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

  • question_answer If tan\[\alpha =\frac{1}{3}\]and\[\mathbf{tan}\frac{\beta }{2}=\frac{1}{2}\], then tan \[\left( \alpha +\beta  \right)\]=..........

    A) 2                                             

    B) \[-1\]             

    C) 3                                             

    D) 1

    Correct Answer: C

    Solution :

    [c] \[tan\alpha =\frac{1}{3}\] \[\And tan\frac{\beta }{2}=\frac{1}{2}\therefore \tan \beta =\frac{2\tan \frac{\beta }{2}}{1-{{\tan }^{2}}\frac{\beta }{2}}\] \[=\frac{2.\frac{1}{2}}{1.\frac{\beta }{2}}=\frac{1}{\frac{3}{4}}=\frac{4}{3}.\] Now, \[\tan \left( \alpha +\beta  \right)=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha .\tan \beta }\] \[=\frac{\frac{1}{3}+\frac{4}{3}}{1-\frac{1}{3}\times \frac{4}{3}}=\frac{\frac{5}{3}}{1-\frac{4}{9}}=\frac{\frac{5}{3}}{\frac{5}{9}}=\frac{5}{3}\times \frac{9}{5}=3\]

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