QCEVault

Further complex numbers — Question 260

Original QCE Vault practice · 5 marks

Q260 · Practice questionTechnology-freeComplex familiar5 marks

QUESTION 260 (5 marks)

The complex coordinates of two adjacent sides of a parallelogram are zz and ww. Its diagonals are represented by z+wz+w and z−wz-w.
a)
Prove the identity ∣z+w∣2+∣z−w∣2=2(∣z∣2+∣w∣2)|z+w|^2+|z-w|^2=2(|z|^2+|w|^2).
[3 marks]
b)
A student says that two perpendicular adjacent sides of lengths 3 and 4 give diagonals with squared lengths summing to 49. Evaluate this statement.
[2 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. Q174 · Original practice · 8 marks
    A nonzero complex number satisfies ∣z∣=1|z|=1. Define w=z+1/zw=z+1/z.
    Further complex numbers