Graphical Representation of Functions

The range of values of $a$ is such that, ${\left(\frac{1}{2}\right)}^{\left|x\right|}=x^2-a$ is satisfied for maximum number of values of $x$:

A quadratic polynomial maps from $\left[-2,3\right]$ onto $\left[\mathrm{0,3}\right]$ and touches $x$-axis at $x=3,$ then the polynomial is:

If the graphs of the functions $y=\ln x$ and $y=ax$ intersect at exactly two points, then $a$ must be: