1. Given data

The numbers are: $-1,\;0,\;1,\;k$ with standard deviation $\sigma = \sqrt{5}$ and $k>0$.

2. Use the formula for standard deviation

For $n$ observations $x_1,x_2,\dots,x_n$, $\sigma^2 = \frac{1}{n}\sum x_i^2 - \left(\frac{1}{n}\sum x_i\right)^2$

Here, $n=4$.

3. Compute the mean

Sum of the numbers: $-1+0+1+k = k$ So the mean is: $\bar{x} = \frac{k}{4}$

4. Compute the mean of squares

$(-1)^2 + 0^2 + 1^2 + k^2 = 1+0+1+k^2 = 2+k^2$ Hence, $\frac{1}{4}\sum x_i^2 = \frac{2+k^2}{4}$

5. Apply the variance formula

Since $\sigma=\sqrt{5}$, $\sigma^2=5$ Therefore, $\frac{2+k^2}{4} - \left(\frac{k}{4}\right)^2 = 5$

Now simplify: $\frac{2+k^2}{4} - \frac{k^2}{16} = 5$ Take LCM $16$: $\frac{4(2+k^2)-k^2}{16} = 5$ $\frac{8+4k^2-k^2}{16} = 5$ $\frac{8+3k^2}{16} = 5$

6. Solve for $k$

$8+3k^2 = 80$ $3k^2 = 72$ $k^2 = 24$ $k = \sqrt{24} = 2\sqrt{6}$ Since $k>0$, we take the positive root.

7. Check with options

$k = 2\sqrt{6}$ So the correct option is: A.

8. Comparison with stored answer

Stored correct answer: A

This matches our derived answer.