Judge each statement in turn.

A — "Gain of an active filter can be more than 1." TRUE. An active filter contains an op-amp (or transistor) powered from a supply, so it can deliver more signal power than it receives. A Sallen–Key low-pass section, for instance, has a passband gain of \(1+R_F/R_1\), which is set greater than unity whenever required.

B — "Gain of a passive filter can be more than 1." FALSE. A passive network of R, L and C contains no energy source, so it can only attenuate: \(|H(j\omega)|\le 1\) in the passband. (An LC network can show a voltage peak above unity at resonance because of Q-multiplication, but there is still no power gain, and the standard statement — and what the question tests — is that a passive filter cannot amplify.)

C — "A Butterworth filter changes the roll-off of the output as its order changes." TRUE. The stopband slope is fixed by the order n:

\(\text{Roll-off}=20n\ \text{dB/decade}=6n\ \text{dB/octave}\)

so a 1st-order section falls at 20 dB/decade, a 2nd-order at 40, a 4th-order at 80, and so on.

D — "The phase at cut-off does not change with order." FALSE. Each pole contributes 45° of lag at the cut-off frequency, so the phase there is

\(\phi = -n\times 45^{\circ}\)

−45° for first order, −90° for second order, −180° for fourth order. The phase at cut-off clearly depends on n. (What is independent of order is the magnitude at cut-off — always −3 dB — which is probably the fact the statement is trying to distort.)

E — "Feedback is not needed in any filter circuit." FALSE. Passive filters indeed use no feedback, but every active filter topology (Sallen–Key, multiple-feedback, state-variable, biquad) relies on feedback around the amplifier to place the poles and set Q. So the sweeping claim "any filter" is wrong.

The true statements are A and C.

Extra context on Butterworth. It is the maximally flat approximation, with magnitude \(|H(j\omega)|=\dfrac{1}{\sqrt{1+(\omega/\omega_c)^{2n}}}\) — no passband ripple, at the price of a gentler transition than a Chebyshev filter of the same order.

Hence, the correct answer is A and C only.