A value of b for which the equations $x^{2} + b x - 1 = 0, x^{2} + x + b = 0$ have one root in common is

$- \sqrt{2}$

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$- i \sqrt{3}$

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$i \sqrt{5}$

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$\sqrt{2}$

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Solution

The correct option is B
$- i \sqrt{3}$

If $a_{1} x^{2} + b_{1} x + c_{1} = 0$
and $a_{2} x^{2} + b_{2} x + c_{2} = 0$
have a common real root, then
$\Rightarrow (a_{1} c_{2} - a_{2} c_{1})^{2} = (b_{1} c_{2} - b_{2} c_{1}) (a_{1} b_{2} - a_{2} b_{1})$
$∴ \begin{matrix} x^{2} + b x - 1 = 0 x^{2} + x + b = 0 \end{matrix}} h a v e a c o m m o n r o o t . \Rightarrow (1 + b)^{2} = (b^{2} + 1) (1 - b) \Rightarrow b^{2} + 2 b + 1 = - b^{3} + b^{2} + 1 - b \Rightarrow b^{3} + 3 b = 0 ∴ b (b^{2} + 3) = 0 \Rightarrow b = 0, \pm \sqrt{3} i$

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