Let $z = 3 + i$ and $w = 2 - 5i$. Find, in the form $x + iy$, (i) $z^2$ (ii) $zw$ (iii) $\frac{w}{z}$ (i) Express $\sqrt{3 - i}$ in modulus-argument form. (ii) Express $\left(\sqrt{3 - i}\right)$ in modulus-argument form. (iii) Hence express $\left(\sqrt{3 - i}\right)^7$ in the form $x + iy$. (c) Find, in modulus-argument form, all solutions of $z^3 = -1$. (d) The equation $|z - 1 - 3i| + |z - 9 - 3i| = 10$ corresponds to an ellipse in the Argand diagram. (i) Write down the complex number corresponding to the centre of the ellipse. (ii) Sketch the ellipse, and state the lengths of the major and minor axes. (iii) Write down the range of $\arg(z)$ for complex numbers $z$ corresponding to points on the ellipse.

SSCE HSC Mathematics Extension 2 Modulus and argument: Let $z = 3 + i$ and $w = 2

Let $z = 3 + i$ and $w = 2 - 5i$ - HSC - SSCE Mathematics Extension 2 - Question 2 - 2006 - Paper 1

Question 2

Let $z = 3 + i$ and $w = 2 - 5i$. Find, in the form $x + iy$, (i) $z^2$ (ii) $zw$ (iii) $\frac{w}{z}$ (i) Express $\sqrt{3 - i}$ in modulus-argument form. ... show full transcript

Worked Solution & Example Answer:Let $z = 3 + i$ and $w = 2 - 5i$ - HSC - SSCE Mathematics Extension 2 - Question 2 - 2006 - Paper 1

Step 1

Find, in the form $x + iy$, (i) $z^2$

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Answer

To find $z^2$ , we calculate:

$z^2 = (3 + i)^2 = 3^2 + 2 \cdot 3 \cdot i + i^2 = 9 + 6i - 1 = 8 + 6i$

Thus, $z^2 = 8 + 6i$ .

Step 2

Find, in the form $x + iy$, (ii) $zw$

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Answer

For $zw$ , we calculate:

$zw = (3 + i)(2 - 5i) = 3 \cdot 2 + 3 \cdot (-5i) + i \cdot 2 + i \cdot (-5i) = 6 - 15i + 2i + 5 = 11 - 13i$

Thus, $zw = 11 - 13i$ .

Step 3

Find, in the form $x + iy$, (iii) $\frac{w}{z}$

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Answer

To compute $\frac{w}{z}$ , we use:

$\frac{w}{z} = \frac{2 - 5i}{3 + i}$

To simplify, multiply by the conjugate:

$\frac{(2 - 5i)(3 - i)}{(3 + i)(3 - i)} = \frac{6 - 2i - 15i + 5}{9 + 1} = \frac{11 - 17i}{10} = 1.1 - 1.7i$

Thus, $\frac{w}{z} = 1.1 - 1.7i$ .

Step 4

Express $\sqrt{3 - i}$ in modulus-argument form (i)

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Answer

First, find the modulus:

$r = |\sqrt{3 - i}| = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}$

Then calculate the argument:

$\theta = \tan^{-1}\left(\frac{-1}{3}\right)$

Thus, the modulus-argument form is:

$\sqrt{3 - i} = \sqrt{10} \left( \cos\theta + i\sin\theta \right)$

Step 5

Express $\left(\sqrt{3 - i}\right)$ in modulus-argument form (ii)

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Answer

Carrying forward, we find:

$\sqrt{3 - i} = \sqrt{10} \left( \cos\theta + i\sin\theta \right)$

Hence, scaling this leads to:

$\left(\sqrt{3 - i}\right) = r^{1/2} \left( \cos\frac{\theta}{2} + i\sin\frac{\theta}{2} \right)$

Step 6

Hence express $\left(\sqrt{3 - i}\right)^7$ in the form $x + iy$ (iii)

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Using De Moivre's theorem:

$\left(\sqrt{3 - i}\right)^7 = \left(r^{1/2}\right)^7 \left(\cos\left(\frac{7\theta}{2}\right) + i\sin\left(\frac{7\theta}{2}\right) \right) = (\sqrt{10})^{7/2} \left( \cos\frac{7\theta}{2} + i\sin\frac{7\theta}{2} \right)$

Thus, simplify the expression accordingly to find $x$ and $y$ .

Step 7

Find, in modulus-argument form, all solutions of $z^3 = -1$ (c)

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Answer

The equation can be rewritten as:

$z^3 = 1 \text{ (where } z = e^{i\pi} )$

The polar form gives us:

$z = \sqrt[3]{1} \left( \cos\left(\frac{\pi + 2k\pi}{3}\right) + i\sin\left(\frac{\pi + 2k\pi}{3}\right) \right), k = 0, 1, 2$

Solving this will yield three distinct complex solutions.

Step 8

Write down the complex number corresponding to the centre of the ellipse (i)

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Answer

The center of the ellipse occurs at the midpoint of the foci. Thus:

$\text{Centre: } \left(\frac{1+9}{2}, \frac{3+3}{2}\right) = (5, 3)$ .

The corresponding complex number is $5 + 3i$ .

Step 9

Sketch the ellipse, and state the lengths of the major and minor axes (ii)

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Given the equation describes an ellipse, its semi-major axis is the distance of 5 units and the semi-minor axis is 2.5 units. The ellipse is centered at $5 + 3i$ .

Step 10

Write down the range of $\arg(z)$ for complex numbers $z$ corresponding to points on the ellipse (iii)

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Answer

The angles (arguments) for points on the ellipse will be bounded between:

$\arg(z) = \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

This indicates the range of possible directions in the Argand diagram.

Let $z = 3 + i$ and $w = 2 - 5i$ - HSC - SSCE Mathematics Extension 2 - Question 2 - 2006 - Paper 1

Worked Solution & Example Answer:Let $z = 3 + i$ and $w = 2 - 5i$ - HSC - SSCE Mathematics Extension 2 - Question 2 - 2006 - Paper 1

Find, in the form $x + iy$, (i) $z^2$

Find, in the form $x + iy$, (ii) $zw$

Find, in the form $x + iy$, (iii) $\frac{w}{z}$

Express $\sqrt{3 - i}$ in modulus-argument form (i)

Express $\left(\sqrt{3 - i}\right)$ in modulus-argument form (ii)

Hence express $\left(\sqrt{3 - i}\right)^7$ in the form $x + iy$ (iii)

Find, in modulus-argument form, all solutions of $z^3 = -1$ (c)

Write down the complex number corresponding to the centre of the ellipse (i)

Sketch the ellipse, and state the lengths of the major and minor axes (ii)

Write down the range of $\arg(z)$ for complex numbers $z$ corresponding to points on the ellipse (iii)

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