First slide

Theory of equations

Question

If the roots of the equation $x^{3} - p x^{2} + q x - r = 0$ are in A.P., then

Moderate

A

$2 p^{3} = 9 p q - 27 r$
B

$2 q^{3} = 9 p q - 27 r$
C

$p^{3} = 9 p q - 27 r$
D

$2 p^{3} = 9 p q + 27 r$

Solution

Let the roots of the given equation be $a - d, a, a + d$
Then,
$(a - d) + a + (a + d) = - \frac{(- p)}{1} \Rightarrow a = \frac{p}{3}$
Since a is a root of the given equation.
$\begin{array}{l} ∴ a^{3} - p a^{2} + q a - r = 0 \\ \Rightarrow \frac{p^{3}}{27} - \frac{p^{3}}{9} + \frac{q p}{3} - r = 0 \Rightarrow 2 p^{3} - 9 p q + 27 r = 0 \end{array}$
This is the required condition

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