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

Answer: (B)

3x2 – 10x – 4 = 0

Answer 2

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).