Sketch the region defined by \(|z|<3\) and \(0 \leq \arg (z-i) \leq \dfrac{\pi}{2}\). (3 marks) --- 8 WORK AREA LINES (style=blank) ---
Complex Numbers, EXT2 N2 2022 HSC 1 MC
Let `R` be the region in the complex plane defined by `1 < text{Re}(z) <= 3` and `(pi)/(6) <= text{Arg}(z) < (pi)/(3)`.
Which diagram best represents the region `R`?
Complex Numbers, EXT2 N2 EQ-Bank 23
Consider the point on the complex plane `z_1 = sqrt3 + 1`.
Sketch the ray given by `text(Arg)(z-z_1) = (5pi)/6` on the Argand diagram below. (2 marks)
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Complex Numbers, EXT2 N2 2020 SPEC2 2
Two complex numbers, `u` and `v`, are defined as `u = −2-i` and `v = −4-3i`.
- Express the relation `|z-u| = |z-v|` in the cartesian form `y = mx + c`, where `m, c ∈ R`. (3 marks)
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- Plot the points that represent `u` and `v` and the relation `|z-u| = |z-v|` on the Argand diagram below. (2 marks)
- State a geometrical interpretation of the graph of `|z-u| = |z-v|` in relation to the points that represent `u` and `v`. (1 mark)
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- Sketch the ray given by `text(Arg)(z-u) = pi/4` on the Argand diagram in part b. (1 mark)
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In Cartesian form, write down the function that describes the ray `text(Arg)(z-u) = pi/4`. (1 mark)
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a. `y = −x-5`
b.
c. `|z-u| = |z-v|\ text(is the graph of the perpendicular bisector of the)`
`text(line joining)\ u and v.`
d.
e. `f: (−2, ∞) ->, f(x) = x + 1`
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a. `text(Let)\ \ z = x + iy`
`z-u = x + 2 + iy + i`
`z-v = x + 4 + iy + 3i`
`|z-u| = |z-v|`
| `(x + 2)^2 + (y + 1)^2` | `= (x + 4)^2 + (y + 3)^2` |
| `x^2 + 4x + 4 + y^2 + 2y + 1` | `= x^2 + 8x + 16 + y^2 + 6y + 9` |
| `-4y` | `= 4x + 20` |
| `y` | `= −x-5` |
| b. |
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c. `|z-u| = |z-v|\ text(is the graph of the perpendicular bisector of the)`
`text(line joining)\ u and v.`
| d. |
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e. `text(Arg)(z-u) = pi/4 =>\ text(gradient) = 1, ytext(-intercept at)\ (0, 1)`
`:. f: (−2, ∞) -> RR, \ f(x) = x + 1`
Complex Numbers, EXT2 N2 2019 HSC 12a
Sketch the region defined by `pi/4 <= text(arg)(z) <= pi/2` and `text(Im)(z) <= 1`. (2 marks)
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`text(Shaded Area): pi/4 <= text(arg)(z) <= pi/2 and text(Im)(z) <= 1`
Complex Numbers, EXT2 N2 2018 HSC 7 MC
Which diagram best represents the solutions to the equation `text(arg)(z) = text(arg)(z + 1-i)`?
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B. |  |
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D. |  |
Complex Numbers, EXT2 N2 2017 HSC 11c
Sketch the region in the Argand diagram where
`-pi/4 <= text(arg)(z) <= 0 and |z-1 + i| <= 1`. (2 marks)
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`|z-1 + i| = 1\ \ text{is a circle with centre (1, –1 )}`
`text{and radius 1.}`
`text(Shaded area:)\ -pi/4 <= text(arg)(z) <= 0\ ∩\ |z-1 + i| <= 1`
Complex Numbers, EXT2 N2 2009 HSC 2d
Sketch the region in the complex plane where the inequalities `| z-1 | <= 2` and `-pi/4 <= text(arg) (z-1) <= pi/4` hold simultaneously. (2 marks)
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Complex Numbers, EXT2 N2 2014 HSC 11c
Sketch the region in the Argand diagram where `|\ z\ | ≤ |\ z-2\ |` and `−pi/4 ≤ text(arg)\ z ≤ pi/4`. (3 marks)
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`text(Let)\ \ z=x+iy`
| `|\ x + iy\ |` | `≤ |\ x-2 + iy\ |` |
| `sqrt(x^2 + y^2)` | `≤ sqrt((x-2)^2 + y^2)` |
| `x^2 + y^2` | `≤ x^2-4x + 4 + y^2` |
| `4x-4` | `≤ 0` |
| `x` | `≤ 1` |
`−pi/4 ≤ text(arg)\ z ≤ pi/4`
