Official Solution

Given:

• The function [z]

• \([z^2] = 2\)

To find: Can [z] be {2, 1, -2}?

Approach:

• To find which values of [z] are possible and which are not, we need to first know the range of possible values of z. Once we know what z can be, we’ll be able to find what [z] can be.

We’ll get an idea of the possible values of z from the fact that \([z^2] = 2\)

Working Out:

• \([z^2] = 2\)

• This means the least integer that is greater than or equal to \(z^2\) is 2.

• We can write: \(1 < z^2 ≤ 2\)

• The above inequality contains 2 inequalities: \(z^2 > 1\) AND \(z^2 ≤ 2\)

So now, we’ll solve these inequalities one by one, and then find the values of z that satisfy both these inequalities

• Solving \(z^2 > 1\)

• \(z^2 – 1 > 0\)

• \((z+1)(z-1) > 0\)

This means, \(z < - 1\) or \(z > 1\) . . . (1)

• Solving \(z^2 ≤ 2\)

• \(z^2 – 2 ≤ 0\)

• \((z + √2)(z-√2) ≤ 0\)

That is, \(-√2 ≤ z ≤ √2\) . . . (2)

Combining (1) and (2):

Combining them to find their overlap zones and finding those values of z that satisfy both the inequalities

• So, either \(-√2 ≤ z < -1\) or \(1 < z ≤ √2\)

• Either \(-√2 ≤ z < -1\)

• \(-1.4 ≤ z < -1\)

• In this case, \([z] = -1\)

• Or \(1 < z ≤ √2\)

• \(1 < z ≤ 1.4 \)

• \([z] = 2\)

Thus, we see that [z] is either equal to -1 or equal to 2. Out of the 3 given values, only value I (which is 2) is therefore possible.

Looking at the answer choices, we see that the correct answer is Option A

Thanks,

Saquib

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