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KVPY Sample Paper KVPY Stream-SX Model Paper-23

  • question_answer
    Let \[S=\left\{ x\in (-\pi ,\pi ):x\ne 0,\pm \frac{\pi }{2} \right\}.\] The sum of all distinct solution of the equation\[\sqrt{3}\sec x+\text{cosec}\,x+2\,(\tan x-\cot x)=0\]in the set S is equal to

    A) \[-\frac{7\pi }{9}\]

    B) \[-\frac{2\pi }{9}\]

    C) \[0\]

    D) \[\frac{5\pi }{9}\]

    Correct Answer: C

    Solution :

    We have,\[\sqrt{3}\sec x+\text{cosec}\,x+2\,(\tan x-\cot x)=0\]
    \[\because \]\[\frac{\sqrt{3}}{\cos x}+\frac{1}{\sin x}+2\left( \frac{\sin x}{\cos x}-\frac{\cos }{\sin x} \right)=0\]\[\Rightarrow \]\[\sqrt{3}\sin x+\cos x+2\,({{\sin }^{2}}x-{{\cos }^{2}}x)=0\]\[\Rightarrow \]\[\sqrt{3}\sin x+\cos x=2\cos 2x\]
    \[\Rightarrow \]\[\cos \left( x-\frac{\pi }{3} \right)=\cos 2x\]\[\Rightarrow \]\[\cos 2x=\cos \left( x-\frac{\pi }{3} \right)\]\[\Rightarrow \]\[2x=2n\pi \pm \left( x-\frac{\pi }{3} \right)\]\[\Rightarrow \]\[x=(6n-1)\frac{\pi }{3}\]or \[(6n+1)\frac{\pi }{9}\]
    \[\Rightarrow \]\[x=-\frac{\pi }{3},\frac{\pi }{9},\frac{7\pi }{9}\]and \[-\frac{5\pi }{9}\]in \[(-\pi \pi )\]
    \[\therefore \] Sum \[=\frac{-\pi }{3}+\frac{\pi }{9}+\frac{7\pi }{9}-\frac{5\pi }{9}=0\]


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