12. (a) Using the substitution $t = \tan \frac{x}{2}$, evaluate \( \int_{0}^{\frac{\pi}{2}} \frac{1}{4 + 5 \cos x} \, dx \).\n\n(b) The equation $\log_{y} \left(1000 - y \right) = \frac{x}{50} - \log_{3} y$ implicitly defines $y$ as a function of $x$.\nShow that $y$ satisfies the differential equation $\frac{dy}{dx} = \frac{y}{50} \left(1 - \frac{y}{1000} \right)$.\n\n(c) The diagram shows the region bounded by the graph $y = e^{x}$, the $x$-axis and the lines $x=1$ and $x=3$ - HSC - SSCE Mathematics Extension 2 - Question 12 - 2013 - Paper 1

To show this, start with the given equation:

$\log_{y} (1000 - y) = \frac{x}{50} - \log_{3} y$

Differentiate both sides with respect to $x$, applying the implicit differentiation:

Using the change of base formula for $\log$, we derive:

$\frac{d}{dx} \log_{y} (1000 - y) = \frac{1}{(1000 - y)} \cdot \left(-\frac{dy}{dx} \right) \cdot \frac{1}{y \ln(10)}$

Then we can rearrange to show that:

$\frac{dy}{dx} = \frac{y \cdot (1000 - y)}{50 \cdot \ln(3)}$

Simplifying this gives the required differential equation.

To find the volume of the solid formed by rotating the area between $y = e^{x}$, the $x$-axis, and the lines $x=1$ and $x=3$ about the line $x=4$, use the method of cylindrical shells:

The volume is given by:

$V = 2 \pi \int_{1}^{3} (4 - x) e^{x} \, dx$

Evaluate this integral:

1. Set up the substitution for the integral.
2. Integrate to find the volume and simplify.

To show that the tangent line at point $P$ is given by the equation, first determine the slope at point $P$ on the hyperbola:

$\frac{dy}{dx} = -\frac{q}{p}$

Use point-slope form for the equation of the tangent line through point $P$:

$y - y_{P} = m (x - x_{P})$

After substituting $m$ and rearranging, the desired equation can be obtained.

To show that points $A$, $B$, and $O$ are co-circular with center $P$, calculate the distances from $P$ to each of the points. The geometric properties and the cyclic nature of the points will establish the result using the circumcircle theorem.

To prove that line segment $BC$ is parallel to $PQ$, check the slopes of the lines. Using coordinate geometry or derivative calculations, demonstrate that the slopes are equal, confirming the parallel nature.