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Question

Let y=(sinx+sin2x+sin3x)2+(cosx+cos2x+cos3x)2 then which of the following(s) is correct?

A
dydx when x=π2 is 2
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B
Value of y when x=π5 is 3+52
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C
Value of y when x=π12 is 1+2+32
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D
y simplifies to (1+2cosx) in [0,π]
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Solution

The correct options are
A dydx when x=π2 is 2
B Value of y when x=π5 is 3+52
C Value of y when x=π12 is 1+2+32
y=(sinx+sin2x+sin3x)2+(cosx+cos2x+cos3x)2
=(sin2x+cos2x)+(sin22x+cos22x)+(sin23x+cos23x)+2(sin2xsinx+cos2xcosx)+2(sin2xsin3x+cos2xcos3x)+2(sinxsin3x+cosxcos3x
=1+1+1+2cos(2xx)+2cos(3x2x)+2cos(3xx)=3+4cosx+2cos2x=4cos2x+4cosx+1
=|2cosx+1|
At x=π2,dydx=2sinx=2sinπ/2=2
At x=π5,y=2cosπ5+1=2×5+14+1=5+32
At x=π12,y=2cosπ12+1=2×3+122+1=1+2+32

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