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28 Aug 2024, 11:07
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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Question Stats:
60% (01:36) correct
40% (00:55) wrong based on 5 sessions
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If |z| ≤ 1, which of the following statements must be true?
Indicate all such statements.
A. \(z^2 ≤ 1\)
B. \(z^2 ≤ z\)
C. \(z^3 ≤ z\)
Indicate all such statements.
A. \(z^2 ≤ 1\)
B. \(z^2 ≤ z\)
C. \(z^3 ≤ z\)
13 Nov 2024, 11:40
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OE
The condition stated in the question, |z| ≤ 1, includes both positive and negative values of z. For example, both \(\frac{1}{2}\) and \(\frac{-1}{2}\) are possible values of z. Keep this in mind as you evaluate each of the inequalities in the answer choices to see whether the inequality must be true.
Choice A: \(z^2 ≤ 1\). First look at what happens for a positive and a negative value of z for which |z| ≤ 1, say, z = 1/2 and z = -1/2. If z = 1/2, then \(z^2 = \frac{1}{4} \). If z = - 1/2 , then \(z^2 = \frac{1}{4}\). So in both these cases it is true that \(z^2 ≤ 1.\)
Since the inequality \(z^2 ≤ 1\) is true for a positive and a negative value of z, try to prove that it is true for all values of z such that \(|z| ≤ 1\). Recall that if \(0 ≤ c ≤ 1\), then \(c^2 ≤ 1\). Since \(0 ≤ |z| ≤ 1\), letting \(c = |z|\) yields \(|z|^2 ≤ 1\). Also, it is always true that \(|z|^2 = z^2\), and so \(z^2 ≤ 1.\)
Choice B: \(z^2 ≤ z\). As before, look at what happens when \(z = \frac{1}{2}\) and when \(z = \frac{-1}{2}\) . If z = 1/2 , then \(z^2 = \frac{1}{4}\) . If z = -1/2 , then z^2 = 1/4 . So when z = 1/2 , the inequality \(z^2 ≤ z\) is true, and when \(z = \frac{-1}{2}\) , the inequality \(z^2 ≤ z\) is false. Terefore, you can conclude that if \(|z| ≤ 1\), it is not necessarily true that \(z^2 ≤ z\).
Choice C: \(z^3 ≤ z\). As before, look at what happens when \(z = \frac{1}{2}\) and when \(z = \frac{-1}{2}\). If \(z = \frac{1}{2}\), then \(z^3 = \frac{1}{8}\). If \(z = \frac{-1}{2}\), then \(z^3 = \frac{-1}{8}\). So when \(z = \frac{1}{2}\), the inequality \(z^3 ≤ z\) is true, and when \(z = \frac{-1}{2}\), the inequality \(z^3 ≤ z\) is false. Therefore, you can conclude that if \(|z| ≤ 1\), it is not necessarily true that \(z^3 ≤ z.\)
Thus when \(|z| ≤ 1\), Choice A, \(z^2 ≤ 1\), must be true, but the other two choices are not necessarily true. Te correct answer consists of Choice A.
The condition stated in the question, |z| ≤ 1, includes both positive and negative values of z. For example, both \(\frac{1}{2}\) and \(\frac{-1}{2}\) are possible values of z. Keep this in mind as you evaluate each of the inequalities in the answer choices to see whether the inequality must be true.
Choice A: \(z^2 ≤ 1\). First look at what happens for a positive and a negative value of z for which |z| ≤ 1, say, z = 1/2 and z = -1/2. If z = 1/2, then \(z^2 = \frac{1}{4} \). If z = - 1/2 , then \(z^2 = \frac{1}{4}\). So in both these cases it is true that \(z^2 ≤ 1.\)
Since the inequality \(z^2 ≤ 1\) is true for a positive and a negative value of z, try to prove that it is true for all values of z such that \(|z| ≤ 1\). Recall that if \(0 ≤ c ≤ 1\), then \(c^2 ≤ 1\). Since \(0 ≤ |z| ≤ 1\), letting \(c = |z|\) yields \(|z|^2 ≤ 1\). Also, it is always true that \(|z|^2 = z^2\), and so \(z^2 ≤ 1.\)
Choice B: \(z^2 ≤ z\). As before, look at what happens when \(z = \frac{1}{2}\) and when \(z = \frac{-1}{2}\) . If z = 1/2 , then \(z^2 = \frac{1}{4}\) . If z = -1/2 , then z^2 = 1/4 . So when z = 1/2 , the inequality \(z^2 ≤ z\) is true, and when \(z = \frac{-1}{2}\) , the inequality \(z^2 ≤ z\) is false. Terefore, you can conclude that if \(|z| ≤ 1\), it is not necessarily true that \(z^2 ≤ z\).
Choice C: \(z^3 ≤ z\). As before, look at what happens when \(z = \frac{1}{2}\) and when \(z = \frac{-1}{2}\). If \(z = \frac{1}{2}\), then \(z^3 = \frac{1}{8}\). If \(z = \frac{-1}{2}\), then \(z^3 = \frac{-1}{8}\). So when \(z = \frac{1}{2}\), the inequality \(z^3 ≤ z\) is true, and when \(z = \frac{-1}{2}\), the inequality \(z^3 ≤ z\) is false. Therefore, you can conclude that if \(|z| ≤ 1\), it is not necessarily true that \(z^3 ≤ z.\)
Thus when \(|z| ≤ 1\), Choice A, \(z^2 ≤ 1\), must be true, but the other two choices are not necessarily true. Te correct answer consists of Choice A.
