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Algebra of Limits

If α,β are ...

Question

If $α, β$ are the roots of the equation $x^{2} + x + 3 = 0$ , then the equation $3 x^{2} + 5 x + 3 = 0$ has root

A

\frac{α}{β}

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B

\frac{β}{α}

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C

\frac{α}{β} + \frac{β}{α}

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D

none of these

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Solution

The correct options are
A $\frac{α}{β}$
B $\frac{β}{α}$
Considering
$x^{2} + x + 3 = 0$
$α + β = - 1$
And $α . β = 3$
Now
$α^{2} + β^{2} = 1 - 2 (3)$
$= - 5$
Hence,
$\frac{α^{2} + β^{2}}{α . β} = \frac{α}{β} + \frac{β}{α} = \frac{- 5}{3}$
Let
$\frac{α}{β} = α^{'}$
and $\frac{β}{α} = β^{'}$
Hence,
$(x - α^{'}) (x - β^{'}) = 0$ be a quadratic equation.
$x^{2} - (α^{'} + β^{'}) + α^{'} . β^{'} = 0$
Substituting the value of roots we get
$x^{2} - \frac{- 5 x}{3} + 1 = 0$
$3 x^{2} + 5 x + 3 = 0$

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