If x > 0 and the range of 1, 2, x, 5, and x^2 equals 7, what is the ap

This is a good question which tests you simultaneously on your knowledge of the statistical measures, range and mean.

The range of a set of values is the difference between the maximum and minimum values.

The set of values given to us is 1, 2, x, 5 and \(x^2\). Depending on the values of x and \(x^2\), the range could have been any value. However, since the question says that the range of the above set of values is 7, this will define the range for x and \(x^2\)
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If x is less than 1, \(x^2\) will also be less than 1 i.e. both x and \(x^2\) will be positive proper fractions (since 0<x<1, and in this range, only positive proper fractions can be found). Also, \(x^2\) < x.
So, the smallest value is \(x^2\) and the biggest value is 5. But, 5 – proper fraction (\(x^2\)) does not give us a range of 7. We can therefore conclude that x is not less than 1.

If x is between 1 and 2, \(x^2\) is between 1 and 4. Now, x is the smallest value and 5 is the largest value. The range is still not 7. So, x is not between 1 and 2.

If x is greater than 5, \(x^2\) is greater than 25. Then, the range will be more than 7, since the biggest value is more than 25, whereas the smallest value is 1.

Therefore, we can safely conclude that x has to lie between 2 and 5 and hence, \(x^2\) will lie between 4 and 25. In this case, \(x^2\) will be the biggest value and 1 will be the smallest value.

Since the range is 7, \(x^2\) – 1 = 7
i.e. \(x^2\) = 8.
So, x = 2\(\sqrt{2}\). We can approximate the value of x to be 2.8 since \(\sqrt{2}\) = 1.414.
Now, Mean = \(\frac{{1+2+2√2+5+8}}{5}\) = \(\frac{18.8}{5}\) = 3.76.
The correct answer is option D.

Hope this helps!

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