Question

the number k is such that tan{arctan(2) + arc tan (20k)} = k. the sum of all possible values of k is -

a) -19/40 b) -21/40 c)0 d)1/5

Answer 1

$$\begin{gathered} \tan \left({\tan }^{-1}\left(2\right)+{\tan }^{-1}\left(20k\right)\right)=k \\ \Rightarrow \tan \left({\tan }^{-1}\frac{2+20k}{1-2\times 20k}\right)=k \\ \Rightarrow \tan \left({\tan }^{-1}\frac{2+20k}{1-40k}\right)=k \\ \Rightarrow \frac{2+20k}{1-40k}=k \\ \Rightarrow 2+20k=k-40k^2 \\ \Rightarrow -40k^2-19k-2=0 \\ \Rightarrow 40k^2+19k+2=0 \\ Sincetheaboveequationisquadratic.Therefore, \\ Sumofroots=-\frac{19}{40} \\ Hence,sumofallpossiblevaluesofkis\mathbf{-}\frac{\mathbf{19}}{\mathbf{40}} \end{gathered}$$