Question

If $\gamma ,\delta$ are the roots of $x^2-3x+a=0,aϵR$, and $\gamma <1<\delta$ then

• A. $aϵ(2,\frac{9}{4})$
• B. $aϵ(-\infty ,2)$
• C. none of these
• D. $aϵ(-\infty ,\frac{9}{4})$

Answer 1

The correct option is B $aϵ(-\infty ,2)$

Let $\gamma \text{and}\delta$ be roots of the equation $x^2-3x+a=0$. The corresponding polynomial of this equation is $f\left(x\right)=x^2-3x+a$

Condition : $\left(i\right)D>0\&\left(ii\right)a\cdot f(k<0)$

Now,
$\left(i\right)D>0\Rightarrow 9-4a>0\Rightarrow a<\frac{9}{4}$

$\left(ii\right)f\left(1\right)<0\Rightarrow 1-3+a<0\Rightarrow a<2$

On taking intersection we get,

$aϵ(-\infty ,2)$