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← JEE Main 2022 · 27 Jul · Shift 2
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JEE Main 2022 · 27 Jul · Shift 2, Mathematics question 3

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

x2−(5+3log⁡35−5log⁡53)x+3(3(log⁡35)13−5(log⁡53)23−1)=0,x^2-\left(5+3^{\sqrt{\log_35}}-5^{\sqrt{\log_53}}\right)x+3\left(3^{(\log_35)^{\frac13}}-5^{(\log_53)^{\frac23}}-1\right)=0,

then the equation, whose roots are α+1β\alpha+\dfrac1\beta and β+1α\beta+\dfrac1\alpha, is :

Choose one option
Answer and worked solution

Official answer: Dropped: full marks to everyone

This question was dropped in NTA's final key.

As printed. 3log⁡35=5log⁡533^{\sqrt{\log_35}}=5^{\sqrt{\log_53}} and 3(log⁡35)1/3=5(log⁡53)2/33^{(\log_35)^{1/3}}=5^{(\log_53)^{2/3}}, so the equation is x2−5x−3=0x^2-5x-3=0. The new roots have sum 5−53=1035-\frac53=\frac{10}3 and product (αβ+1)2αβ=−43\frac{(\alpha\beta+1)^2}{\alpha\beta}=-\frac43, giving 3x2−10x−4=03x^2-10x-4=0, option (B).

Topics:Roots and coefficientsLogarithms
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