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17 Jan 2022, 03:15
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
61% (02:41) correct
39%
(02:20)
wrong
based on 232
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If \(a^2+b^2 = c^2+d^2 = 6\), then which of the following is/are always true?
I. \(ac+bd \leq 6\)
II. \(ab+cd \leq 6\)
III. \(ad+bc \leq 6\)
IV. \(ab+cd \leq 3\)
A. I and II only
B. II and III only
C. III and IV only
D. I, II and III only
E. I, II, III, and IV
Are You Up For the Challenge: 700 Level Questions
I. \(ac+bd \leq 6\)
II. \(ab+cd \leq 6\)
III. \(ad+bc \leq 6\)
IV. \(ab+cd \leq 3\)
A. I and II only
B. II and III only
C. III and IV only
D. I, II and III only
E. I, II, III, and IV
Are You Up For the Challenge: 700 Level Questions
kungfury42
Joined: 07 Jan 2022
Last visit: 31 May 2023
Posts: 580
Given Kudos: 724
Schools: NUS '25 (A)
GMAT 1: 740 Q51 V38
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07 Feb 2022, 20:49
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Hi Bunuel, is the OA correct? The OA says all 4 options are correct. I'm not sure if the 4th one is indeed correct. Here's a case that proves it:
Take a=√5, b=1, c=√5, and d=1
Both a²+b² and c²+d² equal to 6
And ab+cd equals √5*1+√5*1 = 2*√5 = 2*2.236 = 4.472 which is greater than 3 thereby disproving the 4th relation -- ab+cd<=3 is always true
Can you please look into this? Thanks a lot!
Posted from my mobile device
Take a=√5, b=1, c=√5, and d=1
Both a²+b² and c²+d² equal to 6
And ab+cd equals √5*1+√5*1 = 2*√5 = 2*2.236 = 4.472 which is greater than 3 thereby disproving the 4th relation -- ab+cd<=3 is always true
Can you please look into this? Thanks a lot!
Posted from my mobile device
17 Feb 2022, 23:23
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Bunuel
This relation reminds me of pythagorean theorem. We have two right triangles, one will legs a and b and hypotenuse \(\sqrt{6}\)
and other with legs c and d and hypotenuse \(\sqrt{6}\).
Looking at the extreme lengths of the legs, for each triangle, the legs may be anywhere from a little less than \(\sqrt{6}\) and a little more than 0 to both being equal at \(\sqrt{3}\).
I. \(ac+bd \leq 6\)
The maximum value of the product of two sides of different triangles will be \(\sqrt{3}*\sqrt{3}=3\)
The minimum value could be very very small since sqrt(6)*a very small value will give very small value. So maximum value of ac+bd is 6.
True.
II. \(ab+cd \leq 6\)
Again, a, b, c and d all can be \(\sqrt{3}\), so maximum value will be 6. Minimum can be very very small. True.
III. \(ad+bc \leq 6\)
This is no different from statement I. The equations are symmetrical about a and b and c and d so ac + bd is the same as ad + bc. The maximum value will be 6 and minimum will be very very small.
As discussed in statement II, statement IV cannot be true.
