Further Mathematics · Pearson Edexcel 9FM0 · Option FP2 · Topic 4

Further complex numbers

Loci and regions in the Argand diagram, and transformations of the complex plane.

  • 2 sub-topics
  • Taught in Year 13
  • 25 practice questions
  • 2 animated lessons
Practice questions (25) Lessons Worksheets All Edexcel Further Mathematics topics

What the specification says

Every sub-topic Pearson Edexcel can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

  1. 4.1Further loci and regionsY13
  2. 4.2Transformations of the complex planeY13

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Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 25 questions in this topic, free, no account needed.

Further complex numbers

Not needed for Year 12Easy

Transformations of the complex plane · Year 13 · 1 mark

  1. Which transformation of the complex plane is w=3zw = 3z?
    (1)

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  1. Adding a constant translates. Multiplying by a complex number enlarges by its modulus and rotates by its argument.A: An enlargement with scale factor 3, centre the origin

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Further complex numbers

Not needed for Year 12Medium

Further loci · Year 13 · 3 marks

  1. Describe the locus ∣z−1∣=3∣z+1∣|z - 1| = 3|z + 1|.
    (3)
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  1. (x−1)2+y2(x - 1)^2 + y^2=9((x+1)2+y2){}= 9((x + 1)^2 + y^2)
  2. Rearrange to x² + y² + … = 0.
  3. A circle (Apollonius circle).

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Further complex numbers

Not needed for Year 12Hard

A locus in the complex plane · Year 13 · 7 marks

The locus of points zz satisfying ∣z−3∣=3∣z+3∣|z - 3| = 3|z + 3| is a circle CC. The points AA and BB represent 33 and −3-3.
ReImAB
Figure 1
  1. (a)
    Find the centre and the exact radius of CC.
    (5)
  2. (b)
    Sketch CC on an Argand diagram, marking AA and BB.
    (2)

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Part (a)5 marks

  1. Square both sides.zz=x{}= x+iy: (x−3)2{}+ \mathrm{i}y: \ (x - 3)^2+y2{}+ y^2=9[(x+3)2+y2]{}= 9\left[(x + 3)^2 + y^2\right]
  2. Expand and collect.8x2+60x+8y2+72=08 x^{2} + 60 x + 8 y^{2} + 72 = 0
  3. Divide by the coefficient of x2x^2 and complete the square.centre (−154,0)\text{centre } (- \frac{15}{4}, 0)
  4. r=94r = \frac{9}{4}

Part (b)2 marks

  1. See Figure 1.A circle with the given centre, enclosing BB but not AA.

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