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13 Oct 2014, 05:56
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What is the value of \(xy\)?
(1) \(x^2y^2+2xy\pi-3\pi^2 = 0\)
(2) \(xy>-9.5\)
(1) \(x^2y^2+2xy\pi-3\pi^2 = 0\)
(2) \(xy>-9.5\)
13 Oct 2014, 07:47
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Official Solution:
What is the value of \(xy\)?
(1) \(x^2y^2+2xy\pi-3\pi^2 = 0\).
This can be expressed as: \((xy)^2 + 2xy\pi - 3\pi^2 = 0\).
To find the value of \(xy\), we can either directly solve the quadratic for \(xy\) (to simplify, we can denote \(xy\) as \(a\) to get \(a^2 + 2a\pi - 3\pi^2 = 0\) and solve for \(a\)), or factor \((xy)^2 + 2xy\pi - 3\pi^2 = 0\):
\((xy)^2 + 3xy\pi - xy\pi - 3\pi^2 = 0\)
\((xy)(xy + 3\pi) - \pi(xy + 3\pi) = 0\)
\((xy+3\pi)(xy-\pi)=0\)
This gives us two possible solutions: \(xy=-3\pi\) or \(xy=\pi\). This statement is not sufficient to determine a unique solution.
(2) \(xy>-9.5\).
Clearly insufficient.
(1)+(2) From (1), we have two possible values: \(xy = -3\pi \approx -9.45\) and \(xy = \pi \approx 3.14\). Since both values are greater than -9.5, they both satisfy the condition in statement (2). Therefore, even with both statements, we cannot uniquely determine the value of \(xy\). Not sufficient.
Answer: E
What is the value of \(xy\)?
(1) \(x^2y^2+2xy\pi-3\pi^2 = 0\).
This can be expressed as: \((xy)^2 + 2xy\pi - 3\pi^2 = 0\).
To find the value of \(xy\), we can either directly solve the quadratic for \(xy\) (to simplify, we can denote \(xy\) as \(a\) to get \(a^2 + 2a\pi - 3\pi^2 = 0\) and solve for \(a\)), or factor \((xy)^2 + 2xy\pi - 3\pi^2 = 0\):
\((xy)^2 + 3xy\pi - xy\pi - 3\pi^2 = 0\)
\((xy)(xy + 3\pi) - \pi(xy + 3\pi) = 0\)
\((xy+3\pi)(xy-\pi)=0\)
This gives us two possible solutions: \(xy=-3\pi\) or \(xy=\pi\). This statement is not sufficient to determine a unique solution.
(2) \(xy>-9.5\).
Clearly insufficient.
(1)+(2) From (1), we have two possible values: \(xy = -3\pi \approx -9.45\) and \(xy = \pi \approx 3.14\). Since both values are greater than -9.5, they both satisfy the condition in statement (2). Therefore, even with both statements, we cannot uniquely determine the value of \(xy\). Not sufficient.
Answer: E
23 Aug 2017, 07:58
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I think this is a high-quality question and I agree with explanation.
