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Part CCSIR NET December 2024an-antisymmetric-kernel-has-no-real-eigenvalue

An antisymmetric kernel has no real eigenvalue

The integral equation u(x)=f(x)+(2/π)∫0πsin⁡(x−t)u(t)u(x) = f(x) + (2/\pi)\int_{0}^\pi \sin(x - t)u(t)dt has a unique solution if

  1. A.f(x) = cos x✓ Correct
  2. B.f(x) = cos 5x✓ Correct
  3. C.f(x) = sin x✓ Correct
  4. D.f(x) = sin 5x✓ 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 base field or ring.

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50 are analysed free — try those first.

The trap it tests

Base field or ring

The answer changes with the field or ring you are working over.

Drill statements like this

Related counterexample: Every integral equation of the second kind has a unique solution

More on this topic

The chapter behind this: Integral equations: Fredholm vs Volterra — free to read

From Linear Integral Equations › Fredholm and Volterra equations

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