Signal System Basics
Lets check whether option A is the correct choice or not. 


When signal x(t) goes through system A, the magnitudes of all frequency components get multiplied by +j or -j correspondingly. Hence, the output response alpha will look like:





The magnitude of alpha response get multiplied by j in the next step.





This response beta is then added with the original signal x(t) in the next step. The response beta is just a -ve mirror image of x(t) around -fc band. Hence all negative frequency components are diminished to zero after addition. But the band of frequencies around +fc are added up and doubled in magnitude.





Now if you look at the output response y(t), it is nothing but the frequency shifted version of gamma. Hence the function of system B would be to frequency shift the gamma  by -fc, with unity gain. Time to take a look at the following properties of fourier transform.




From the second property, we can infer that gamma(t) has to be multipled by e^jwt to obtain y(t). Hence the function of system B would be just that. Now we know:
$$e^{jwt} = cos(wt) + j sin(wt)$$
So the corresponding frequency domain representation can be obtained as:

 

Therefore, in the frequency domain, e^jwt should have to look like:



Ha finally but not yet !! This aint the transfer function of B. This is just the fourier transform of e^jwt. We wont get y(jw) by multiplying e^jwt's frequency response with gamma . It has to be instead convolved in frequency domain. In fact, we cannot characterise B with a transfer function at all. Because it is NOT an LTI System. It is a time varying system, as system B multiplies gamma(t) by a function of t. So whoever asked this question might have wrongly taken e^jwt's fourier transform as the "transfer function" of B.