Further Mathematics · OCR H245 · Topic 4.02

Complex numbers

The language of complex numbers, basic operations, solution of equations, Argand diagrams, Euler's formula, loci, De Moivre's theorem and more.

  • 9 sub-topics
  • 5 in Year 12, 4 in Year 13
  • 178 practice questions
  • 9 animated lessons
Practice questions (178) Lessons Worksheets coming soon All OCR Further Mathematics topics

What the specification says

Every sub-topic OCR 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. 4.02a-dThe language of complex numbersY12
  2. 4.02e-fBasic operationsY12
  3. 4.02g-jSolution of equationsY12
  4. 4.02k-mArgand diagramsY12
  5. 4.02nEuler's formulaY13
  6. 4.02o-pLociY12
  7. 4.02qDe Moivre's theoremY13
  8. 4.02rnth rootsY13
  9. 4.02sRoots of unityY13

Try 3 questions

Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 178 questions in this topic, free, no account needed.

Complex numbers

Easy

Quadratic with complex roots · Year 12 · 2 marks

  1. Solve z2+8z+41=0z^2 + 8z + 41 = 0.
    (2)
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  1. Quadratic formula.z=−8±−1002z = \frac{-8 \pm \sqrt{-100}}{2}
  2. zz=−4{}= -4+5i, −4{}+ 5\mathrm{i}, \ -4−5i{}- 5\mathrm{i}

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Complex numbers

Medium

Polynomial with complex roots · Year 12 · 4 marks

f(z)=z3+9z2+43z+75f(z) = z^{3} + 9z^{2} + 43z + 75. Given that −3+4i-3 + 4\mathrm{i} is a root:
  1. Find the other roots.
    (4)

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  1. Real coefficients: roots come in conjugate pairs.−3−4i-3 - 4\mathrm{i} is a root.
  2. Multiply the conjugate factors.(z−(−3+4i))(z−(−3−4i))(z - (-3 + 4\mathrm{i}))(z - (-3 - 4\mathrm{i}))=z2+6z+25{}= z^{2} + 6z + 25
  3. Divide to find the remaining linear factor.z=−3z = -3

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Complex numbers

Hard

Perpendicular bisector locus · Year 12 · 3 marks

  1. Find the Cartesian equation of ∣z−2∣=∣z−4i∣|z - 2| = |z - 4\mathrm{i}|.
    (3)
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  1. Square both distances.(x−2)2+y2=x2+(y−4)2(x - 2)^2 + y^2 = x^2 + (y - 4)^2
  2. −4x+4=−8y+16-4x + 4 = -8y + 16
  3. The perpendicular bisector of the two points.y=12x+32y = \tfrac{1}{2}x + \tfrac{3}{2}

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