KVPY Sample Paper KVPY Stream-SX Model Paper-5

  • question_answer
    If \[\sin \alpha +cos\beta =\frac{1}{\sqrt{2}}\] and \[\cos \alpha +\sin \beta =\sqrt{\frac{2}{3}},\] then the value of \[\tan \left( \frac{\alpha -\beta }{2} \right)\] is

    A) \[3\sqrt{3}-6\]               

    B) \[\sqrt{3}-2\]

    C) \[4\sqrt{3}-7\]   

    D) \[4\sqrt{3}+7\]

    Correct Answer: C

    Solution :

    \[\sin \alpha +\cos \beta =\frac{1}{\sqrt{2}}\]                  ? (i)
    \[\cos \alpha +\sin \beta =\frac{\sqrt{2}}{\sqrt{3}}\]                     ? (ii)
    On subtracting Eq. (ii) from Eq. (i), we get \[(\sin \alpha -\sin \beta )+(\cos \beta -\cos \alpha )=\frac{1}{\sqrt{2}}-\frac{\sqrt{2}}{\sqrt{3}}\]
    \[2\cos \frac{\alpha +\beta }{2}\sin \left( \frac{\alpha -\beta }{2} \right)+2\sin \frac{\alpha +\beta }{2}\]
    \[\sin \frac{\alpha -\beta }{2}=\frac{\sqrt{3}-2}{\sqrt{6}}\]
    \[2\sin \left( \frac{\alpha +\beta }{2} \right)\left[ \cos \frac{\alpha +\beta }{2}+\sin \frac{\alpha +\beta }{2} \right]\]
    \[=\frac{\sqrt{3}-2}{\sqrt{6}}\] ? (iii)
    On adding Eqs. (i) and (ii), we get \[2\sin \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}+2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}\] \[=\frac{\sqrt{3}+2}{\sqrt{6}}\]
    \[2\cos \left( \frac{\alpha -\beta }{2} \right)\left[ \sin \frac{\alpha +\beta }{2}+\cos \frac{\alpha +\beta }{2} \right]\]
    \[=\frac{\sqrt{3}+2}{\sqrt{6}}\] ? (iv)
    On dividing Eq. (iii) by Eq. (iv), we get \[\tan \left( \frac{\alpha -\beta }{2} \right)=\frac{\sqrt{3}-2}{\sqrt{3}+2}=4\sqrt{3}-7\]

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