Part CCSIR NET December 2024the-two-hypotheses-fix-the-order-of-vanishing

The two hypotheses fix the order of vanishing

Let $p (z) = a_{0} + a_{1} z + \dots + a_{n} z^{n}$ be a polynomial of degree ≥ 3. Suppose that p(0) = 0 and p′(0) ≠ 0. Let $f : C$ \ {0} $\to C$ be defined by $f (z) = p (z) / z^{2}$ . Which of the following statements are true?

A.f has a removable singularity at z = 0.
B.f has a simple pole at z = 0.✓
C. $\int_{C} (p (z) / z)$ dz = 0, where C is any counterclockwise oriented smooth closed curve in $C$ .✓
D. $\int_{γ} f (z)$ dz $= 2 π$ ia $_{1}$ , where $γ =$ { $z \in C$ : |z| = 2025} is a counterclockwise oriented circle.✓

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.

See pricing

50 are analysed free — try those first.

Related counterexample: ∮γ f = 0 implies f is holomorphic inside γ

The two hypotheses fix the order of vanishing

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