Question

If α, β are the roots of the equation
\(x^2-(5+3^{\sqrt{log_35}}-5^{\sqrt{log_53}})+3(3^{(log_35)^{\frac{1}{3}}}-5^{(log_53)^{\frac{2}{3}}}-1) = 0\)
then the equation, whose roots are α + 1/β and β + 1/α , is

• A. 3x² – 20x – 12 = 0
• B. 3x² – 10x – 4 = 0
• C. 3x² – 10x + 2 = 0
• D. 3x² – 20x + 16 = 0

Answer 1

The Correct Option is B

The correct answer is (B) : 3x² – 10x – 4 = 0
\(3^{\sqrt{\log_{3}5}} - 5^{\sqrt{\log_{5}3}} = 3^{\sqrt{\log_{3}5}} - (3\log_{3}5)^{\sqrt{\log_{5}3}}= 0\)
\(3^{(\log_{3}5)^{\frac{1}{3}}} - 5^{(\log_{5}3)^{\frac{2}{3}}} = 5^{(\log_{5}3)^{\frac{2}{3}}} - 5^{(\log_{5}3)^{\frac{2}{3}}} = 0\)
Note: In the given equation ‘x’ is missing.
So α, β are the roots of x² – 5x + 3(-1) = 0
\(α + β + \frac{1}{α} + \frac{1}{β} = (α+β) + \frac{α+β}{αβ}\)
\(= 5-\frac{5}{3}\)
\( = \frac{10}{3}\)
\((α+\frac{1}{β})(β+\frac{1}{α}) = 2+αβ+\frac{1}{αβ}\)
\( = 2-3-\frac{1}{3}=\frac{-4}{3}\)
So Equation must be option (B).