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If $\cos^{-1}\alpha+\cos^{-1}\beta+\cos^{-1}\gamma=3\pi$,then $\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)$ equals\[(A)\;0\quad(B)\quad 1\quad(C)\quad 6\quad(D)\quad 12\]

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Toolbox:
  • Principal interval of cos is [\(0,\pi\)]
  • \(cos\pi=-1\)
Ans - ( C ) 6
Max value of \( cos^{-1}x=\pi\) in the principal interval
and given that
\(cos^{-1}\alpha+cos^{-1}\beta+cos^{-1}\gamma=3\pi\)
 
\( \Rightarrow cos^{-1}\alpha = cos^{-1}\beta = cos^{-1}\gamma=\pi\)
\( \Rightarrow \alpha = \beta = \gamma =cos\pi= 1\)
\(\alpha(\beta+\gamma)+\beta(\gamma+\alpha)+\gamma(\alpha+\beta)=\)
\(-1(-2)-1(-2)-1(-2)=2+2+2=6\)

 

answered Feb 18, 2013 by thanvigandhi_1
edited Mar 9, 2013 by rvidyagovindarajan_1
 

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