If $\alpha,\beta$ are the roots of the equation $x^{2}-\left(5+3^{\sqrt{\log_{3}5}}-5^{\sqrt{\log_{5}3}}\right)x+3\left(3^{(\log_{3}5)^{1/3}}-5^{(\log_{5}3)^{2/3}}-1\right)=0$, then the equation whose roots are $\alpha+\frac{1}{\beta}$ and $\beta+\frac{1}{\alpha}$ is:

JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Quadratic Equations and Inequalities

α, β

are the roots of the equation

x^{2} - (5 + 3^{l o g_{3} 5} - 5^{l o g_{5} 3}) x + 3 (3^{(l o g_{3} 5)^{1/3}} - 5^{(l o g_{5} 3)^{2/3}} - 1) = 0

, then the equation whose roots are

α + \frac{1}{β}

and

β + \frac{1}{α}

is:

Answer: (B)

Using

3^{l o g_{3} 5} = 5^{l o g_{5} 3}

and

3^{(l o g_{3} 5)^{1/3}} = 5^{(l o g_{5} 3)^{2/3}}

, the bracketed differences vanish and the equation collapses to

x^{2} - 5 x - 3 = 0

, so

α + β = 5

and

α β = - 3

. The required roots are

α + \frac{1}{β} = \frac{α β + 1}{β} = \frac{- 2}{β}

and

β + \frac{1}{α} = \frac{- 2}{α}

. Their sum is

- 2 \cdot \frac{α + β}{α β} = - 2 \cdot \frac{5}{- 3} = \frac{10}{3}

and their product is

\frac{4}{α β} = \frac{4}{- 3} = - \frac{4}{3}

, so the monic equation

x^{2} - \frac{10}{3} x - \frac{4}{3} = 0

clears to

3 x^{2} - 10 x - 4 = 0

JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Single correct option