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Joined: 07 Jun 2014
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GRE 1: Q167 V156
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If |z| < 1, which of the following statements must be true
[#permalink]
Updated on: 09 Jan 2020, 10:14
Updated on: 09 Jan 2020, 10:14
7
Expert Reply
10
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Question Stats:
49% (01:06) correct
50% (00:59) wrong based on 571 sessions
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If \(|z| \leq 1\), 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. \(z^3 ≤ z\)
Practice Questions
Question: 15
Page: 64
Question: 15
Page: 64
_________________
Sandy
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Re: If |z| < 1, which of the following statements must be true
[#permalink]
13 Jan 2020, 07:00
13 Jan 2020, 07:00
4
Expert Reply
sandy wrote:
If \(|z| \leq 1\), 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. \(z^3 ≤ z\)
Practice Questions
Question: 15
Page: 64
Question: 15
Page: 64
Now that the original question has been fixed, here's my full solution:
When solving inequalities involving ABSOLUTE VALUE, there are 2 things you need to know:
Rule #1: If |something| < k, then –k < something < k
Rule #2: If |something| > k, then EITHER something > k OR something < -k
Note: these rules assume that k is positive
----------------------
GIVEN: |z| ≤ 1
So, from Rule #1 above, we can write: -1 ≤ z ≤ 1
Now let's check the three statements:
A. \(z^2 ≤ 1\)
If -1 ≤ z ≤ 1, then it MUST be the case that \(z^2 ≤ 1\)
Statement A is true
B. \(z^2 ≤ z\)
If -1 ≤ z ≤ 1, then it COULD be the case that \(z = -1\)
Now plug \(z = -1\) into \(z^2 ≤ z\) to get: \((-1)^2 ≤ -1\)
Simplify to get: \(1 ≤ -1\)
Doesn't work!
Statement B is NOT true
C. \(z^3 ≤ z\)
If -1 ≤ z ≤ 1, then it COULD be the case that \(z = -0.5\)
Now plug \(z = -0.5\) into \(z^2 ≤ z\) to get: \((-0.5)^3 ≤ -0.5\)
Simplify to get: \(-0.125 ≤ -0.5\)
Doesn't work!
Statement C is NOT true
Answer: A
Cheers,
Brent
_________________
Re: If |z| < 1, which of the following statements must be true
[#permalink]
24 Mar 2019, 08:33
24 Mar 2019, 08:33
7
1
Bookmarks
|z| < 1 means -1 < z < 1
so I will pick a number in that range: z = -1/2
Case A.
z^2 < 1
(-1/2)^2 < 1
1/4 < 1 True
Case B.
z^2 < z
(-1/2)^2 < (-1/2)
1/4 < -1/2 False
Case C.
z^3 < z
(-1/2)^3 < (-1/2)
-1/8 < -1/2 True
Now I pick another number in that range, such as z=1/2
Case C.
z^3 < z
(1/2)^3 < (-1/2)
1/8 < -1/2 False
So, A is the answer
_________________
so I will pick a number in that range: z = -1/2
Case A.
z^2 < 1
(-1/2)^2 < 1
1/4 < 1 True
Case B.
z^2 < z
(-1/2)^2 < (-1/2)
1/4 < -1/2 False
Case C.
z^3 < z
(-1/2)^3 < (-1/2)
-1/8 < -1/2 True
Now I pick another number in that range, such as z=1/2
Case C.
z^3 < z
(1/2)^3 < (-1/2)
1/8 < -1/2 False
So, A is the answer
_________________
Diana
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General Discussion
Re: If |z| < 1, which of the following statements must be true
[#permalink]
15 May 2016, 20:40
15 May 2016, 20:40
5
|z| < 1 means z is between -1 and 1
i) z^2 will always be less than or equal to 1 if z is between -1 and 1. Hence true
ii) z^2 <= z. This true if z between 0 and 1, but false if z is between -1 and 0
iii) z^3 <= z This true if z between 0 and 1, but false if z is between -1 and 0
Hence the answer is A
i) z^2 will always be less than or equal to 1 if z is between -1 and 1. Hence true
ii) z^2 <= z. This true if z between 0 and 1, but false if z is between -1 and 0
iii) z^3 <= z This true if z between 0 and 1, but false if z is between -1 and 0
Hence the answer is A
Re: If |z| < 1, which of the following statements must be true
[#permalink]
05 Jun 2019, 12:01
05 Jun 2019, 12:01
3
Hi Diana,
Now I pick another number in that range, such as z=1/2
Case C.
z^3 < z
(1/2)^3 < (-1/2)
1/8 < -1/2 False
So, A is the answer
Should it not be 1/8 < 1/2? Where did the negative sign appear from?
Now I pick another number in that range, such as z=1/2
Case C.
z^3 < z
(1/2)^3 < (-1/2)
1/8 < -1/2 False
So, A is the answer
Should it not be 1/8 < 1/2? Where did the negative sign appear from?
