Question

The smallest integral value of $p$ for which the inequality $(p-3)x^2-2px+3(p-2)>0$ is satisfied for all real values of $x$ is

• A. $8$
• B. $7$
• C. $6$
• D. $5$

Answer 1

Answer: (B)

$7$

$f\left(x\right)>0$ when $D<0$ and $p>3$
$\Rightarrow 4p^2-12(p-3)(p-2)<0$
$\Rightarrow 2p^2-15p+18>0$
$\Rightarrow (p-6)(2p-3)>0$
$\Rightarrow p<\frac{3}{2}$ or $p>6$
Hence, smallest integral value of $p$ is $7$.