Question

Statement 1 : lf $x\cos \theta =y\cos ({120}^0+\theta )=z\cos (\theta +{240}^0)$, then $xy+yz+xz=0$
Statement 11 :Value of $\cos \alpha +\cos ({120}^0+\alpha )+\cos ({120}^0-\alpha )=0$
Then which of the above statements is correct?

• A. Only $I$
• B. Only $II$
• C. Both $I$ & $II$
• D. Neither $I$ nor $II$

Answer 1

Answer: (C)

Both $I$ & $II$

Statement - 1
Let $xcos\theta =ycos(120+\theta )=2cos(\theta +240)=k$
$\therefore cos\theta +cos(120+\theta )+cos(\theta +240)=k(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})$
$\therefore cos\theta +2(-cos\theta )cos60=k(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})$
$cos\theta -cos\theta =kxyz(xy+xz+yz)$
$k\left(xyz\right)(xy+yz+zx)=0$
$\therefore xy+yz+zx=0$ True
Statement - 2
$cos\alpha +cos(120-\alpha )+cos(120+\alpha )$
$=cos\alpha +\left(2cos120cos\alpha \right)$
$=cos\alpha -cos\alpha =0$ True.
Both 1 and 2 are True.