KVPY Sample Paper KVPY Stream-SX Model Paper-29

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 :

\[a=\sqrt{3}+1\]

\[b=\sqrt{3}-1\]

\[\frac{\sin A}{\sin B}=\frac{\sqrt{3}+1}{\sqrt{3}-1}\]

\[=\frac{3+1+2\sqrt{3}}{2}=2+\sqrt{3}\]

\[\frac{\sin A}{\sin (120-A)}=\sqrt{3}+2\]

\[\frac{\sin A}{\sin 12\operatorname{cosA}-cos12sinA}=\sqrt{3}+2\]

\[\frac{1}{\frac{\sqrt{3}}{2}\cot A+\frac{1}{2}}=\sqrt{3}+2.\]

\[\frac{\sqrt{3}\cot A+1}{2}=\frac{1}{\sqrt{3}+2}=\frac{\sqrt{3}-1}{-1}\]

\[\frac{\sqrt{3}\cot A+1}{2}=-\sqrt{3}+2\]

\[\sqrt{3}\cot A=4-2\sqrt{3}-1\]

\[\sqrt{3}\cot A=3-2\sqrt{3}\]

\[\cot A=\sqrt{3}-2\]

\[-\cot A=2-\sqrt{3}=\tan 15\]

\[\therefore \] \[A=105{}^\circ \]

\[\therefore \] \[B=15{}^\circ .\]