Aussie Maths & Science Teachers: Save your time with SmarterEd
It is given that \(\abs{z-1+i}=2\).
What is the maximum possible value of \(\abs{z}\)?
\(\abs{z-1+i}=2 \ \Rightarrow \ \text{circle centre} \ \ (1,-i), \ \ \text {radius}=2\)
\(OA=\sqrt{1^2+1^2}=\sqrt{2}\)
\(AB=2 \ \ \text{(radius)}\)
\(\therefore \abs{z}_{\text{max}}=2+\sqrt{2}\)
\(\Rightarrow C\)
A shaded region on a complex plane is shown.
Which relation best describes the region shaded on the complex plane?
The graph of the circle given by `|z-2-sqrt3i| = 1`, where `z ∈ C`, is shown below.
For points on this circle, find the maximum value of `|z|`. (2 marks)
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`text(Centre of circle at)\ (2, sqrt3)`
`text(Radius = 1)`
`text(By Pythagoras, line from origin to the centre of the circle)`
`d = sqrt(2^2 + sqrt(3)^2) = sqrt7`
`:. |z|_text(max) = sqrt7 + 1`
Sketch the region in the complex plane where the inequalities
`| z + overset_z | ≤ 1` and `| z-i | ≤ 1`
hold simultaneously. (3 marks)
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`| z + overset_z | ≤ 1 `
| `| x + i y + x-iy |` | `≤ 1` |
| `| 2x |` | `≤ 1` |
| `| x |` | `≤ frac{1}{2}` |
`| z-i | ≤ 1 \ => \ text{Circle, radius = 1 , centre (0, 1)`
Sketch the region on the Argand diagram where the inequalities
`| z-overset_z | < 2` and `| z-1 | >=1`
hold simultaneously. (3 marks)
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| `| z-overset_z |` | `< 2` |
| `| x + i y-(x-i y) |` | `< 2` |
| `| 2 i y |` | `< 2` |
| `| y |` | `< 1` |
`| z-1 | = 1 \ => \ text{Circle, radius = 1, centre (1, 0)}`
`:.\ text(Graph:)\ | z-overset_z |<2 \ ∩ \ | z-1 | >= 1`
Sketch the region in the Argand diagram where
`-pi/4 <= text(arg)(z) <= 0 and |z-1 + i| <= 1`. (2 marks)
`|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`
Which complex number lies in the region `2 < |z-1| < 3`?
`|z-1| = 2\ \ text(is a circle with centre)\ \ (1, 0),\ text(radius 2)`
`|z-1| = 3\ \ text(is a circle with centre)\ \ (1, 0),\ text(radius 3)`
`:.\ text(Only)\ \ 3-i\ \ text(lies within the two circles,)`
`text(i.e. satisfies)\ \ 2 < |z-1| < 3`
`=> D`
The point `P` on the Argand diagram represents the complex number `z`, where `z` satisfies
`1/z + 1/bar z = 1.`
Give a geometrical description of the locus of `P` as `z` varies. (3 marks)
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The complex number `z` satisfies `| z-1 | = 1.`
What is the greatest distance that `z` can be from the point `i` on the Argand diagram?
`| z-1 | = 1\ \ text{is a circle, centre (1,0), radius 1.}`
`text(Consider the graph:)`
`text(Distance from)\ \(0, i)\ \ text(to the centre)=sqrt2`
`:.\ text(Greatest distance)=sqrt2 + text(radius)=sqrt2+1`
`=> D`
Which region on the Argand diagram is defined by `pi/4 <= | z-1 | <= pi/3?`
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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Sketch the region in the complex plane where the inequalities `1 ≤ |\ z\ | ≤ 2` and `0 ≤ z + bar z ≤ 3` hold simultaneously. (2 marks)
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| `text(Consider)\ \ \ \ ` | `1≤|\ z\ |≤2` |
| `1≤x^2+y^2≤4` |
`z + bar z = (x+iy)+(x-iy)=2x`
| `text(Consider)\ \ \ \ ` | `0≤z + bar z≤3` |
| `0≤2x≤3` | |
| `0≤x≤3/2` |
Shade the region on the Argand diagram where the two inequalities
`|\ z + 2\ | ≥ 2` and `|\ z-i\ | ≤ 1` both hold. (2 marks)
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`|\ z + 2\ | ≥ 2 and |\ z-i\ | ≤ 1`
| `(x + 2)^2 + y^2` | `≥ 4` |
| `x^2 + (y-1)^2` | `≤ 1` |
