Question

Let A and $B$ denote the statements
$A:\cos \alpha +\cos \beta +\cos \gamma =0$
$B$ : $\sin \alpha +\sin \beta +\sin \gamma =0$
If $\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )=-3/2$, then

• A. A is true and B is false
• B. A is false and B is true
• C. both A and B are true
• D. both A and B are false

Answer 1

Answer: (C)

both A and B are true

$\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )=-3/2$
$\Rightarrow 2\cos (\beta -\gamma )+2\cos (\gamma -\alpha )+2\cos (\alpha -\beta )=-3$
$\Rightarrow 2\Sigma \cos \beta \cos \gamma +2\Sigma \sin \alpha \sin \beta +3=0$
$\Rightarrow (\cos \alpha +\cos \beta +\cos \gamma {)}^2+(\sin \alpha +\sin \beta +\sin \gamma {)}^2=0$
$\Rightarrow \cos \alpha +\cos \beta +\cos \gamma =0$ and $\sin \alpha +\sin \alpha +\sin \gamma =0$
Hence, option 'C' is correct.