Step 2 2004 q2

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asianism
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#1
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#1
Is this question only true when x is not a complex number?
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asianism
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#2
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#2
question below
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asianism
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#3
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#3
Solution below. Neither solution nor question mentions complex numbers. 2-a^2/4 is greater than 0 but if x is complex than (x-a/2)^2 could be negative so they haven't proved it would definitely give a number greater than 0.
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robertgallacher
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#4
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Yes, eg. x=i, α=2

x^2 - α|x| + 2
= i^2 - 2|i| + 2
= -1 - 2(1) + 2
= -1 < 0

Hence there would exist a value
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DFranklin
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#5
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#5
(Original post by asianism)
Solution below. Neither solution nor question mentions complex numbers. 2-a^2/4 is greater than 0 but if x is complex than (x-a/2)^2 could be negative so they haven't proved it would definitely give a number greater than 0.
Context matters. x is obviously intended to be real here.
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asianism
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#6
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(Original post by DFranklin)
Context matters. x is obviously intended to be real here.
How would I tell if context matters? This is a STEP 2 question which involves further maths content like complex numbers...
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DFranklin
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#7
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(Original post by asianism)
How would I tell if context matters? This is a STEP 2 question which involves further maths content like complex numbers...
Well, context *always* matters, even if usually it's obvious enough that no-one remarks on it.
Being explicit, in this case the following observations are pretty much immediate:

(!) The variable name is x, rather than z, or w.
(2) There is no mention of complex numbers or \mathbb{C} anywhere in the question.
(3) The result is very obviously not true for complex numbers.
(4) The full question talks about intervals, which wouldn't make sense if x was allowed to be complex.

I'll grant you that the way |\alpha| is used might make you momentarily doubt whether you were supposed to working in \mathbb{C} rather than \mathbb{R}, but the deciding factor here would be that the question makes no sense under that assumption (due to (3) and (4)).
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