Complex Numbers And Quadratic Equations question 613
Question: If $ \alpha ,\beta $ are the roots of the equation $ x^{2}+2x+4=0, $ then $ \frac{1}{{{\alpha }^{3}}}+\frac{1}{{{\beta }^{3}}} $ is equal to [Kerala (Engg.) 2002]
Options:
A) $ -\frac{1}{2} $
B) $ \frac{1}{2} $
C) 32
D) $ \frac{1}{4} $
Show Answer
Answer:
Correct Answer: D
Solution:
Here, $ \alpha +\beta =-2 $ and $ \alpha \beta =4 $
$ \therefore \frac{1}{{{\alpha }^{3}}}+\frac{1}{{{\beta }^{3}}}=\frac{{{\alpha }^{3}}+{{\beta }^{3}}}{{{(\alpha \beta )}^{3}}} $ $ =\frac{{{(\alpha +\beta )}^{3}}-3\alpha \beta (\alpha +\beta )}{{{(\alpha \beta )}^{3}}} $ $ =\frac{{{(-2)}^{3}}-3(-2)(4)}{{{(4)}^{3}}} $ = $ \frac{16}{64}=\frac{1}{4} $ .
BETA
