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

The two hypotheses fix the order of vanishing

Let p(z)=a0+a1z+⋯+anznp(z) = a_{0} + a_{1}z + \cdots + a_{n}z^{n} be a polynomial of degree ≥ 3. Suppose that p(0) = 0 and p′(0) ≠ 0. Let f:Cf : \mathbb{C} \ {0} →C\to \mathbb{C} be defined by f(z)=p(z)/z2f(z) = p(z)/z^{2}. Which of the following statements are true?

  1. A.f has a removable singularity at z = 0.
  2. B.f has a simple pole at z = 0.✓ Correct
  3. C.∫C(p(z)/z)\int_C (p(z)/z)dz = 0, where C is any counterclockwise oriented smooth closed curve in C\mathbb{C}.✓ Correct
  4. D.∫γf(z)\int_\gamma f(z)dz =2π= 2\piia1_{1}, where γ=\gamma = {z∈Cz \in \mathbb{C} : |z| = 2025} is a counterclockwise oriented circle.✓ Correct

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 execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

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

More on this topic

The chapter behind this: Residue theorem — the five standard patterns — free to read

From Singularities and Residues › Residue theorem and standard contour integrals

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