For the first set, the mean is $\frac{2+4+6+8+10}{5}=\frac{30}{5}=6$. For the second set, the mean is $\frac{2+4+6+8+12}{5}=\frac{32}{5}=6.4$, so the mean increases. Now compare the medians. In each ordered set, the middle value is the third number: $6$. So the median stays the same. Therefore, the statement that must be true is that the mean increases, but the median stays the same.

The plotted points follow the linear pattern $y=2x+2$. To compare the standard deviations of the $x$-values and $y$-values, use the fact that adding a constant does not change standard deviation, while multiplying every value by a constant multiplies the standard deviation by the absolute value of that constant.

The $x$-values are $1,2,3,4,5$.
The corresponding $y$-values are $4,6,8,10,12$, which can be written as $y=2x+2$.

Starting from the $x$-values:
- multiplying by $2$ gives $2,4,6,8,10$, so the standard deviation becomes $2S_x$
- adding $2$ gives $4,6,8,10,12$, so the standard deviation stays the same

Therefore, $S_y=2S_x$, and

Let the data set be $4,7,7,10,x$. Since the range is 8, the difference between the greatest and least values must be 8.

Start by checking where $x$ could be.
- If $x>10$, then the minimum is 4 and the maximum is $x$, so the range is $x-4=8$. This gives $x=12$.
- If $x<4$, then the maximum is 10 and the minimum is $x$, so the range is $10-x=8$. This gives $x=2$.

So the range condition leaves only two possible values: $x=12$ or $x=2$.

Now use the condition that the mean is greater than the median.

If $x=12$, the ordered data are $4,7,7,10,12$. The median is the middle value, which is 7. The mean is

Since $8>7$, this works.

If $x=2$, the ordered data are $2,4,7,7,10$. The median is still 7. The mean is

Since $6>7$ is false, this does not work.

Therefore, the only possible value is $x=12$, so the statement that must be true is $x=12$.

Doubling the width of each rectangular panel doubles its area, because area = width $\times$ height. So the original areas 18, 20, 20, 24, and 28 become 36, 40, 40, 48, and 56. Every data value in the set has been multiplied by 2. When every value in a data set is multiplied by the same constant, the standard deviation is also multiplied by the absolute value of that constant. Therefore, the new standard deviation is $2s$.

The quiz scores are the $y$-values of the plotted points: $70,75,75,80,90$. These are already in order. Since there are 5 scores, the median is the middle value, which is the 3rd score. The 3rd score is $75$, so the correct answer is $75$.