JEE Main & Advanced JEE Main Solved Paper-2014

  • question_answer
    The angle between the lines whose direction cosines satisfy the equations \[\ell +m+n=0\]and\[{{\ell }^{2}}={{m}^{2}}+{{n}^{2}}\]is :   JEE Main  Solved  Paper-2014

    A) \[\frac{\pi }{3}\]                              

    B) \[\frac{\pi }{4}\]

    C) \[\frac{\pi }{6}\]                                              

    D) \[\frac{\pi }{2}\]

    Correct Answer: A

    Solution :

                    \[{{\ell }^{2}}+m+n=0;\]                                \[{{\ell }^{2}}={{m}^{2}}+{{n}^{2}}\] \[\Rightarrow \]\[\ell =-(m+n)\] \[\Rightarrow \]\[{{m}^{2}}+{{n}^{2}}+2mn={{m}^{2}}+{{n}^{2}}\] \[\Rightarrow \]\[mn=0\]\[\Rightarrow \]\[m=0\]or\[n=0\]
    Case I Case II
    \[m=0\] \[n=0\]
    \[\ell +n=0\] \[\ell +m=0\]
    \[\Rightarrow \]\[\ell =k\] \[\ell =k\]
    \[m=0\] \[m=-k\]
    \[n=-k\] \[n=0\]
    \[\ell =1/\sqrt{2}\] \[\ell =1/\sqrt{2}\]
    \[m=0\] \[m=-1/\sqrt{2}\]
    \[n=-1/\sqrt{2}\] \[n=0\]
    \[\Rightarrow \]\[\cos \theta =\frac{1}{2}\]\[\Rightarrow \]\[\theta =\pi /3.\]

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