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Question

Find the value(s) of k for which the equation sinx+cos(k+x)+cos(kx)=2 has real solutions.

A
nππ3knπ+π3,nI
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B
nπ2π3knπ+2π3,nI
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C
nππ6knπ+π6,nI
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D
nπ3π4knπ+3π4,nI
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Solution

The correct option is C nππ6knπ+π6,nI
Given, sinx+cos(k+x)+cos(kx)=2
This equation is of the form acosx+bsinx=c
Here a=2cosk,b=1 and c=2
Since for real solutions, |c|a2+b2
|2|1+4cos2k21+4cos2k
cos2k34sin2k14
sin2k=140[sink+12][sink12]0
12sink12nππ6knπ+π6

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