question_answer

In \[\Delta \]ABC. \[\angle \]B = \[90{}^\circ \] and AB: BC = 2: 1. The value of sin A + cot C is

A) \[3+\sqrt{5}\]               

B) \[\frac{2+\sqrt{5}}{2\sqrt{5}}\]

C) \[2+\sqrt{5}\]                           

D) \[3\sqrt{5}\]

Correct Answer: B

Solution :

\[\frac{AB}{BC}=\frac{2}{1}\] \[\Rightarrow AB=2k,BC=k\] \[\therefore AC=\sqrt{{{(2k)}^{2}}+{{k}^{2}}}=\sqrt{5{{k}^{2}}}=\sqrt{5k}\] \[\therefore \] sin A + cot C=\[\frac{BC}{AC}+\frac{BC}{AB}=\frac{K}{\sqrt{5}k}+\frac{k}{2k}\] \[=\frac{1}{\sqrt{5}}+\frac{1}{2}=\frac{2+\sqrt{5}}{2\sqrt{5}}\]