Hello! Although I cannot directly sketch the graph for you, I can guide you through the process of sketching the graph yourself.

To sketch the first 0.5 ms of the continuous signal x(t) = cos(2 * pi * 10000 * t), follow these steps:

1. Label the x-axis as 't' (time in milliseconds) and the y-axis as 'x(t)' (amplitude).
2. The period of the signal can be calculated using the formula T = 1/f, where f is the frequency. In this case, f = 10000 Hz, so T = 1/10000 = 0.1 ms.
3. Divide the 0.5 ms time range into intervals of 0.1 ms (the period of the signal). This means you will have 5 intervals: [0, 0.1], [0.1, 0.2], [0.2, 0.3], [0.3, 0.4], and [0.4, 0.5] ms.
4. At the beginning of each interval (0, 0.1, 0.2, 0.3, and 0.4 ms), the value of the cosine function will be 1, since cos(0) = 1.
5. At the midpoint of each interval (0.05, 0.15, 0.25, 0.35, and 0.45 ms), the value of the cosine function will be -1, since cos(pi) = -1.
6. Connect the points with a smooth curve that follows the shape of a cosine wave.

Regarding the aliased frequency, we need to know the sampling frequency (f_s) to determine if aliasing occurs. In general, if the sampling frequency is less than twice the signal frequency (i.e., f_s < 2 * f), aliasing will occur. The aliased frequency can be found using the formula:

aliased_frequency = |f_s * k - f| or |f_s * k + f|

where k is an integer such that the result is within the range of [0, f_s/2].
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