Sampling low pass filtered white noise
The output of your filter is what is sometimes called band-limited white noise. In
your particular case, the autocorrelation function of the output noise is a sinc
function whose zeroes are every $100$ microseconds, that is, samples taken at
the rate of $10^4$ samples/second are uncorrelated.  Your samples at $3\times 10^4$
samples per second are closer together and thus are correlated.  In fact,
the correlation coefficient between successive samples is 
$\displaystyle \frac{\sin(\pi/3)}{\pi/3} = \frac{3\sqrt{3}}{2\pi} \approx 0.827$

Note added after question was edited: Samples taken $0.03$ milliseconds apart
are at a frequency of $33.33\ldots$ kHz, not $30$ kHz as you say in the
part of the question that you typed in yourself. Regardless, the answer (B) is
incorrect and the reasoning given in support of answer (B)
is bogus. What is being sampled is
not white noise but filtered white noise, and Answer A is correct.

For (B) to be the correct answer,
the sampling rate must be a sub-multiple of $10$ kilosamples per second,
that is the samples must be spaced $100$ microseconds apart
or an integer multiple of $100$ microseconds
apart.  Sampling every $0.3$ milliseconds, that is, every $300$ microseconds 
meets this criterion; sampling every $0.03$ milliseconds does not.
Now sampling at $10$ kilosamples/second  makes the samples uncorrelated 
(answer (C)) and to get from this to the stronger result that
the samples are statistically independent (answer (B)), we need
further assumptions about the noise. The standard assumption is
that the noise is Gaussian. 


  In summary, (B) would be the correct answer
  if the sampling were done every $0.3$ milliseconds (since the problem
  statement already includes the assertion that the noise is Gaussian)
  and (C) would be the correct answer if it did not say that the noise
  is Gaussian since we could not the make the specialization from
  uncorrelated to independent.
  But when the sampling is done every $0.03$ milliseconds, then (A)
  is the correct answer.