Part BCSIR NET June 2024residues-can-cancel

Residues can cancel

Let a, b be two real numbers such that a < 0 < b. For a positive real number r, define $γ_{r} (t) =$ re^(it) (where $t \in [0, 2 π])$ and $I_{r} = (1/2 π i) \int_{(} γ_{r}) (z^{2} + 1) / ((z - a) (z - b))$ dz. Which of the following statements is necessarily true?

A.I_r ≠ 0 if r > max{|a|, b}
B.I_r ≠ 0 if r < max{|a|, b}
C.I_r = 0 if r > max{|a|, b} and |a| = b✓
D.I_r = 0 if |a| < r < b

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

Residues can cancel

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