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Question

All possible values of $$a$$, so that $$6$$ lies between the roots of the equation $$x^2 + 2(a - 3)x + 9 = 0$$ is


A
(,2)(2,)
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B
(,34)
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C
(2,)
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D
none of these
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Solution

The correct option is B $$\left(- \infty, -\dfrac{3}{4}\right)$$
Given equation,
$$x^2 + 2(a - 3)x + 9 = 0$$
Condition for a number 6 to lie between the roots is 
$$f(6) < 0 \ldots\ldots (\because a.f(k)<0)$$
$$\Rightarrow 36 + 12(a - 3) + 9 < 0$$
$$\Rightarrow 12a + 9 < 0$$
$$\Rightarrow a < \displaystyle -\frac{3}{4}$$  
$$\displaystyle a \in \left( - \infty , - \frac{3}{4}\right)$$

Mathematics

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