Question

If $x^2-4x+{\log }_{\frac{1}{2}}A=0$ does not have two distinct real roots, then maximum value of $A$ is:

• A. $\frac{1}{4}$
• B. $\frac{1}{16}$
• C. $\frac{1}{8}$
• D. none of these

Answer 1

The correct option is B $\frac{1}{16}$

If the given equation $x^2-4x+{\log }_{1/2}A=0$ does not have two distinct real roots, then discriminant $\left(D\right)$ will be less than $0$.

So,

$D=b^2-4ac<0$

$(-4{)}^2-4\times 1\times {\log }_{1/2}A<0$

$16<4{\log }_{1/2}A$

$4<{\log }_{1/2}A$

$(1/2{)}^4>A$

$1/16>A$

So, maximum value of $A$ can be $1/16$