A line $$L:y=mx+3$$ meets $$y$$-axis at R$$(0, 3)$$ and the arc of the parabola $${y^2} = 16x,$$ $$0 \le y \le 6$$ at the point $$F\left( {{x_0},{y_0}} \right)$$. The tangent to the parabola at $$F\left( {{x_0},{y_0}} \right)$$ intersects the $$y$$-axis at $$G\left( {0,{y_1}} \right)$$. The slope $$m$$ of the line $$L$$ is chosen such that the area of the triangle $$EFG$$ has a local maximum.

Match List $$I$$ with List $$II$$ and select the correct answer using the code given below the lists:

List $$I$$
P.$$\,\,\,m = $$
Q.$$\,\,\,$$Maximum area of $$\Delta EFG$$ is
R.$$\,\,\,$$ $${y_0} = $$
S.$$\,\,\,$$ $${y_1} = $$

List $$II$$
1.$$\,\,\,$$ $${1 \over 2}$$
2.$$\,\,\,$$ $$4$$
3.$$\,\,\,$$ $$2$$
4.$$\,\,\,$$ $$1$$

Circle (s) touching x-axis at a distance 3 from the origin and having an intercept of length $$2\sqrt 7 $$ on y-axis is (are)

If $${3^x}\, = \,{4^{x - 1}},$$ then $$x\, = $$

Let $$S = {S_1} \cap {S_2} \cap {S_3}$$, where $${S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\nolimits} \left[ {{{z - 1 + \sqrt 3 i} \over {1 - \sqrt 3 i}}} \right] > 0} \right\}$$ and $${S_3} = \left\{ {z \in C:{\mathop{\rm Re}\nolimits} z > 0} \right\}\,$$.

$$\,\mathop {\min }\limits_{z \in S} \left| {1 - 3i - z} \right| = $$