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Part BCSIR NET June 2025two-arcs-share-the-same-endpoints-so-check-an-interior-point

Two arcs share the same endpoints so check an interior point

Let X be the image of the interval [0, 1] under the Möbius transformation f(z) = (z − i)/(z + i). Which of the following statements is true?

  1. A.X is the line segment joining −1 and −i.
  2. B.X = {eiθe^{i\theta} | θ∈[π,3π/2]\theta \in [\pi, 3\pi/2]}.✓ Correct
  3. C.X is the line segment joining −1 to 1.
  4. D.X = {eiθe^{i\theta} | θ∈[−π/2,π]\theta \in [-\pi/2, \pi]}.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests invariants don't determine.

See pricing

50 are analysed free — try those first.

The trap it tests

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Drill statements like this

Related counterexample: ℂ and the unit disc are biholomorphic (both are simply connected)

More on this topic

The chapter behind this: Conformal maps and Möbius transformations — free to read

From Zeros and Mappings › Conformal maps, Möbius transformations, Schwarz lemma

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