First slide

Theory of equations

Question

If $α$ and $β$ be the roots of $x^{2} + px + q = 0$ then, $\frac{(ωα + ω^{2} β) (ω^{2} α + ωβ)}{\frac{α^{2}}{β} + \frac{β^{2}}{α}} isequal to$

Moderate

A

$- \frac{q}{p}$
B

$α β$
C

$- \frac{p}{q}$
D

$ω$

Solution

Since, $α$ and $β$ are the roots of the equation $x^{2} + px + q = 0$ , therefore $α + β = - p and αβ = q$ , Now, $(ωα + ω^{2} β) (ω^{2} α + ωβ)$
$\begin{array}{lr} = α^{2} + β^{2} + (ω^{4} + ω^{2}) αβ (∵ ω^{3} = 1) \\ = α^{2} + β^{2} - αβ (∵ ω + ω^{2} = - 1) \\ = (α + β)^{2} - 3 αβ = p^{2} - 3 q \end{array}$
Also, $\begin{aligned} \frac{α^{2}}{β} + \frac{β^{2}}{α} = \frac{α^{3} + β^{3}}{αβ} \\ = \frac{(α + β)^{3} - 3 αβ (α + β)}{αβ} = \frac{p (3 q - p^{2})}{q} \end{aligned}$
$∴ The given expression = \frac{(p^{2} - 3 q)}{\frac{p (3 q - p^{2})}{q}} = - \frac{q}{p}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App