Are the following statements true or false?

? True False  1. If $S$ is the unit sphere centered at the origin, oriented outward and the flux integral $\displaystyle \iint_S \vec{F} \cdot d\vec{A}$ is zero, then $\vec{F}(x,y,z)$ is perpendicular to $\vec{r} = \langle x,y,z \rangle$ at every point of $S$.

? True False  2. If $S_1$ is a rectangle with area 1 and $S_2$ is a rectangle with area 2, then $\displaystyle 2 \iint_{S_1} \vec{F} \cdot d\vec{A} = \iint_{S_2} \vec{F} \cdot d\vec{A}$

? True False  3. If $S$ is an open-ended circular cylinder centered about the z-axis, oriented away from the z-axis, and $\vec{F} = \langle x,y,0 \rangle$, then the flux of $\vec{F}$ through $S$ is positive.

? True False  4. If $S$ is an open-ended circular cylinder centered about the z-axis, oriented away from the z-axis, and $\vec{F} = \langle 3,-2,6 \rangle$, then the flux of $\vec{F}$ through $S$ is zero.

? True False  5. If $S$ is the unit sphere centered at the origin, oriented outward and the flux integral $\displaystyle \iint_S \vec{F} \cdot d\vec{A}$ is zero, then $\vec{F} = \vec{0}$.

? True False  6. The area vector of a flat, oriented surface is parallel to the surface.

? True False  7. If $S$ is the cube bounded by the six planes $x = \pm 3$, $y = \pm 3$, $z = \pm 3$, oriented outward, and $\vec{F} = 2 \vec{i} - \vec{k}$, then the flux of $\vec{F}$ through $S$ is zero.

? True False  8. If $S$ is the unit sphere centered at the origin, oriented outward and $\vec{F} = x \vec{i} + y \vec{j} + z\vec{k} = \vec{r}$, then the flux integral $\displaystyle \iint_S \vec{F} \cdot d\vec{A}$ is positive.