Multiple choice

If $\dfrac {1}{\sqrt{\alpha}}$ and $\dfrac {1}{\sqrt{\beta}}$ are the roots of the equation, $ax^2+bx+1=0$ $(a\neq 0, a, b \in R)$, then the equation, $x(x+b^3)+(a^3-3abx)=0$ has roots

  1. $\alpha^{{3}/{2}}$ and $\beta^{{3}/{2}}$
  2. $\alpha \beta^{{1}/{2}}$ and $\alpha^{{1}/{2}}\beta$
  3. $\sqrt {\alpha \beta}$ and $\alpha \beta$
  4. $\alpha^{-{3}/{2}}$ and $\beta^{- {3}/{2}}$
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Explanation

Given 1/sqrt(alpha) and 1/sqrt(beta) are roots of ax^2 + bx + 1 = 0. This implies alpha and beta are roots of a transformed equation. Through algebraic manipulation of the coefficients and roots, the roots of the second equation are alpha^(3/2) and beta^(3/2).

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