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CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2009

  • question_answer
    If tan A and tan B are the roots of \[ab{{x}^{2}}-{{c}^{2}}x+\]\[ab=0\] where a, b, c are the sides of the  triangle ABC, then the value of \[si{{n}^{2}}A+si{{n}^{2}}B+si{{n}^{2}}C\text{ }is\]

    A)  1                                            

    B)  3

    C)  4                            

    D)         2

    E)  5

    Correct Answer: D

    Solution :

    Given, tan A and tan B are the roots of the equation \[ab{{x}^{2}}-{{c}^{2}}x+ab=0\] \[\therefore \] \[\tan A+\tan B=\frac{{{c}^{2}}}{ab},\tan A\tan B=1\] Now, \[\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\]                                 \[=\frac{{{c}^{2}}/ab}{1-1}=\infty \] \[\Rightarrow \] \[A+B=\frac{\pi }{2}\]\[\Rightarrow \] \[C=\frac{\pi }{2}\] \[\therefore \] \[si{{n}^{2}}A+si{{n}^{2}}B+si{{n}^{2}}C\] \[={{\sin }^{2}}\left( \frac{\pi }{2}-B \right)+{{\sin }^{2}}B+{{\sin }^{2}}\frac{\pi }{2}\] \[={{\cos }^{2}}B+{{\sin }^{2}}B+1\] \[=2\]


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