Exams Joint Entrance Examination Mathematics Complex Numbers Question 179 Question 179 of 455 Medium MCQ
If α , β \alpha,\beta α , β are the roots of the equation (after simplifying the given logarithmic expression, which reduces to) x 2 − 5 x − 3 = 0 x^2-5x-3=0 x 2 − 5 x − 3 = 0 , then the equation whose roots are α + 1 β \alpha+\dfrac1\beta α + β 1 and β + 1 α \beta+\dfrac1\alpha β + α 1 , is:
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α + β = 5 , α β = − 3 \alpha+\beta=5,\ \alpha\beta=-3 α + β = 5 , α β = − 3 .
\[\left(\alpha+\frac1\beta\right)+\left(\beta+\frac1\alpha\right)=(\alpha+\beta)+\frac{\alpha+\beta}{\alpha\beta}=5+\frac5{-3}=\frac{10}3.\]
\[\left(\alpha+\frac1\beta\right)\left(\beta+\frac1\alpha\right)=\alpha\beta+2+\frac1{\alpha\beta}=-3+2-\frac13=-\frac43.\]
Required equation: x 2 − 10 3 x − 4 3 = 0 ⟹ 3 x 2 − 10 x − 4 = 0. x^2-\dfrac{10}3x-\dfrac43=0 \implies 3x^2-10x-4=0. x 2 − 3 10 x − 3 4 = 0 ⟹ 3 x 2 − 10 x − 4 = 0. Answer the question to unlock