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JEE Main & Advanced Mathematics Three Dimensional Geometry Question Bank Critical Thinking

  • question_answer
    The angle between two diagonals of a cube will be [MP PET 1996, 2000; RPET 2000, 02; UPSEAT 2004]

    A) \[{{\sin }^{-1}}1/3\]

    B) \[{{\cos }^{-1}}1/3\]

    C) Variable

    D) None of these

    Correct Answer: B

    Solution :

    • Let the cube be of side 'a'                   
    • \[O\,(0,\,\,0,\,\,0),\,\,D\,(a,\,\,a,\,\,a),\,\,B\,(0,\,\,a,\,\,0),\,\,G\,(a,\,\,0,\,\,a)\]                   
    • Then equation of OD and BG are \[\frac{x}{a}=\frac{y}{a}=\frac{z}{a}\] and \[\frac{x}{a}=\frac{y-a}{-a}=\frac{z}{a}\] respectively.        
    • Hence, angle between OD and BG is                   
    • \[{{\cos }^{-1}}\left( \frac{{{a}^{2}}-{{a}^{2}}+{{a}^{2}}}{\sqrt{3{{a}^{2}}}.\,\sqrt{3{{a}^{2}}}} \right)={{\cos }^{-1}}\,\left( \frac{1}{3} \right)\]. Note: Students should remember this question as a fact.


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