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A-LevelMathematicsComplex numbers9709/32 · Feb/Mar 2020 · Q1010 Marks

The complex numbers v and w satisfy the equations v + iw = 5 and (1 + 2i)v - w = 3i. Solve the equations for v and w, giving your answers in the form x + iy, where x and y are real.

  1. (a)The complex numbers v and w satisfy the equations v + iw = 5 and (1 + 2i)v - w = 3i. Solve the equations for v and w, giving your answers in the form x + iy, where x and y are real.[6]
  2. (b)(i)On an Argand diagram, sketch the locus of points representing complex numbers z satisfying |z - 2 - 3i| = 1.[2]
  3. (b)(ii)Calculate the least value of arg z for points on this locus.[2]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2020). The mark or mark type is shown in brackets.

  1. [M1] Solve for v or w
  2. [M1] Use i² = -1
  3. [A1] Obtain v = 2i/(1+i) or w = (5+7i)/(-1+i)
  4. [M1] Multiply numerator and denominator by the conjugate of the denominator
  5. [A1] Obtain v = 1-i
  6. [A1] Obtain w = 1-6i
  7. [B1] Show a circle with centre 2 + 3i
  8. [B1] Show a circle with radius 1 and centre not at the origin
  9. [M1] Carry out a complete method for finding the least value of arg z
  10. [A1] Obtain answer 40.2° or 0.702 radians

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About this question

Topic
Complex numbers, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source
A-Level Mathematics, Feb/Mar 2020 Paper 3 Variant 2, question 10.

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