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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.
admin
2017-10-16
42
问题
If |z| ≤ 1, which of the following statements must be true? Indicate all such statements.
选项
A、z
2
≤l
B、z
2
≤z
C、z
3
≤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 and
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=
. If z = 1/2, then z
2
= 1/4. If z =
, then z
2
=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 =1/2 and when z=
. If z =1/2, then z
2
=1/4. If z =
, then z
2
=1/4. So when z =1/2, the inequality z
2
≤z is true, and when z =
, the inequality z
2
≤z is false. Therefore 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 =1/2 and when z =
. If z =1/2, then z
3
=1/8. If z =
, then z
3
=
. So when z =1/2, the inequality z
3
≤z is true, and when z =
, 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. 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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