Complex Numbers And Quadratic Equations question 459
Question: If $ \alpha ,\beta $ are the roots of the equation $ u^{2}-2u+2=0 $ and if $ \cot \theta =x+1 $ , then $ [{{(x+\alpha )}^{n}}-{{(x+\beta )}^{n}}]/[\alpha -\beta ] $ is equal to
Options:
A) $ \frac{\sin n\theta }{{{\sin }^{n}}\theta } $
B) $ \frac{\cos n\theta }{{{\cos }^{n}}\theta } $
C) $ \frac{\sin n\theta }{{{\cos }^{n}}\theta } $
D) $ \frac{\cos n\theta }{{{\sin }^{n}}\theta } $
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Answer:
Correct Answer: A
Solution:
[a] $ u^{2}-2u+2=0\Rightarrow u=1\pm i $
$ \Rightarrow \frac{{{(x+\alpha )}^{2}}-{{(x+\beta )}^{n}}}{\alpha -\beta } $ $ =\frac{{{[(cot\theta -1)+(1+i)]}^{n}}-{{[(cot\theta -1)+(1-i)]}^{n}}}{2i} $ $ (\therefore cot\theta -1=x) $ $ =\frac{{{(cos\theta +isin\theta )}^{n}}-{{(cos\theta -isin\theta )}^{n}}}{{{\sin }^{n}}\theta 2i} $ $ =\frac{2i\sin n\theta }{{{\sin }^{n}}\theta 2i}=\frac{\sin n\theta }{{{\sin }^{n}}\theta } $
BETA
