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23 Feb 2019, 05:11
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
37% (02:15) correct
63%
(02:31)
wrong
based on 139
sessions
History
Date
Time
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Not Attempted Yet
If the sum of squares of two numbers is 97, then which one of the following cannot be their product?
1. 64
2. 16
3. −32
4. 48
5. 24
1. 64
2. 16
3. −32
4. 48
5. 24
Updated on: 28 Feb 2019, 15:12
Originally posted by AnthonyRitz on 28 Feb 2019, 01:55.
Last edited by AnthonyRitz on 28 Feb 2019, 15:12, edited 3 times in total.
Last edited by AnthonyRitz on 28 Feb 2019, 15:12, edited 3 times in total.
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I see this as a special quadratics problem.
\(A^2+B^2=97\), and we're asked about \(AB\). This should ring your special quadratics alarm bells for sure.
Note that \((A \pm B)^2 = A^2 \pm 2AB+B^2 \geq 0\). (Squares are never negative.)
This means that \(A^2+B^2 \geq \mp 2AB\), so \(\frac{97}{2} \geq |AB|\).
And since this means that \(AB\) definitely cannot be \(64\), we're done.
(For those of you citing the AM-GM inequality, yes, AM-GM is out of GMAT scope, and yes, I basically just re-derived the two-unknown case of it. However, this doesn't mean that one needs to know AM-GM, per se, to solve this problem, so I'm not entirely convinced that this problem is out of GMAT scope.)
\(A^2+B^2=97\), and we're asked about \(AB\). This should ring your special quadratics alarm bells for sure.
Note that \((A \pm B)^2 = A^2 \pm 2AB+B^2 \geq 0\). (Squares are never negative.)
This means that \(A^2+B^2 \geq \mp 2AB\), so \(\frac{97}{2} \geq |AB|\).
And since this means that \(AB\) definitely cannot be \(64\), we're done.
(For those of you citing the AM-GM inequality, yes, AM-GM is out of GMAT scope, and yes, I basically just re-derived the two-unknown case of it. However, this doesn't mean that one needs to know AM-GM, per se, to solve this problem, so I'm not entirely convinced that this problem is out of GMAT scope.)
24 Feb 2019, 18:53
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OhsostudiousMJ
This question is out of scope. To solve it, we have to know Inequality of arithmetic and geometric means.
a + b ≥ 2√ab
So, let's assume these two numbers are a and b. Then we have:
a² + b² = 97 ≥ 2√a² b² = 2ab
=> 2ab ≤ 97, ab ≤ 97/2 = 48.5
A.
