Verified Solution • 2025–26 Session moderate

QJEEM22Ma9d7a: JEE Main 2022 Mathematics - Class 11 JEEM22Ma9d7a Solution: If are the roots of the equation , then the equation, whose…

If α,β\alpha, \beta are the roots of the equation

x2−(5+3log⁡35−5log⁡53)x+3(3(log⁡35)13−5(log⁡53)23−1)=0x^{2}-\left(5+3^{\sqrt{\log _{3} 5}}-5^{\sqrt{\log _{5} 3}}\right)x+3\left(3^{\left(\log _{3} 5\right)^{\frac{1}{3}}}-5^{\left(\log _{5} 3\right)^{\frac{2}{3}}}-1\right)=0 ,

then the equation, whose roots are α+1β\alpha+\frac{1}{\beta} and β+1α\beta+\frac{1}{\alpha} , is :

Multiple Choice Options

  • A 3x2−20x−12=03 x^{2}-20 x-12=0
  • B 3x2−10x−4=03 x^{2}-10 x-4=0
  • C 3x2−10x+2=03 x^{2}-10 x+2=0
  • D 3x2−20x+16=03 x^{2}-20 x+16=0

Step-by-Step Verified Solution

3log⁡35−5log⁡53=3log⁡35−(3log⁡35)log⁡53{3^{\sqrt {{{\log }_3}5} }} - {5^{\sqrt {{{\log }_5}3} }} = {3^{\sqrt {{{\log }_3}5} }} - {\left( {{3^{{{\log }_3}5}}} \right)^{\sqrt {{{\log }_5}3} }}

3(log⁡35)13−5(log⁡53)23=5(log⁡53)23−5(log⁡53)23=0{3^{{{\left( {{{\log }_3}5} \right)}^{{1 \over 3}}}}} - {5^{{{\left( {{{\log }_5}3} \right)}^{{2 \over 3}}}}} = {5^{{{\left( {{{\log }_5}3} \right)}^{{2 \over 3}}}}} - {5^{{{\left( {{{\log }_5}3} \right)}^{{2 \over 3}}}}} = 0

Note : In the given equation 'x' is missing.

So JEE Main 2022 (Online) 27th July Evening Shift Mathematics - Quadratic Equation and Inequalities Question 54 English Explanation

α+β+1α+1β=(α+β)+α+βαβ\alpha + \beta + {1 \over \alpha } + {1 \over \beta } = (\alpha + \beta ) + {{\alpha + \beta } \over {\alpha \beta }}

=5−53=103= 5 - {5 \over 3} = {{10} \over 3}

(α+1β)(β+1α)=2+αβ+1αβ=2−3−13=−43\left( {\alpha + {1 \over \beta }} \right)\left( {\beta + {1 \over \alpha }} \right) = 2 + \alpha \beta + {1 \over {\alpha \beta }} = 2 - 3 - {1 \over 3} = {{ - 4} \over 3}

So Equation must be option (B).

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