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Question

The arithmetic mean of the roots of the equation4cos3x-4cos2x-cos(315šœ‹+x)=1 in the interval (0,315) is equal to?


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Solution

We know thatcos[(2n+1)Ļ€+x]=ā€“cosxnāˆˆI

So, cos315Ļ€+x=-cosx

Therefore, the given equation becomes

4cos3x-4cos2x+cosx=1

ā‡’4cos2x(cosxā€“1)+cosxā€“1=0ā‡’(4cos2x+1)(cosxā€“1)=0

cosx=1orcos2x=-14(not possible)

So,

cosx=1ā‡’cosx=cos0Ā°ā‡’x=2nĻ€nāˆˆI

Now, we see that 100Ļ€<315<101Ļ€

āˆ“x=2Ļ€,4Ļ€,6Ļ€,8Ļ€,...100Ļ€[āˆµ0<x<315]

We have , there are 50 terms which are in A.P.

Sum of 50terms,

S=502Ɨ(2Ļ€+100Ļ€)=25Ɨ102Ļ€

Thus, Arithmetic mean =25+102Ļ€50=51Ļ€

Hence, The arithmetic mean of roots of the given equation is 51Ļ€


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