Straight-line graphs

A curve has equation $y = x^2 \ln(3x^2 + 5)$.

The normal to the curve at the point where $x = -1$ meets the $x$-axis at the point $P$ and the $y$-axis at the point $Q$.

Find the area of the triangle $OPQ$, where $O$ is the origin.
Give your answer correct to 3 significant figures.

The coordinates of points $A$, $B$, $C$ and $D$ are as follows.

The line $L$ has equation $y = 11x - 75$.
The perpendicular bisector of the line $AB$ meets $L$ at the point $E$.

Find the area of triangle $CDE$.

The line $y = 3x + 4$ meets the curve $y = 2x^2 + 8x + 1$ at two points $A$ and $B$.

Find the equation of the perpendicular bisector of $AB$, giving your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers.

A curve has equation $y = 5\mathrm{e}^{2x - 1} + \mathrm{e}$. The tangent to the curve at the point where $x = 1$ cuts the $x$-axis at the point $P$.

Find the equation of the tangent in the form $y = mx + c$, where $m$ and $c$ are exact values, and hence find the $x$-coordinate of $P$.

The diagram shows part of the curve $y = 32x - 4x^2 - 48$ and the line $AB$.
The curve and the line $AB$ meet the $x$-axis at $A$ and meet again at the point $B(5, 12)$.
The line $CD$ extended is parallel to the $y$-axis and passes through the maximum point of the curve.
Find the area of the shaded region.

The tangent to the curve $y = \frac{\sqrt{x+1}}{x}$ at the point where $x = 3$ meets the line $y = x - 16$ at the point $A$. Find the coordinates of $A$.

Variables $x$ and $y$ are such that when $\lg y$ is plotted against $\sqrt{x}$ a straight line passing through the points $(1, 5)$ and $(2.5, 8)$ is obtained. Show that $y = A \times b^{\sqrt{x}}$ where $A$ and $b$ are constants to be found.