MCQ
$\int\limits_{\frac{\pi }{6}}^{\frac{{5\pi }}{6}} {\left( {\frac{1}{2}{{(3\sin \theta )}^2} - \frac{1}{2}{{(1 + \sin \theta )}^2}} \right)\,d\theta } $
- A$\pi -\sqrt 3 $
- ✓$\pi$
- C$\pi -2\sqrt 3 $
- D$\pi +\sqrt 3 $
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$a+b+c $$ =x $ ; $a+b \omega+c \omega^2 $$ =y $ ; $a+b \omega^2+c \omega $$ =z .$
Then the value of $\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}$ is
$(A)$ $x^2+2 \sqrt{3} y=3+\sqrt{3}$
$(B)$ $x^2-2 \sqrt{3} y=3+\sqrt{3}$
$(C)$ $x^2+2 \sqrt{3} y=3-\sqrt{3}$
$(D)$ $x^2-2 \sqrt{3} y=3-\sqrt{3}$