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09 Jun 2024, 19:39
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
70% (01:33) correct
30%
(01:36)
wrong
based on 663
sessions
History
Date
Time
Result
Not Attempted Yet
If m < p < r < t < z and if the product mprtz is positive, which of the following products must be positive?
I. \(mp\)
II. \(rt\)
III. \(tz\)
A. None
B. I only
C. I and II only
D. I and III only
E. I, II, and III
I. \(mp\)
II. \(rt\)
III. \(tz\)
A. None
B. I only
C. I and II only
D. I and III only
E. I, II, and III
02 Sep 2024, 13:30
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Victor314
If m < p < r < t < z and if the product mprtz is positive, which of the following products must be positive?
I. \(mp\)
II. \(rt\)
III. \(tz\)
A. None
B. I only
C. I and II only
D. I and III only
E. I, II, and III
Given that the product of five numbers, \(m\), \(p\), \(r\), \(t\), and \(z\), is positive, there must be an even number of negative numbers among them (0, 2, or 4 negative numbers). Since it is also given that \(m < p < r < t < z\), we can have the following three cases:I. \(mp\)
II. \(rt\)
III. \(tz\)
A. None
B. I only
C. I and II only
D. I and III only
E. I, II, and III
\(m\; |\; p\; |\; r\; |\; t\; |\; z\)
\(+ | + | + | + | +\) (no negative numbers)
\(- | - | + | + | +\) (two negative numbers)
\(- | - | - | - | +\) (four negative numbers)
I. \(mp>0\)
This option is true for each of the three cases. Therefore, this option is always true.
II. \(rt>0\)
This option is true for each of the three cases. Therefore, this option is always true.
III. \(tz>0\)
If we have the third case, then this option is not true. Eliminate.
Consequently, only options I and II are always true, and thus the answer is C.
Answer: C.
General Discussion
05 Jul 2024, 05:19
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for \(mprtz\) to be positive there must be an even number of negative numbers
The following are the possibilities for \(m < p < r < t < z\)
1.
\(-m*-p*r*t*z\)
\(mp, rt\) & \(tz\) are positive
2.
\(-m*-p*-r*-t*z\)
\(mp, rt\) are positive, but \(tz\) is negative.
So \(mp, rt\) must be positive
Answer is C
The following are the possibilities for \(m < p < r < t < z\)
1.
\(-m*-p*r*t*z\)
\(mp, rt\) & \(tz\) are positive
2.
\(-m*-p*-r*-t*z\)
\(mp, rt\) are positive, but \(tz\) is negative.
So \(mp, rt\) must be positive
Answer is C
