Question

The range of $p$ for which the number $6$ lies between the roots of $x^2+2(p-3)x+9=0$ is

• A. $(-\infty ,-\frac{3}{4})$
• B. $(-\infty ,-\frac{3}{2})$
• C. $(-\infty ,0)$
• D. $(6,\infty )$

Answer 1

The correct option is A $(-\infty ,-\frac{3}{4})$

Let $f\left(x\right)=x^2+2(p-3)x+9$ whose roots are $\alpha ,\beta .$
Given: $\alpha <6<\beta$

$\therefore f\left(6\right)<0\Rightarrow 36+12(p-3)+9<0$
$\Rightarrow 12p+9<0\Rightarrow p\in (-\infty ,-\frac{3}{4})$