QCEVault

Further complex numbers — Question 174

Original QCE Vault practice · 8 marks

Q174 · Practice questionTechnology-freeComplex unfamiliar8 marks

QUESTION 174 (8 marks)

A nonzero complex number satisfies ∣z∣=1|z|=1. Define w=z+1/zw=z+1/z.
a)
Show that w is real and determine its full possible range.
[3 marks]
b)
Solve w=1w=1 for z in exact Cartesian form.
[2 marks]
c)
An animation uses z(t)=cis⁡(2t)z(t)=\operatorname{cis}(2t) and w(t)=z(t)+1/z(t)w(t)=z(t)+1/z(t), with 0≤t≤π0\le t\le\pi. Treat w(t)w(t) as displacement in metres. Determine its greatest speed and every time attaining it.
[3 marks]
Question linkSyllabus coverage

Related questions

  1. Q148 · Original practice · 1 mark
    The four roots of z4=−16z^4=-16 are shown on the Argand plane. For each root, a new complex number is defined by
    w=z+4z.w=z+\frac{4}{z}.
    Which statement describes the set of distinct values of ww?
    Further complex numbers
  2. Q153 · Original practice · 1 mark
    The points AA, BB and CC represent the three roots of z3=−8iz^3=-8i on the Argand plane. The point PP represents z=2z=2. What is the exact value of PA×PB×PCPA\times PB\times PC?
    Further complex numbers
  3. Q173 · Original practice · 5 marks
    Four beacons in a light installation have complex coordinates given by the solutions of (z−1−i)4=16(z-1-i)^4=16. They are represented on the Argand plane. The dashed circle has centre C=1+iC=1+i.
    Further complex numbers
  4. Q175 · Original practice · 1 mark
    The polynomial p(z)=z2−2iz−2p(z)=z^2-2iz-2 has zero 1+i1+i. Which is its other zero?
    Further complex numbers