11th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-1

  • question_answer If \[\alpha +\beta =\frac{\pi }{2}\]and \[\beta +\gamma =\alpha \]then tan a equal to.

    A) \[2tan\text{ }\beta tan\,\gamma \]           

    B) \[tan\text{ }\beta +2\text{ }tan\text{ }\gamma \]

    C) \[tan\text{ }\beta -2\text{ }tan\text{ }\gamma \]               

    D) \[2\left( tan\text{ }\beta +tan\text{ }\gamma \text{ } \right)\]

    Correct Answer: B

    Solution :

    [b] \[\because \alpha +\beta =\frac{\pi }{2}\Rightarrow \alpha =\frac{\pi }{2}-\beta \] \[\Rightarrow tan\alpha =tan\left( \frac{\pi }{2}-\beta  \right)=cot\beta =\frac{1}{\tan \beta }\] \[\Rightarrow tan\alpha \,tan\beta =1\]               ............(i) \[\therefore \alpha =\beta +\gamma \] \[tan\alpha =tan\left( \beta +\gamma  \right)=\frac{tan\beta +\tan \gamma }{1-\tan \beta .tan\gamma }\] \[\Rightarrow \] \[tan\beta +tan\gamma =tan\alpha -tan\alpha .tan\beta .\tan \gamma \] \[=tan\alpha -1.tan\beta \left[ from\left( i \right) \right]\] \[\Rightarrow \] \[tan\beta +2.\tan \gamma =tan\,\alpha \]

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