KVPY Sample Paper KVPY Stream-SX Model Paper-29

  • question_answer
    With the usual notation, in \[\Delta ABC,\] if \[\angle A+\angle B=120{}^\circ ,\] \[a=\sqrt{3}-1,\] then the ratio \[\angle A:\angle B,\] is:

    A) \[7:1\]   

    B) \[5:3\]

    C) \[9:7\]   

    D) \[3:1\]

    Correct Answer: A

    Solution :

    \[\frac{\sin A}{\sin B}=\frac{\sqrt{3}+1}{\sqrt{3}-1}\]
    \[\frac{\sin A}{\sin (120-A)}=\sqrt{3}+2\]
    \[\frac{\sin A}{\sin 12\operatorname{cosA}-cos12sinA}=\sqrt{3}+2\]
    \[\frac{1}{\frac{\sqrt{3}}{2}\cot A+\frac{1}{2}}=\sqrt{3}+2.\]
    \[\frac{\sqrt{3}\cot A+1}{2}=\frac{1}{\sqrt{3}+2}=\frac{\sqrt{3}-1}{-1}\]
    \[\frac{\sqrt{3}\cot A+1}{2}=-\sqrt{3}+2\]
    \[\sqrt{3}\cot A=4-2\sqrt{3}-1\]
    \[\sqrt{3}\cot A=3-2\sqrt{3}\]
    \[\cot A=\sqrt{3}-2\]
    \[-\cot A=2-\sqrt{3}=\tan 15\]
    \[\therefore \]      \[A=105{}^\circ \]
    \[\therefore \]      \[B=15{}^\circ .\]

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