The given continuous-time signal is $x(t) = \cos(200\pi t)$. This signal is sampled at instants $t = n/400, n = 0, 1, \ldots ,7$ to obtain the discrete-time signal x[n].
The discrete-time signal (x[n]) can be written as:
$x[n] = \cos(200\pi \cdot n/400) = \cos(\pi/2 \cdot n)$
The 8-point DFT of (x[n]) is defined as:
$X[k] = \sum_{n=0}^{7} x[n]e^{-j2\pi nk/8}, k=0,1,\ldots,7$
Substituting (x[n]) into the equation, we get:
$X[k] = \sum_{n=0}^{7} \cos(\pi/2 \cdot n) \cdot e^{-j2\pi nk/8}$
We can calculate the DFT for each (k) value. However, we know that $\cos(n \cdot \pi/2))$ is 0 for odd (n) and 1 for even (n). Therefore, the non-zero terms in the sum are for even (n) (0, 2, 4, 6).
For these terms, $e^{-j2\pi nk/8}$ equals 1 when (k) is a multiple of 4. Therefore, the only non-zero terms in the DFT are when (k) equals 0 or 4.
So, the correct answer is: Only X[0] and X[4] are non-zero. However, this option is not given in the choices. There seems to be a mistake in the question or the provided options.