NEW HERE? LEARN MORE
closeclose
Last visit was: 26 Apr 2024, 06:27 It is currently 26 Apr 2024, 06:27
Decision Tracker
My Rewards
New posts
New comers' posts

Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

If |x^2 + y^2 + z^2| < 4, is |x^2 + z^2| < 1 ?

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 92933
Own Kudos [?]: 619168 [9]
Given Kudos: 81609
Send PM
CAT Tests Forum Quiz Mode
If |x^2 + y^2 + z^2| < 4, is |x^2 + z^2| < 1 ? [#permalink] New post   14 Aug 2017, 01:59
9
Bookmarks
Expert Reply
00:00
A
B
C
D
E

Difficulty:

75% (hard)

Question Stats:

50% (02:10) correct 50% (02:12) wrong based on 150 sessions
Hide Show timer Statistics
If \(|x^2 + y^2 + z^2| < 4\), is \(|x^2 + z^2| < 1\) ?


(1) \(|x^2 + y^2| < 1\)

(2) \(|y^2 + z^2 | < 1\)
ShowHide Answer
Official Answer
E
User avatar
L
broall
Retired Moderator
Joined: 10 Oct 2016
Status:Long way to go!
Posts: 1144
Own Kudos [?]: 6122 [0]
Given Kudos: 65
Location: Viet Nam
Send PM
GMAT ToolKit User
Re: If |x^2 + y^2 + z^2| < 4, is |x^2 + z^2| < 1 ? [#permalink] New post   14 Aug 2017, 02:09
Bunuel wrote:
If \(|x^2 + y^2 + z^2| < 4\), is \(|x^2 + z^2| < 1\) ?


(1) \(|x^2 + y^2| < 1\)

(2) \(|y^2 + z^2 | < 1\)


\(|x^2 + y^2 + z^2| < 4 \iff x^2 + y^2 + z^2 < 4\)
\(|x^2 + z^2| < 1 \iff x^2+z^2<1\)

(1) \(x^2+y^2 < 1 \implies x^2 < 1\)

If \(z^2 = 3\) then \(x^2+z^2>1\).
If \(z^2 = 0\) then \(x^2+z^2<1\).

Insufficient.

(2) \(y^2+z^2 < 1 \implies z^2 < 1\)

If \(x^2 = 3\) then \(x^2+z^2>1\).
If \(x^2 = 0\) then \(x^2+z^2<1\).

Insufficient.

Combine (1) and (2)
\(x^2+y^2+y^2+z^2<2 \implies x^2+2y^2+z^2<2 \implies x^2+z^2<2\)

We could have \(x^2+z^2=1\) or \(x^2+z^2<1\) or \(x^2+z^2>1\). Insufficient.

The answer is E.
User avatar
L
Mo2men
RSM Erasmus Moderator
Joined: 26 Mar 2013
Posts: 2461
Own Kudos [?]: 1360 [0]
Given Kudos: 641
Concentration: Operations, Strategy
Schools: Erasmus (II)
Send PM
Reviews Badge
If |x^2 + y^2 + z^2| < 4, is |x^2 + z^2| < 1 ? [#permalink] New post   14 Aug 2017, 23:40
If \(|x^2 + y^2 + z^2| < 4\), is \(|x^2 + z^2| < 1\) ?


(1) \(|x^2 + y^2| < 1\)

Let x^2 = y^2 =0 & z^2 =1/2........Answer to question is Yes

Let x^2 = 1/2 & y^2 =0 & z^2 =1/2........Answer to question is No

Insufficient

(2) \(|y^2 + z^2 | < 1\)

Let x^2 = y^2 =0 & z^2 =1/2........Answer to question is Yes

Let x^2 = 1/2 & y^2 =0 & z^2 =1/2........Answer to question is No

Insufficient

Combining 1 & 2

Same examples above for Statementa 1 & 2 prove Insufficient

Answer: E
avatar
B
Dsgshanky
Intern
Intern
Joined: 17 Apr 2017
Posts: 6
Own Kudos [?]: 0 [0]
Given Kudos: 20
Send PM
GMAT ToolKit User
If |x^2 + y^2 + z^2| < 4, is |x^2 + z^2| < 1 ? [#permalink] New post   15 Aug 2017, 05:19
1. x^2 +y^2 <1, this implies x^2 + y^2 lies in [0,1), and x^2 lies in [0,1)
hence, z^2 < 4 - (x^2 +y^2), implies z^2 lies between [0,4)
also, since x^2 is always positive so x^2+z^2 lies between [0+x^2, 4+x^2) thus min is 0, and max is 5, insufficient.

2. y^2 +z^2 <1, this implies, y^2 +z^2 lies in [0,1) also, z^2 lies in [0,1)
given x^2 < 4- (y^2 +z^2), implies x^2 lies in [0,4)

for the same reason as 1, insufficient.

answer E.
User avatar
D
stonecold
Alum
Joined: 12 Aug 2015
Posts: 2282
Own Kudos [?]: 3131 [3]
Given Kudos: 893
Schools: Boston U '20 (M)
GRE 1: Q169 V154
Send PM
GMAT ToolKit User
Re: If |x^2 + y^2 + z^2| < 4, is |x^2 + z^2| < 1 ? [#permalink] New post   15 Aug 2017, 07:19
1
Kudos
2
Bookmarks
Waoooo, Some really really byzantine solutions above.

Let me try to put this in a laconic way => in such questions -> ALWAYS (and I mean ALWAYS) try and use test cases or else the question can be pretty painful.

Statement 1 =>
Test case 1 => 0,0,0 => |x^2+z^2|<1 is true
Test case 2 => 0.9,0,0.9 => |x^2+z^2|<1 is false

Statment 2 =>
Test case 1 => 0,0,0 => |x^2+z^2|<1 is true
Test case 2 => 0.9,0,0.9 => |x^2+z^2|<1 is false



Combining them =>

Test case 1 => 0,0,0 => |x^2+z^2|<1 is true
Test case 2 => 0.9,0,0.9 => |x^2+z^2|<1 is false


Hence E.

Notice how the test case is the same which makes the job 10x times easier :)
User avatar
G
Madhavi1990
Senior Manager
Senior Manager
Joined: 15 Jan 2017
Posts: 259
Own Kudos [?]: 85 [0]
Given Kudos: 932
Send PM
Re: If |x^2 + y^2 + z^2| < 4, is |x^2 + z^2| < 1 ? [#permalink] New post   19 Sep 2017, 00:08
St 1: |x sq + y sq| <1
|+ve| < 1 --> either it adds to 0; or tend to 0.9 but less than 1

--> but we don't know exact value of x or y

St 2: |y sq + z sq| < 1
|+ve| < 1 --> either it adds to 0; or tend to 0.9 but less than 1
--> but we don't know exact value of x or y

1) + 2) if we take max value |0.49+0.64+0.81|< 4
--> we still don't know the exact values of x,y,z square

so Ans E
User avatar
bumpbot
Non-Human User
Joined: 09 Sep 2013
Posts: 32685
Own Kudos [?]: 822 [0]
Given Kudos: 0
Send PM
Re: If |x^2 + y^2 + z^2| < 4, is |x^2 + z^2| < 1 ? [#permalink] New post   07 Feb 2022, 19:26
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
GMAT Club Bot
gmatclubot
Re: If |x^2 + y^2 + z^2| < 4, is |x^2 + z^2| < 1 ? [#permalink]
07 Feb 2022, 19:26
Moderator:
Bunuel
Math Expert
92933 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions