A) 6\[\pi \]
B) 7\[\pi \]
C) 8\[\pi \]
D) 4\[\pi \]
Correct Answer: A
Solution :
[a] \[3+3\cos \theta =2-2{{\cos }^{2}}\theta \] \[\Rightarrow 2{{\cos }^{2}}\theta +3\cos \theta +1=0\] \[\Rightarrow (2cos\theta +1)(cos\theta +1)=0\] \[\therefore \cos \theta =-1or\cos \theta =-\frac{1}{2}\] if \[\cos \theta =-1\] then \[\theta =\pi ,3\pi ,5\pi ,7\pi ,9\pi ,...\] if \[\cos \theta =-1/2\] then \[\theta =\frac{2\pi }{3},\frac{4\pi }{3},\frac{8\pi }{3},\frac{10\pi }{3},\frac{14\pi }{3},...\] Solutions in increasing order are \[0<\frac{2\pi }{3}<\pi <\frac{4\pi }{3}<\frac{8\pi }{3}<3\pi <\frac{10\pi }{3}<\frac{14\pi }{3}<5\pi ,..\] \[\therefore \,{{\theta }_{3}}+{{\theta }_{7}}=\frac{4\pi }{3}+\frac{14\pi }{3}=6\pi \]
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