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

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a

- \sqrt{2}

b

- i \sqrt{3}

c

i \sqrt{5}

d

\sqrt{2}

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detailed solution

Correct option is B

We know that, if $a_{1} x^{2} + b_{1} x + c_{1} = 0$ and $a_{2} x^{2} + b_{2}^{2} x + c_{2} = 0$ have a common real root, then ${(a_{1} c_{2} - a_{2} c_{1})}^{2} = (b_{1} c_{2} - b_{2} c_{1}) (a_{1} b_{2} - a_{2} b_{1})$ then ${(a_{1} c_{2} - a_{2} c_{1})}^{2} = (b_{1} c_{2} - b_{2} c_{1}) (a_{1} b_{2} - a_{2} b_{1})$
$\begin{array}{l} \Rightarrow (1 + b)^{2} = (b^{2} + 1) (1 - b) \\ \Rightarrow b^{2} + 2 b + 1 = b^{2} - b^{3} + 1 - b \\ \Rightarrow b^{3} + 3 b = 0 \Rightarrow b (b^{2} + 3) = 0 \\ \Rightarrow b = 0, \pm i \sqrt{3} \end{array}$

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