Re: If |z| < 1, which of the following statements must be true
[#permalink]
18 Jun 2019, 06:58
18 Jun 2019, 06:58
1
rapsjade wrote:
|z| < 1 means z is between -1 and 1
i) z^2 will always be less than or equal to 1 if z is between -1 and 1. Hence true
ii) z^2 <= z. This true if z between 0 and 1, but false if z is between -1 and 0
iii) z^3 <= z This true if z between 0 and 1, but false if z is between -1 and 0
Hence the answer is A
i) z^2 will always be less than or equal to 1 if z is between -1 and 1. Hence true
ii) z^2 <= z. This true if z between 0 and 1, but false if z is between -1 and 0
iii) z^3 <= z This true if z between 0 and 1, but false if z is between -1 and 0
Hence the answer is A
I need little clarification:
if z is between -1 and 1 , then how Z^2 is less than or equal to 1.
Z^2 can be less than 1, but how come it becomes equal to 1?
pls explain .
Re: If |z| < 1, which of the following statements must be true
[#permalink]
18 Jun 2019, 11:03
18 Jun 2019, 11:03
Expert Reply
You are right unless the extremes are included.
Re: If |z| < 1, which of the following statements must be true
[#permalink]
09 Jan 2020, 02:23
09 Jan 2020, 02:23
1
sandy wrote:
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. \(z^3 ≤ z\)
Practice Questions
Question: 15
Page: 64
Question: 15
Page: 64
Please can the author check the options
Attachments
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Re: If |z| < 1, which of the following statements must be true
[#permalink]
09 Jan 2020, 10:14
09 Jan 2020, 10:14
Expert Reply
Thanks. There was a typo in the stem. It was not \(<\) but \(\leq\)
Regards
Regards
Manager
Joined: 23 Dec 2019
Posts: 50
Concentration: Finance, Economics
GMAT 1: 690 Q47 V38
GRE 1: Q165 V155
Re: If |z| < 1, which of the following statements must be true
[#permalink]
14 Jan 2020, 20:30
14 Jan 2020, 20:30
Think about z= -1/2 and 1/2.
if z=-1/2, Only A fits
A. (-1/2)^2=1/4 which is less than 1. A fits
B. (-1/2)^2= Positive which is definite greater than a negative value (-1/2). B out
C. (-1/2)^3= -1/8 or -12.50% which Greater than -50% (i.e. -1/2). C out
Even if z=1/2, A fits.
if z=-1/2, Only A fits
A. (-1/2)^2=1/4 which is less than 1. A fits
B. (-1/2)^2= Positive which is definite greater than a negative value (-1/2). B out
C. (-1/2)^3= -1/8 or -12.50% which Greater than -50% (i.e. -1/2). C out
Even if z=1/2, A fits.
Re: If |z| < 1, which of the following statements must be true
[#permalink]
10 May 2020, 14:48
10 May 2020, 14:48
2
Case C.
z^3 < z
(-1/2)^3 < (-1/2)
-1/8 < -1/2 True
No. That's FALSE. - 1/8 is -0.125. - 1/2 is -0.5. -0.125 is GREATER than -0.5. Hence, -1/8 > -1/2.
z^3 < z
(-1/2)^3 < (-1/2)
-1/8 < -1/2 True
No. That's FALSE. - 1/8 is -0.125. - 1/2 is -0.5. -0.125 is GREATER than -0.5. Hence, -1/8 > -1/2.
Re: If |z| < 1, which of the following statements must be true
[#permalink]
30 Nov 2021, 00:38
30 Nov 2021, 00:38
dcmartinezp wrote:
|z| < 1 means -1 < z < 1
so I will pick a number in that range: z = -1/2
Case A.
z^2 < 1
(-1/2)^2 < 1
1/4 < 1 True
Case B.
z^2 < z
(-1/2)^2 < (-1/2)
1/4 < -1/2 False
Case C.
z^3 < z
(-1/2)^3 < (-1/2)
-1/8 < -1/2 True
Now I pick another number in that range, such as z=1/2
Case C.
z^3 < z
(1/2)^3 < (-1/2)
1/8 < -1/2 False
So, A is the answer
so I will pick a number in that range: z = -1/2
Case A.
z^2 < 1
(-1/2)^2 < 1
1/4 < 1 True
Case B.
z^2 < z
(-1/2)^2 < (-1/2)
1/4 < -1/2 False
Case C.
z^3 < z
(-1/2)^3 < (-1/2)
-1/8 < -1/2 True
Now I pick another number in that range, such as z=1/2
Case C.
z^3 < z
(1/2)^3 < (-1/2)
1/8 < -1/2 False
So, A is the answer
Posted from my mobile device
Re: If |z| < 1, which of the following statements must be true
[#permalink]
28 Jun 2022, 10:45
28 Jun 2022, 10:45
1
as Z is between -1 and 1.
try using the number z=0
only option "A" holds true for z=0, others do not, because the condition is of MUST in the question.
try using the number z=0
only option "A" holds true for z=0, others do not, because the condition is of MUST in the question.
gmatclubot
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