Worksheet: Regions in the Complex Plane

In this worksheet, we will practice using loci to identify regions in the complex plane.

Q1:

Which of the following represents the region of the complex plane defined by |𝑧+12𝑖|>3?

  • A(b)
  • B(a)
  • C(c)
  • D(d)

Q2:

Which of the following systems of inequalities in terms of the complex number 𝑧 describes the region in an Argand diagram that is enclosed inside a rectangle of vertices 2+3𝑖, 7+3𝑖, 72𝑖, and 22𝑖 and includes all the sides of the rectangle?

  • A2(𝑧)32(𝑧)7ReIm
  • B2(𝑧)33(𝑧)7ReIm
  • C2(𝑧)72(𝑧)3ReIm
  • D2(𝑧)72(𝑧)2ReIm
  • E2(𝑧)72(𝑧)3ReIm

Q3:

Which of the following represents the region of the complex plane defined by |𝑧2𝑖3|2?

  • A(c)
  • B(d)
  • C(b)
  • D(a)

Q4:

The shaded region in the following figure can be described as the intersection of two regions, each described by an inequality. Write both these inequalities in terms of 𝑧.

  • A|𝑧12𝑖|5, 2𝜋3<(𝑧12𝑖)<7𝜋4arg
  • B|𝑧+1+2𝑖|5, 2𝜋3<(𝑧12𝑖)7𝜋4arg
  • C|𝑧+1+2𝑖|5, 𝜋3<(𝑧+1+2𝑖)<7𝜋4arg
  • D|𝑧12𝑖|5, 2𝜋3<(𝑧12𝑖)7𝜋4arg
  • E|𝑧+1+2𝑖|5, 𝜋3<(𝑧+1+2𝑖)7𝜋4arg

Q5:

Which region satisfies the inequality |𝑧6+2𝑖||𝑧2+4𝑖|?

  • A𝐵
  • B𝐴
  • C𝐶
  • D𝐴𝐵
  • E𝐵𝐶

Q6:

Which of the following shaded regions represents the locus of the point 𝑧 satisfying the system of inequalities |𝑧1+𝑖|1, |𝑧1|>|𝑧1+𝑖|?

  • A(C)
  • B(A)
  • C(B)
  • D(D)

Q7:

The figure shows a region in the complex plane.

Write an algebraic description of the shaded region.

  • A|𝑧4𝑖|23
  • B|𝑧4𝑖|213
  • C|𝑧+1+4𝑖|213
  • D|𝑧+4+𝑖|213
  • E|𝑧+4+𝑖|23

Q8:

Which of the following represents the region of the complex plane defined by 𝜋2(𝑧+32𝑖)<𝜋4?arg

  • Ae
  • Bd
  • Cb
  • Da
  • Ec

Q9:

The shaded region in the following figure can be algebraically described by 𝐴𝐵𝐶, where 𝐴={𝑧(𝑧)<𝑎},𝐵={𝑧|𝑧||𝑧𝑧|},𝐶={𝑧|𝑧||𝑧𝑧|}.Im

Find the values of 𝑎, 𝑧, and 𝑧, where 𝑎 and 𝑧,𝑧.

  • A𝑎=2, 𝑧=33𝑖, 𝑧=18131213𝑖
  • B𝑎=2, 𝑧=33𝑖, 𝑧=1813+1213𝑖
  • C𝑎=2, 𝑧=33𝑖, 𝑧=36132413𝑖
  • D𝑎=2, 𝑧=3𝑖, 𝑧=3613+2413𝑖
  • E𝑎=3, 𝑧=3𝑖, 𝑧=36132413𝑖

Q10:

We define the regions 𝐴, 𝐵, and 𝐶 in the complex plane as 𝐴={𝑧𝕔(𝑧)<4},𝐵={𝑧𝕔|𝑧||𝑧812𝑖|},𝐶={𝑧𝕔|𝑧65𝑖|<5}.Re

Which of the following figures could represent the region of the complex plane defined by 𝐴𝐵𝐶?

  • Aa
  • Be
  • Cc
  • Db
  • Ed

Q11:

The complex number 𝑧 satisfies the following conditions: |𝑧|2|𝑧+129𝑖|,|𝑧2𝑖||𝑧+6+4𝑖|,(𝑧)<12.Im

Which of the following shaded regions represents the locus of 𝑧?

  • A
  • B
  • C
  • D

Find the area of the locus of 𝑧.

  • A25𝜋2
  • B191𝜋4
  • C75𝜋2
  • D9𝜋4

Q12:

Consider the following regions in the complex plane: 𝐴={𝑧2|𝑧|<4},𝐵={𝑧(𝑧)(𝑧)<0}.ReIm

The region 𝑅 is defined as 𝑅=𝐴𝐵.

Which of the following shaded regions represents 𝑅 on an Argand diagram?

  • A
  • B
  • C
  • D

Find the area of the region 𝑅.

  • A9𝜋
  • B12𝜋
  • C4𝜋
  • D18𝜋
  • E6𝜋

Q13:

On an Argand diagram, points 𝑃 and 𝑄 represent the complex numbers 33 and 53 respectively, where they are on the circles with centers 𝐶 and 𝐶. 𝐶 represents the number 3+4𝑖 and 𝐶 the number 34𝑖.

Describe the shaded region using two simultaneous inequalities which involve the modulus of complex numbers.

  • A|𝑧34𝑖|8|𝑧3+4𝑖|8
  • B|𝑧34𝑖|8|𝑧3+4𝑖|8
  • C||𝑧34𝑖||8||𝑧3+4𝑖||8
  • D||𝑧34𝑖||8||𝑧3+4𝑖||8
  • E||𝑧34𝑖||22||𝑧3+4𝑖||22

Describe the shaded region using one double inequality which involves the argument of a complex number.

  • A𝜋3<𝑧3+4𝑖𝑧34𝑖<2𝜋3arg
  • B𝜋3<𝑧3+4𝑖𝑧34𝑖<2𝜋3arg
  • C𝜋3𝑧+34𝑖𝑧+3+4𝑖2𝜋3arg
  • D𝜋3𝑧+34𝑖𝑧+3+4𝑖2𝜋3arg
  • E𝜋3𝑧3+4𝑖𝑧34𝑖2𝜋3arg

Q14:

Which of the following shaded regions represents the locus of the point 𝑧 satisfying the inequality 2|𝑧+2+𝑖|<4?

  • A(A)
  • B(B)
  • C(C)
  • D(D)

Q15:

Which of the following shaded regions represents the region of the complex plane described by 3𝜋4<(𝑧2+𝑖)7𝜋6arg?

  • A(c)
  • B(d)
  • C(b)
  • D(a)

Q16:

The complex number 𝑧 satisfies the following conditions:

  1. |𝑧|2|𝑧+129𝑖|,
  2. |𝑧2𝑖||𝑧+6+4𝑖|,
  3. Im(𝑧)<12.

Represent the region on an Argand diagram.

  • A
  • B
  • C
  • D
  • E

Q17:

On an Argand diagram, sketch the region represented by 𝜋2(𝑧+32𝑖)<𝜋4arg.

  • A
  • B
  • C
  • D
  • E

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