x=5π3x=2π3+π3=7π3
x=π3,π9,5π9,7π9

Sum of distinct solutions=π3+π9+5π9+7π9
=0

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