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Question

lf $$ax^{2}-(2a+3)x +(3+5a)=0$$ has no real roots, then $$a$$ lies in the inverval


A
(34,34)
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B
(,34)
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C
(,34)
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D
(,34)(34, )
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Solution

The correct option is D $$\left(-\infty, -\displaystyle \frac{3}{4}\right)\cup\left(\dfrac{3}{4},\ \infty\right)$$
If $$ax^{2}+bx+c$$ has no real roots, then
$$\Rightarrow b^{2}-4ac< 0$$
$$\Rightarrow (2a+3)^{2}-4(a)(3+5a)< 0.$$
$$\Rightarrow 4a^{2}+9+12a - 12a -20a^{2}< 0$$
$$\Rightarrow 9-16a^{2}< 0$$
$$\Rightarrow \left(-\infty, -\displaystyle \frac{3}{4}\right) \cup\left(\dfrac{3}{4},\ \infty\right)$$

Mathematics

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