Complex Numbers And Quadratic Equations question 829
Question: If $ z=r{e^{i\theta }}, $ then $ |e^{iz}| $ = [Kerala (Engg.) 2005]
Options:
A) $ {e^{r\sin \theta }} $
B) $ {e^{-r\sin \theta }} $
C) $ {e^{-r\cos \theta }} $
D) $ {e^{r\cos \theta }} $
Show Answer
Answer:
Correct Answer: B
Solution:
If $ z=r{e^{i\theta }}=r(\cos \theta +i\sin \theta ) $
Þ $ iz=ir(\cos \theta +i\sin \theta )=-r\sin \theta +ir\cos \theta $ or $ e^{iz}={e^{(-r\sin \theta +ir\cos \theta )}}={e^{-\sin \theta }}{e^{ri\cos \theta }} $ or $ |e^{iz}|=|{e^{-r\sin \theta }}||{e^{ri\cos \theta }}| $ $ ={e^{-r\sin \theta }}|{e^{ir\cos \theta }}| $ $ ={e^{-r\sin \theta }}{{[{{{\cos }^{2}}(r\cos \theta )+{{\sin }^{2}}(r\cos \theta )}]}^{1/2}}={e^{-r\sin \theta }} $
BETA
