Question

If $x^2-2x+{\log }_{\frac{1}{2}}p=0$ does not have two distinct real roots, then the maximum value of $p$ is

• A. $2$
• B. $1$
• C. $\frac{1}{4}$
• D. $\frac{1}{2}$

Answer 1

The correct option is D $\frac{1}{2}$

$x^2-2x+{\log }_{\frac{1}{2}}p=0$
It does not have two distinct real roots means
It can have either equal real roots or imaginary roots.
So,
$D\le 0$
$\Rightarrow 4-4{\log }_{\frac{1}{2}}p\le 0\Rightarrow 4+4{\log }_{2}p\le 0\Rightarrow {\log }_{2}p\le -1\Rightarrow p\le 2^{-1}\Rightarrow p\le \frac{1}{2}$