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09 Nov 2021, 07:25
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56% (01:27) correct
44%
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Alice is measuring the temperature at noon for four consecutive days for observation. She records the temperatures each day. What is the mode of the recorded temperatures?
(1) The product of no two recorded temperatures is positive.
(2) The range of the recorded temperatures is 0.
(1) The product of no two recorded temperatures is positive.
(2) The range of the recorded temperatures is 0.
09 Nov 2021, 07:25
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Official Solution:
Alice is measuring the temperature at noon for four consecutive days for observation. She records the temperatures each day. What is the mode of the recorded temperatures?
Assume the recorded temperatures were \(\{a, \ b, \ c, \ d \}\).
(1) The product of no two recorded temperatures is positive.
The above means that the product of temperatures in all two-temperature groups possible from the list is 0 or negative: \(ab \leq 0\), \(ac \leq 0\), \(ad \leq 0\), \(bc \leq 0\), ...
Can the list contain two or more positive temperatures? No. Because in this case we can have a group of two positive temperatures which will give the positive product.
Can the list contain two or more negative temperatures? No. Because in this case we can have a group of two negative temperatures which will give the positive product.
So, the list should contain at most one positive temperature, at most one negative temperature and the remaining temperatures must be 0. We can have the following four cases:
\(\{0, \ 0, \ 0, \ 0 \}\)
\(\{negative, \ 0, \ 0, \ 0 \}\)
\(\{0, \ 0, \ 0, \ positive \}\)
\(\{negative, \ 0, \ 0, \ positive \}\)
The mode of each of the possible lists above is 0. Sufficient.
(2) The range of the recorded temperatures is 0.
The range of 0 means that the list contains four equal temperatures, but we don't know what temperature this is. Not sufficient.
Answer: A
Alice is measuring the temperature at noon for four consecutive days for observation. She records the temperatures each day. What is the mode of the recorded temperatures?
Assume the recorded temperatures were \(\{a, \ b, \ c, \ d \}\).
(1) The product of no two recorded temperatures is positive.
The above means that the product of temperatures in all two-temperature groups possible from the list is 0 or negative: \(ab \leq 0\), \(ac \leq 0\), \(ad \leq 0\), \(bc \leq 0\), ...
Can the list contain two or more positive temperatures? No. Because in this case we can have a group of two positive temperatures which will give the positive product.
Can the list contain two or more negative temperatures? No. Because in this case we can have a group of two negative temperatures which will give the positive product.
So, the list should contain at most one positive temperature, at most one negative temperature and the remaining temperatures must be 0. We can have the following four cases:
\(\{0, \ 0, \ 0, \ 0 \}\)
\(\{negative, \ 0, \ 0, \ 0 \}\)
\(\{0, \ 0, \ 0, \ positive \}\)
\(\{negative, \ 0, \ 0, \ positive \}\)
The mode of each of the possible lists above is 0. Sufficient.
(2) The range of the recorded temperatures is 0.
The range of 0 means that the list contains four equal temperatures, but we don't know what temperature this is. Not sufficient.
Answer: A
18 Jan 2024, 00:52
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Bunuel
Is it not the case that if a set contains all the same elements there exists no mode?
https://www150.statcan.gc.ca/n1/edu/pow ... 73-eng.htm
all the best and thanks!
