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JEE Main & Advanced Sample Paper JEE Main Sample Paper-45

  • question_answer
    The maximum number of solutions of the equation\[5\,\,\cos \,x+\frac{5}{2\,\cos \,x}-5\,=2\,{{\cos }^{2}}\,x+\frac{1}{2\,{{\cos }^{2}}x},\]\[x\,\in [0,\,2\pi ]\]

    A)  3                                

    B)  4                  

    C)  5                                

    D)  2

    Correct Answer: B

    Solution :

     Let \[\cos \,\,x=t,\] so that given equation reduces to \[5t+\frac{5}{2t}-5=2{{t}^{2}}+\frac{1}{2{{t}^{2}}}\] \[\Rightarrow \] \[4{{t}^{4}}-10{{t}^{3}}+10{{t}^{2}}-5t+1=0\] \[\Rightarrow \] \[(t-1)\,(2t-1)\,[{{t}^{2}}+{{(t-1)}^{2}}]=0\] \[\Rightarrow \]            \[t=1,\,\,\frac{1}{2}\] \[\Rightarrow \]            \[\cos \,x=1,\,\frac{1}{2}\] \[\Rightarrow \]            \[x=0,\,\,\frac{\pi }{3},\,\frac{5\pi }{3},\,2\pi \]


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