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Question

lf both the roots of the equation $${x}^{2}+{x}+{a}=0$$ exceed $${a}$$, then


A
2<a<3
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B
a>3
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C
3<a<3
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D
a<2
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Solution

The correct option is D $$ a<-2$$
For both roots to exist,

$$D>0$$

$$\Rightarrow 1-4a>0$$

$$a<\dfrac{1}{4}$$

$$\Rightarrow a\in (-\infty ,\dfrac{1}{4})$$

For both roots to exceed $$a$$, $$f(a)>0$$

$$\Rightarrow a^{2}+2a>0$$

$$a(a+2)>0$$

$$a\in (-\infty ,-2)\bigcup (0,-\infty )$$

$$\therefore a\in (-\infty ,-2)$$ for roots to exists and exceed $$a$$.

$$\Rightarrow a<-2$$

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