2. 外语
  3. If |z| ≤ 1, which of the following statements must be true? Indicate all such statements.

If |z| ≤ 1, which of the following statements must be true? Indicate all such statements.

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问题 If |z| ≤ 1, which of the following statements must be true? Indicate all such statements.
选项 A、z2≤l
B、z2≤z
C、z3≤z
答案A
解析 The condition stated in the question, |z|≤1, includes both positive and negative values of z. For example, both 1/2 andare 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: z2≤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=. If z = 1/2, then z2= 1/4. If z =, then z2=1/4. So in both these cases it is true that z2≤1.
Since the inequality z2≤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 c2≤1. Since 0≤\z\≤1, letting c = \z\ yields \z\2≤1. Also, it is always true that \z\2= z2, and so z2≤1.
Choice B: z2≤z. As before, look at what happens when z =1/2 and when z=. If z =1/2, then z2=1/4. If z =, then z2=1/4. So when z =1/2, the inequality z2≤z is true, and when z =, the inequality z2≤z is false. Therefore you can conclude that if |z|≤1, it is not necessarily true that z2≤z.
Choice C: z3≤z. As before, look at what happens when z =1/2 and when z =. If z =1/2, then z3=1/8. If z =, then z3=. So when z =1/2, the inequality z3≤z is true, and when z =, the inequality z3≤ z is false. Therefore, you can conclude that if |z|≤1, it is not necessarily true that z3≤z.
Thus when \z\≤1, Choice A, z2≤1, must be true, but the other two choices are not necessarily true. The correct answer consists of Choice A.
This explanation uses the following strategies.
Strategy 8: Search for a Mathematical Relationship
Strategy 10: Trial and Error
Strategy 13: Determine Whether a Conclusion Follows from the Information Given
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