Carcass wrote:
If \(1 < p < 3\), then which of the following could be true?
(I) \(p^2 < 2p\)
(II) \(p^2 = 2p\)
(III) \(p^2 > 2p\)
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III
Algebraically.
Given that p is positive from 1 < p < 3
p^2 < 2p
p^2 - 2p < 0
p (p - 2) < 0 possible if p is less than 2.
Also p < 2 which is already given from the statement.
p^2 = 2p
p^2 - 2p = 0
p (p-2) = 0
Either p = 0 or 2, we know that p cannot be 0 so it must be 2
p = 2 (possible)
The last option as p^2 > 2p
p^2 - 2p > 0
p (p-2) > 0 possible if p > 2
Or p > 2 (true according to the stem)
Answer choice E.