A) \[\left( 0,\frac{\pi }{4} \right]\]
B) \[\left[ \frac{\pi }{6},\frac{\pi }{3} \right]\]
C) \[\left[ \frac{\pi }{4},\frac{\pi }{2} \right]\]
D) \[\left( \frac{\pi }{3},\frac{\pi }{2} \right]\]
Correct Answer: C
Solution :
It makes \[\theta \]with x and y-axes. \[l=\cos \theta ,m=\cos \theta ,n=\cos (\pi -2\theta )\] we have \[{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1\] \[\Rightarrow \]\[{{\cos }^{2}}\theta +{{\cos }^{2}}\theta +{{\cos }^{2}}(\pi -2\theta )=1\] \[\Rightarrow \]\[2{{\cos }^{2}}\theta +{{(-\cos 2\theta )}^{2}}=1\] \[\Rightarrow \]\[2{{\cos }^{2}}\theta -1+{{\cos }^{2}}2\theta =0\] \[\Rightarrow \]\[\cos 2\theta -[1+cos2\theta ]=0\] \[\Rightarrow \]\[\cos 2\theta =0\]or\[\cos 2\theta =-1\] \[\Rightarrow \]\[2\theta =\pi /2\]or\[2\theta =\pi \] \[\Rightarrow \]\[\theta =\pi /4\]or\[\theta =\frac{\pi }{2}\]\[\Rightarrow \]\[\theta =\left[ \frac{\pi }{4},\frac{\pi }{2} \right]\]
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