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Let A and B denote the statements \[ \] A: $ \cos \alpha + \cos \beta + \cos \gamma = 0$ \[\] B: $ \sin \alpha + \sin \beta + \sin \gamma = 0$ \[\] If $ \cos(\beta- \alpha) + \cos( \gamma- \alpha) + \cos( \alpha - \beta)= -\large\frac{3}{2}$, then

$\begin {array} {1 1} (A)\;A\: is \: true\: and\: B\: is\: false & \quad (B)\;A \: is\: false\: and\: B\: is\: true \\ (C)\;Both\: A\: and\: B\: are\: true & \quad (D)\;Both\: A \: and \: B \: are\: false \end {array}$

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Ans : (C)
$ \cos(\beta- \alpha) + \cos(\gamma- \alpha) + \cos(\alpha- \beta)= -\large\frac{3}{2}$
$2[ \cos(\beta- \alpha) + \cos(\gamma- \alpha) + \cos(\alpha- \beta)]+3=0$
$ 2[\cos(\beta- \alpha) + \cos(\gamma- \alpha) + \cos(\alpha- \beta)]+ \sin^2 \alpha+\cos^2 \alpha+ \sin^2 \beta+ \cos^2 \beta+\sin^2 \gamma+\cos^2 \gamma=0$
$ (\sin \alpha + \sin \beta + \sin \gamma)2+( \cos \alpha + \cos \beta + \cos \gamma)2=0$
answered Jan 22, 2014 by thanvigandhi_1
 

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