Find $X(j\omega)$ after sampling of $2\cos(2000\pi t)+\sin(5000\pi t)$ at 5 kHz sampling rate

Question

The Fourier transform of the first term has two spikes at -2000pi and 2000pi of magnitudes 2pi for both.

The Fourier transform of the second term has two spikes at 5000pi and -5000pi having magnitudes $-j\pi$ and $j\pi$ respectively.

Since sampling is at 5000Hz, the -2000pi and 2000pi spikes repeat at 3000pi and -3000pi respectively, while the -5000pi and 5000pi spikes repeat at 0.

Now we've got the spectrum in the big-Omega domain. We divide this by 5000Hz to get the spectrum in -pi to pi domain

So the answer I've derived has seven spikes, at -pi, pi, -2/5pi, 2/5pi, -3/5pi, 3/5pi, and 0. However, none of the options have seven spikes. What am I doing wrong?

EDIT - I forgot to scale by $\frac{1}{T}$. But my problem still remains. Only option 1 and 3 have spikes at -pi, pi, -0.4pi and 0.4pi. But neither of these options have spikes at 3/5pi, -3/5pi, and zero.

Answer 1

Look at your signal and figure out what sampling rate you need to avoid aliasing. In this case, you are sampling at twice the maximum frequency component so you should not expect aliasing, but you work leads to aliasing components showing up at plus/minus $\frac{3\pi}{5}$.

Remember that the DTFT is periodic with period $2\pi$, and try re-working the problem.