Consider the complex numbers $z = -2 - 2i$ and $w = 3 + i$ - HSC - SSCE Mathematics Extension 2 - Question 11 - 2014 - Paper 1

To find $\frac{z}{w}$:

1. Compute $\frac{z}{w} = \frac{-2 - 2i}{3 + i}$.
2. Multiply by the conjugate: $\frac{z}{w} \cdot \frac{3 - i}{3 - i} = \frac{(-2 -2i)(3 - i)}{(3 + i)(3 - i)}$.
3. Denominator calculation: $(3 + i)(3 - i) = 9 + 1 = 10$.
4. Numerator expansion: $(-2 - 2i)(3 - i) = -6 + 2 + (-6 - 2)i = -4 - 8i$.
5. So, we have: $\frac{z}{w} = \frac{-4 - 8i}{10} = -\frac{2}{5} - \frac{4}{5}i$.

To evaluate the integral:

1. Use integration by parts, let:
   • $u = 3x - 1$
   • $dv = \cos(\pi x) \, dx$.
2. Thus, $du = 3 \, dx$ and $v = \frac{1}{\pi} \sin(\pi x)$.
3. Applying integration by parts: $\int u \, dv = uv - \int v \, du$.
4. Evaluating gives: $\left[ (3x - 1)\frac{1}{\pi}\sin(\pi x) \right]_0^{1/2} - \int_0^{1/2} \frac{1}{\pi} \sin(\pi x)(3) \, dx$.
5. This evaluates to the value after calculations.

To sketch the specified region:

1. The inequality $|z| \leq 2$ describes a circle centered at the origin with radius 2.
2. The argument condition describes the angle constraint, which represents the sector of the circle.
3. Mark the angles $-\frac{\pi}{4}$ and $\frac{\pi}{4}$ on the Argand plane and shade the region within the circle that falls between these angles.

To sketch the graph:

1. Find intercepts by setting $y = 0$: $x^2 - 1 = 0 \Rightarrow x = \pm 1$.
2. For vertical asymptote, evaluate when $x^2 = 0 \Rightarrow x = 0$ is undefined.
3. The horizontal line test as $x \to \infty$ gives $y \to 1$. Sketch the graph with calculated points and asymptotes.

To find the volume:

1. The volume $V$ of the solid of revolution is given by: $V = 2\pi \int_a^b (radius)(height) \, dx$.
2. In this case, the radius is the y-value from the curve and we rotate around the x-axis. Define height as the resulting function from $y = 6 - y$.
3. Calculate the volume by setting appropriate limits and evaluate the integral.