InterviewSolution
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InterviewSolution
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If a = costheta + i sin theta, b = cos 2theta - i sin 2theta, c = cos 3theta + i sin3thetaandif |{:(a,,b,,c),(b,,c,,a),(c,,a,,b):}| = 0then |
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Answer» `theta=2 k pi, k in Z` `=-(a+b+c)(a^(2)+b^(2)+c^(2)-ab-bc-ca)` `=(1)/(2) (a+b+c)[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)]=0` `rArr a+b+c=0"or" a=b=c ` `" if" a+b+c =0 , ` we have `costheta + cos 2theta + cos 3theta =0` and `sintheta-sin 2theta + sin 3theta = 0` or`cos 2 theta (2 cos theta + 1)=0` and`sin 2 theta (1 -2 cos theta ) =0 .............(1)` which is notpossibleas `cos 2 theta = 0`given`sin 2 theta ne 0, cos theta ne 1/2` and`cos theta = 1/2` gives `sin 2 theta ne 0, cos thetane 1/2` . Therefore , Eq. `(1)` does nothold SIMULTANEOUSLY .Therefore , `a+b+c ne 0` `rArr a=b=c` `:. e^(itheta)=e^(-2itheta) =e^(3itheta)` whichis satisfied only byby `e^(i theta) =1`, i.e., `cos theta =1, sin theta =0` so `theta = 2k pi , k in Z` |
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