Answer
1. the function is symmetric with respect to the polar axis.
2. the function is not symmetric with respect to the pole.
3. the function is not symmetric with respect to the line $\theta=\frac{\pi}{2}$.
Work Step by Step
1. Test the polar equation for symmetry with respect to the polar axis : replace $\theta$ with $-\theta$, we have
$r=\frac{5}{1+3cos(-\theta)}=\frac{5}{1+3cos(\theta)}$ which is the same as the original, so the function is symmetric with respect to the polar axis.
2. Test the polar equation for symmetry with respect to the pole : replace $\theta$ with $\pi+\theta$, we have
$r=\frac{5}{1+3cos(\pi+\theta)}=\frac{5}{1-3cos(\theta)}=$ which is different from the original, so the function is not symmetric with respect to the pole.
3. Test the polar equation for symmetry with respect to the line $\theta=\frac{\pi}{2}$: replace $\theta$ with $\pi-\theta$, we have $r=\frac{5}{1+3cos(\pi-\theta)}=\frac{5}{1-3cos(\theta)}$ which is different from the original, so the function is not symmetric with respect to the line $\theta=\frac{\pi}{2}$.