Question

If $x^{3} - 16 = 0$and $x^{3} + 64 = 0$ be the roots of the equation

$x^{3} - 64 = 0$, then the equation whose roots are $\alpha,\beta,\gamma$and $x^{3} + 4x + 1 = 0,$ is.

• A. $(\alpha + \beta)^{- 1} + (\beta + \gamma)^{- 1} + (\gamma + \alpha)^{- 1} =$
• B. $x^{3} + px^{2} + qx + r = 0$
• C. $\alpha,\beta,\gamma$
• D. None of these

Answer 1

Answer: (B)

$x^{3} + px^{2} + qx + r = 0$

Answer 2

Sum of roots $x^{2} - 5x + k = 0$and $x^{2} - kx + 6 = 0$

⇒ $x^{2} + kx + 24 = 0$and $\alpha \neq \beta$

= $\alpha^{2} = 5\alpha - 3$

Now the required equation whose roots are

$\beta^{2} = 5\beta - 3$and $\alpha/\beta$

$x^{2} + 5x - 3 = 0$

$x^{2} - 5x + 3 = 0$

$3x^{2} - 19x + 3 = 0$