Let $z = 2 - i ext{√}3$ and $w = 1 + i ext{√}3$ - HSC - SSCE Mathematics Extension 2 - Question 11 - 2013 - Paper 1

To express $w$ in modulus-argument form, we calculate its modulus:

$|w| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2.$

Next, we find the argument:

$\arg(w) = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}.$

Therefore, the modulus-argument form of $w$ is:

$w = 2\text{cis}\left(\frac{\pi}{3}\right).$

From the modulus-argument form, we can find:

$w^{24} = \left(2 \text{cis}\left(\frac{\pi}{3}\right)\right)^{24} = 2^{24} \text{cis}(24 \cdot \frac{\pi}{3}) = 16777216 \text{cis}(8\pi).$

Since $ext{cis}(8\pi) = 1$, the simplest form of $w^{24}$ is:

$w^{24} = 16777216.$

To find $A$, $B$, and $C$, we can write:

$x^2 + 8x + 11 = A(x^2 + 2) + (Bx + C)(x - 3).$

Expanding the right-hand side gives:

$Ax^2 + 2A + Bx^2 - 3Bx + Cx - 3C = (A + B)x^2 + (C - 3B)x + (2A - 3C).$

By equating coefficients:

1. $A + B = 1$
2. $C - 3B = 8$
3. $2A - 3C = 11$

Solving this system, we find:

$A = 4, B = -3, C = -1.$

To factorise $z^2 + 4iz + 5$, we use the quadratic formula:

$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$

where $a = 1$, $b = 4i$, and $c = 5$:

1. Calculate the discriminant: $b^2 - 4ac = (4i)^2 - 20 = -16 - 20 = -36.$
2. Compute the roots:

$z = \frac{-4i \pm \sqrt{-36}}{2} = 2i \pm 3.$

Thus, the factorization is:

$(z - (3 + 2i))(z - (3 - 2i)).$

To evaluate this integral, we use the substitution $x = \sin \theta$, then $dx = \cos \theta \, d\theta$.

0 to 1 maps to 0 to $\frac{\pi}{2}$:

$\int_0^{\frac{\pi}{2}} \sin^3 \theta \sqrt{1 - \sin^2 \theta} \cos \theta \, d\theta = \int_0^{\frac{\pi}{2}} \sin^3 \theta \cos^2 \theta \, d\theta.$

Using the reduction formula or direct integration:

The value of the integral is:

$\frac{3}{8}.$

The given inequality can be rewritten as:

$2z^2 \leq 8 \quad \implies \quad z^2 \leq 4.$

This inequality represents a circle of radius 2 centered at the origin in the Argand diagram. Therefore, the region is the area inside and on the circle defined by:

$|z| \leq 2.$

The sketch should represent a filled circle with radius 2.