Straight-line graphs

The tangent to the curve $y = (3x-1)^{\frac{1}{3}}$ at the point where $x = 3$ meets the coordinate axes at the points $A$ and $B$. The point with coordinates $(a, a)$ lies on the perpendicular bisector of the line $AB$. Find the exact value of $a$.

The tangent to the curve $y = \mathrm{e}^x(2x + 5)^{\frac{1}{2}}$ at the point where $x = 2$ meets the $x$-axis at the point $X$ and the $y$-axis at the point $Y$. Find the coordinates of the mid-point of $XY$, giving your answer in exact form.

When $\ln(y + 2)$ is plotted against $x^2$ a straight line graph is obtained. The line passes through the points $(2.25, 9.37)$ and $(4.75, 3.92)$. Find $y$ in terms of $x$.

The perpendicular bisector of the line joining the points $\left(-3, \frac{2}{3}\right)$ and $\left(6, -\frac{7}{3}\right)$ passes through the point $(2, k)$. Find the value of $k$.

The normal to the curve $y = \tan\left(3x + \frac{\pi}{2}\right)$ at the point $P$ with coordinates $(p, -1)$, where $0 < p \leq \frac{\pi}{6}$, meets the $x$-axis at the point $A$ and the $y$-axis at the point $B$. Find the exact coordinates of the mid-point of $AB$.