A) \[20\]
B) \[2\sqrt{5}\]
C) \[2\sqrt{7}\]
D) \[4\sqrt{2}\]
Correct Answer: B
Solution :
By quadratic formula, the roots of this equation are: \[\alpha ,\beta =\frac{\lambda -2\pm \sqrt{4-4\lambda +{{\lambda }^{2}}-40+4\lambda }}{2}=\frac{\lambda -2\pm \sqrt{{{\lambda }^{2}}-36}}{2}\]The magnitude of the difference of the roots is clearly\[|\sqrt{{{\lambda }^{2}}-36}|\] We have, \[{{\alpha }^{3}}+{{\beta }^{3}}=\frac{{{(\lambda -2)}^{3}}}{4}+\frac{3(\lambda -2)({{\lambda }^{2}}-36)}{4}=\frac{(\lambda -2)(4{{\lambda }^{2}}-4\lambda -104)}{4}=(\lambda -2)({{\lambda }^{2}}-\lambda -26)\] This function attains its minimum value at \[\lambda =4\]. Thus, the magnitude of the difference of the roots is clearly \[|i\sqrt{20}|=2\sqrt{5}\]. So the correct answer is option B.
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