If every pair from among the equation $x^{2} + p x + q r = 0, x^{2} + q x + p r = 0$ and $x^{2} + r x + p q = 0$ has a common root, then the sum of the three common roots is

2 (p + q + r)

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p + q + r

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- (p + q + r)

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p q r

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Solution

The correct option is B $p + q + r$
Let, $a$ be the common root then satisfies the root in all the three equation
we get,
$\begin{matrix} a^{2} + a p + q r = 0........ (1) a^{2} + a q + p r = 0........ (2) a^{2} + r a + p q = 0........ (3) \end{matrix}$
solving these, we get
$\begin{matrix} p a + q r = a q + r p a (p - q) = r (p - q) a = r a l s o, a q + r p = a r + p q a (q - r) = p (q - r) a = q s i m i l a r l y, a = q \end{matrix}$
the sum of common roots $= p + q + r$

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