1. Object motion

The particle oscillates on the $x$-axis about $x_0=10\,\text{cm}$ with amplitude $2\,\text{cm}$. So the object position is $x=10+2\cos(\omega t)$ Hence the object moves between $u=8\,\text{cm} \quad \text{and} \quad u=12\,\text{cm}$ where $u$ is the object distance from the mirror.

2. Mirror formula

For a concave mirror of focal length $f=5\,\text{cm}$, using the standard mirror relation for magnitudes, $\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$ So, $\frac{1}{5}=\frac{1}{u}+\frac{1}{v}$ which gives $\frac{1}{v}=\frac{1}{5}-\frac{1}{u}=\frac{u-5}{5u}$ Therefore, $v=\frac{5u}{u-5}$

Since $u=10+2\cos(\omega t),$ we get $v(t)=\frac{5(10+2\cos\omega t)}{5+2\cos\omega t}$

This is a nonlinear function of $\cos(\omega t)$.

3. Is image motion periodic?

Because $v(t)$ is a definite single-valued function of $\cos(\omega t)$, and $\cos(\omega t)$ is periodic with period $T=\frac{2\pi}{\omega},$ we have $v(t+T)=v(t).$ So the image also executes periodic motion.

Thus:

• A is correct
• B is incorrect

4. Turning points of the image

The turning points occur when the object is at its extremes:

• For $u=8\,\text{cm}$, $v_1=\frac{5\cdot 8}{8-5}=\frac{40}{3}\,\text{cm}$

• For $u=12\,\text{cm}$, $v_2=\frac{5\cdot 12}{12-5}=\frac{60}{7}\,\text{cm}$

The image of the mean position $u=10\,\text{cm}$ is $v_0=\frac{5\cdot 10}{10-5}=10\,\text{cm}$

Now compare distances from $v_0$: $v_1-v_0=\frac{40}{3}-10=\frac{10}{3}$ $v_0-v_2=10-\frac{60}{7}=\frac{10}{7}$ These are not equal, so the turning points are asymmetric about the image of the central point.

Thus:

• C is correct

5. Distance between turning points

$\Delta v=\left|\frac{40}{3}-\frac{60}{7}\right|$ $=\left|\frac{280-180}{21}\right|=\frac{100}{21}\,\text{cm}$

Thus:

• D is correct

6. Final evaluation of options

• A: True
• B: False
• C: True
• D: True

So the correct statements are $\boxed{A,\ C,\ D}$

7. Comparison with stored answer

Stored correct answer: $B$

But from the mirror equation, the image position is a periodic function of time, not non-periodic. Also statements $C$ and $D$ are directly verified numerically.

Therefore, I do not agree with the stored answer.