To sketch the first 0.5 ms of x(t) = cos(2pi10000*t), we need to plot the waveform for the time range of 0 to 0.0005 seconds. The sampling frequency for this signal is 100000 Hz.

Label the x-axis as time (in seconds) and the y-axis as amplitude. For the x-axis, we can use the following scale: 0 s, 0.00005 s, 0.0001 s, 0.00015 s, 0.0002 s, 0.00025 s, 0.0003 s, 0.00035 s, 0.0004 s, 0.00045 s, and 0.0005 s. For the y-axis, we can use the range of -1 to 1.

Using the above information, we can sketch the waveform as shown below:

     ^
1.00 |                                              .   
     |                                         .        
     |                                     .            
     |                                 .                
     |                             .                    
     |                         .                        
     |                     .                            
     |                 .                                
     |             .                                    
     |         .                                        
-1.00 +-----------------------------------------------> 0.000 0.00005 0.0001 0.00015 0.0002 0.00025 Time (in seconds)

From the above plot, we can see that the waveform has a frequency of 10000 Hz and a period of 0.0001 seconds.

Determine the aliased frequency f: If the signal is sampled at a frequency lower than the Nyquist frequency (i.e. 2 times the maximum frequency component in the signal), then aliasing can occur. In this case, if the signal is sampled at a frequency of 200
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