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Updated on: 29 May 2023, 07:55
Originally posted by ChandlerBong on 03 Apr 2023, 06:55.
Last edited by ChandlerBong on 29 May 2023, 07:55, edited 1 time in total.
Last edited by ChandlerBong on 29 May 2023, 07:55, edited 1 time in total.
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A
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Difficulty:
95%
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Question Stats:
24% (03:05) correct
76%
(02:54)
wrong
based on 25
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Four angles are formed by the intersection of line X and line Y. Is any of these angles less than 80?
(1) The product of four angles < 2\(^{12}\) * 5\(^{6}\).
(2) The product of any two distinct angles is less than 8100.
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04 Apr 2023, 23:54
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ChandlerBong
Since the angles will be positive, and p + q = 180,
it means the product pq will take the maximum value when p = q = 90. i.e. the maximum value of the product will be 8100.
Also note that as the difference between the angles keeps increasing, the product will keep decreasing. The product of 89 and 91 is less than 8100 but more than the product of 88 and 92 etc.
We want to know whether either of these angles is less than 80 degrees.
(2) The product of any two distinct angles is less than 8100.
Statement 2 tells us something we already mostly know. It just adds the info that the two angles p and q are not 90 each i.e. p and q are distinct (since the product is less than 8100). They could be 89 and 91 or 88 and 92 or 70 and 110 etc and their product will be less than 8100. We don't know then whether any angle is less than 80.
Not sufficient alone.
(1) The product of four angles < 2^12*5^6.
So \(pq < 2^6 * 5^3\)
2^6 * 5^3 = 8000
A product of 8000 is obtained when the angles are 80 and 100 (as close as possible such that the sum is 180)
Since the product needs to be less than 8000, one angle will be less than 80 and the other will be greater than 100 e.g. 79 and 101.
Sufficient alone.
Answer (A)
