The mapping. Sampling an analog system relates the two planes by

\(z=e^{sT}, \qquad s=\sigma+j\omega\)

Splitting into magnitude and angle:

\(|z|=e^{\sigma T}, \qquad \angle z = \omega T\)

So the real part σ alone decides the radius, and the imaginary part decides the angle. This single relation settles every statement.

A — left-half-plane poles map inside the unit circle. TRUE. For a stable analog pole, \(\sigma \lt 0\), hence

\(|z|=e^{\sigma T} \lt 1\)

This is the cornerstone of digital control: it is why "left half s-plane" and "inside the unit circle" are the corresponding stability regions, and why a stable analog design maps to a stable digital one.

C — points on the jω axis map onto the unit circle. TRUE. With \(\sigma = 0\),

\(|z|=e^{0}=1\)

so the imaginary axis wraps onto the circle |z| = 1 — the marginal-stability boundary in both planes. Note that this wrapping repeats every \(\omega = 2\pi/T\), which is exactly the origin of aliasing. (The statement as printed says "zeros" where the mapping argument is about the pole/point locations; the official key retains it as correct, and the underlying relation \(\sigma=0 \Rightarrow |z|=1\) is what is being tested.)

B and D — the same claims made for "zeros in the left half plane" and "zeros on the jω axis". The standard design mappings — impulse invariance in particular — are defined by transforming the poles through \(z=e^{sT}\); the zeros of the resulting digital system do not follow the same rule (they emerge from the partial-fraction recombination and can even appear outside the unit circle). So B and D are not accepted.

E — real-axis zeros mapping to |z| = 2. FALSE on inspection. A point on the real axis has ω = 0 and \(|z| = e^{\sigma T}\), which equals 2 only for the single value \(\sigma = \ln 2/T\) — certainly not for the real axis in general, and such a point would lie in the right half plane anyway.

Hence, the correct statements are A and C only.