Write $i^8$ in the form $a + ib$ where $a$ and $b$ are real. Write $-2 + 3i$ in the form $a + ib$ where $a$ and $b$ are real. The points $P$ and $Q$ on the Argand diagram represent the complex numbers $z$ and $w$ respectively. Copy the diagram into your writing booklet, and mark on it the following points: (i) the point $R$ representing $iz$ (ii) the point $S$ representing $w$ (iii) the point $T$ representing $z + w$. Sketch the region in the complex plane where the inequalities $|z - 1| eq 2$ and $-rac{ au}{4} eq ext{arg}(z - 1) eq rac{ au}{4}$ hold simultaneously. Find all the 5th roots of $-1$ in modulus-argument form. Sketch the 5th roots of $-1$ on an Argand diagram. Find the square roots of $3 + 4i$. Hence, or otherwise, solve the equation $z^2 + iz - 1 = i$.

SSCE HSC Mathematics Extension 2 Roots of complex numbers: Write $i^8$ in the form $a + ib$

Write $i^8$ in the form $a + ib$ where $a$ and $b$ are real - HSC - SSCE Mathematics Extension 2 - Question 2 - 2009 - Paper 1

Question 2

Write $i^8$ in the form $a + ib$ where $a$ and $b$ are real. Write $-2 + 3i$ in the form $a + ib$ where $a$ and $b$ are real. The points $P$ and $Q$ on the Argand ... show full transcript

Worked Solution & Example Answer:Write $i^8$ in the form $a + ib$ where $a$ and $b$ are real - HSC - SSCE Mathematics Extension 2 - Question 2 - 2009 - Paper 1

Step 1

Write $i^8$ in the form $a + ib$ where $a$ and $b$ are real.

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Answer

To find $i^8$ , we first note that $i = ext{cis}( rac{ au}{2})$ . Thus, we can apply De Moivre's Theorem: $i^8 = ( ext{cis}( rac{ au}{2}))^8 = ext{cis}(4 au) = 1.$ Therefore, in the form $a + ib$ , we have $1 + 0i$ (where $a = 1$ and $b = 0$ ).

Step 2

Write $-2 + 3i$ in the form $a + ib$ where $a$ and $b$ are real.

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The expression $-2 + 3i$ is already in the form $a + ib$ , thus $a = -2$ and $b = 3$ .

Step 3

Copy the diagram into your writing booklet, and mark on it the following points: (i) the point $R$ representing $iz$

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To find the point $R$ representing $iz$ , we multiply the coordinates of point $P$ by $i$ , resulting in a 90-degree rotation of point $P$ around the origin.

Step 4

(ii) the point $S$ representing $w$

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Directly plot point $Q$ on the Argand diagram to represent $w$ .

Step 5

(iii) the point $T$ representing $z + w$

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Point $T$ is determined by the vector addition of points $P$ and $Q$ . Plot $T$ at the resultant position of the sum of the coordinates of $P$ and $Q$ .

Step 6

Sketch the region in the complex plane where the inequalities $|z - 1| eq 2$ and $-rac{ au}{4} eq ext{arg}(z - 1) eq rac{ au}{4}$ hold simultaneously.

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To sketch this region, identify the circle centered at $(1, 0)$ with radius $2$ ; however, exclude this circle to represent $|z - 1| eq 2$ . For the argument, represent the sector limited by angles $- rac{ au}{4}$ and $rac{ au}{4}$ . The desired region is the area outside the circle, constrained between these angle bounds.

Step 7

Find all the 5th roots of $-1$ in modulus-argument form.

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Expressing $-1$ in modulus-argument form gives: $-1 = 1 ext{cis}( au).$ To find the 5th roots, apply De Moivre's theorem: $z_k = 1^{1/5} ext{ cis} igg( rac{ au + 2k au}{5} igg), k = 0, 1, 2, 3, 4.$ Calculating these gives: $z_k = ext{cis} igg( rac{(2k + 1) au}{5} igg) ext{ for } k = 0, 1, 2, 3, 4.$

Step 8

(ii) Sketch the 5th roots of $-1$ on an Argand diagram.

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Plot each of the 5th roots identified previously in the Argand diagram. Each root should be spaced evenly on the unit circle, separated by an angle of $rac{2 au}{5}$ .

Step 9

Find the square roots of $3 + 4i$.

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To find the square roots, assume a solution of the form $z = x + yi$ . Thus: $z^2 = 3 + 4i.$ Hence, we have:

ightarrow x^2 - y^2 + 2xyi = 3 + 4i.$$ This leads to two equations: 1. $x^2 - y^2 = 3$ 2. $2xy = 4$ From the second equation, we get $xy = 2$. Substitute $y = rac{2}{x}$ into the first equation: $$ x^2 - igg( rac{2}{x}igg)^2 = 3 ightarrow x^4 - 3x^2 - 4 = 0.$$ Letting $u = x^2$ gives: $$ u^2 - 3u - 4 = 0.$$ Solving using the quadratic formula provides the possible values for $u$, and subsequently for $x$ and $y$.

Step 10

Hence, or otherwise, solve the equation $z^2 + iz - 1 = i$.

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To solve the equation, rearrange to get: $z^2 + iz - (1 + i) = 0.$ Apply the quadratic formula: $z = rac{-b ext{ extpm } ext{sqr}(b^2 - 4ac)}{2a}$ with $a = 1$ , $b = i$ , and $c = -(1 + i)$ , substituting and simplifying will yield the solutions.

Write $i^8$ in the form $a + ib$ where $a$ and $b$ are real - HSC - SSCE Mathematics Extension 2 - Question 2 - 2009 - Paper 1

Worked Solution & Example Answer:Write $i^8$ in the form $a + ib$ where $a$ and $b$ are real - HSC - SSCE Mathematics Extension 2 - Question 2 - 2009 - Paper 1

Write $i^8$ in the form $a + ib$ where $a$ and $b$ are real.

Write $-2 + 3i$ in the form $a + ib$ where $a$ and $b$ are real.

Copy the diagram into your writing booklet, and mark on it the following points: (i) the point $R$ representing $iz$

(ii) the point $S$ representing $w$

(iii) the point $T$ representing $z + w$

Sketch the region in the complex plane where the inequalities $|z - 1| eq 2$ and $- rac{ au}{4} eq ext{arg}(z - 1) eq rac{ au}{4}$ hold simultaneously.

Find all the 5th roots of $-1$ in modulus-argument form.

(ii) Sketch the 5th roots of $-1$ on an Argand diagram.

Find the square roots of $3 + 4i$.

Hence, or otherwise, solve the equation $z^2 + iz - 1 = i$.

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Sketch the region in the complex plane where the inequalities $|z - 1| eq 2$ and $-rac{ au}{4} eq ext{arg}(z - 1) eq rac{ au}{4}$ hold simultaneously.