Part BCSIR NET December 2024the-kernel-inverts-a-second-derivative

The kernel inverts a second derivative

If u is a solution of the integral equation $u(x) = \lambda\int_{0}^{1}K(x, t)u(t)$ dt, where K(x, t) := x(1 − t) for 0 ≤ x ≤ t ≤ 1 and t(1 − x) for 0 ≤ t ≤ x ≤ 1, then

A. $d^{2}u/$ dx $^{2} + \lambda{}u = 0, u(0) = 0 = u(1)$ ✓ Correct
B. $d^{2}u/$ dx $^{2} - \lambda{}u = 0, u(0) = 0 = u(1)$
C.du/dx $+ \lambda{}u = 0, u(0) = 0 = u(1)$
D.du/dx $- \lambda{}u = 0, u(0) = 0 = u(1)$

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 standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: Every boundary value problem has a Green's function

The kernel inverts a second derivative

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