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
Quant Expert
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