Let $f (x) = x^{2} + b_{1} x + c_{1}$ , $g (x) = x^{2} + b_{2} x + c_{2}$ . Real roots of $f (x) = 0$ be $α, β$ and real roots of $g (x) = 0$ be $α + δ, β + δ$ .The least value of $f (x)$ be $\frac{- 1}{4}$ , least value of $g (x)$ occurs at $x = \frac{7}{2}$

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the least value of $g (x)$ is $\frac{- 1}{4}$

the value of $b_{2}$ is -7

The value of $c_{2}$ is 15

the difference of the root of $g (x) = 0$ is 1

answer is A, B, D.

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Detailed Solution

$| α - β | = | (α + δ) - (β + δ) |$ difference of roots are equal least value of $f (x) = \frac{4 a c - b^{2}}{4 a} = \frac{- 1}{4}$
$\Rightarrow \frac{4 c_{1} - b_{1}^{2}}{4} = \frac{- 1}{4} \Rightarrow b_{1}^{2} - 4 c_{1} = 1$
and $b_{2}^{2} - 4 c_{2} = 1$ Here $b_{2} = - 7$ $\Rightarrow 49 - 4 c_{2} = 1$ $\Rightarrow c_{2} = 12$
$g (x)$ is $x^{2} - 7 x + 12 = 0$ $\Rightarrow$ $x = 3, 4$