Must the SD of a 'bell shaped' distribution be less than the SD of a skewed distribution with the same range? 
10) Which of the following distributions will have the smallest
  standard deviation, assuming that none contain outliers?
  a) A uniform distribution of integers with a mean of 5 and a range of 10.
  b) A bell-shaped distribution of integers with a mean of 5 and a range of 10.
  c) A right-skewed distribution of integers with a mean of 4 and a range of 10.
  d) All three distributions have equal standard deviation.
  e) The answer cannot be determined from the information given.

The answer is given as (b) but I really can't figure out a simple reason why.
 A: While you should be able to argue that the sd of (b) should be less than that of (a), you can't coherently argue, based on the given information, that it will be less than that of (c); it may be, but it also may not.
Here's an example which should suggest to you that the answer should not be the general statement implied by choosing (b). Here are histograms of samples, one skewed, and one more bell-shaped:

They have the right mean and range (answers calculated in R):
> mean(xb);mean(xc)
[1] 5
[1] 4
> diff(range(xb));diff(range(xc))
[1] 10
[1] 10

Yet the standard deviation of the skewed one is smaller:
> sd(xb);sd(xc)
[1] 2.100015
[1] 1.853432

Here I generate a bell-shaped distribution that looks much more normal (rather than just generically bell-shaped) than that red one above, still with the same outcome. Here the light grey bars are the normal-looking sample while the red-bordered bars are the skew sample:

> max(xb2)-min(xb2);mean(xb2);sd(xb2)  # the normal-looking data
[1] 10
[1] 5
[1] 1.51216
> max(xc2)-min(xc2);mean(xc2);sd(xc2)  # the skewed data
[1] 10
[1] 4
[1] 1.499837

(Indeed, the standard deviation of a skewed sample could be made much smaller than that of a bell-shaped sample, but it suffices to produce one where it's smaller at all)
Such examples eliminate (b) and (d) as answers. Similarly, it's easy to construct examples where (c) is contradicted. (Similar examples may be also be constructed with continuous distributions rather than samples; indeed my examples began by drawing samples from distributions with the desired properties and then tweaking the resulting samples a little to produce sample results with those properties.)
Since we can produce examples that contradict answers (a) to (d), we're left with (e) as the only plausible answer.
A: Remember that standard deviation is all about average distance from the mean. That the distribution is bell shaped means that moving away from the mean in either direction decreases the likelihood of new observations. With a strong right skew, you could move away from the mean into a higher likelihood region, which would lead to higher average deviations from the mean in typical situations.
But I think you could argue (e), especially if you are able to find a concrete counterexample.
