Bunuel wrote:
Given that \(x^3*y > 0\) and that \(x^2*y^3 < 0\) which of the following statements must be true?
I. \(x < 0\)
II. \(x < y < 0\)
III. \(y^3 < x^2\)
A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
(i) \(x^3*y > 0\)
Either both \(x\) and \(y\) must be negative or both must be positive for the above result to be true.
(ii) \(x^2*y^3 < 0\)
Since square of any number must be positive, \(y\) must be negative for the above result to be true.
From (i) and (ii), it can be concluded that both \(x\) and \(y\) are negative.
I) This is true from the above.
II) This need not necessarily be true. For e.g., let \(x = y = -2\), then \((-2)^{2} * (-2)^{3} = 4 * -8 = -32\), but here \(x\) is not less than \(y\).
III) This must be true as both \(x\) and \(y\) are negative numbers.
So, only I and III are true. Ans - D.
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