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Joined: 01 Nov 2017
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If the sum of squares of two numbers is 97, then which one of the foll
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23 Feb 2019, 05:11
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31% (02:53) correct 69% (02:33) wrong based on 75 sessions
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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
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Re: If the sum of squares of two numbers is 97, then which one of the foll
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24 Feb 2019, 01:38
Can someone please explain me the answer?
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Re: If the sum of squares of two numbers is 97, then which one of the foll
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24 Feb 2019, 11:09
Is the question like which one could be their product?
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If the sum of squares of two numbers is 97, then which one of the foll
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24 Feb 2019, 18:53
OhsostudiousMJ wrote: Can someone please explain me the answer? This question is out of scope. To solve it, we have to know Inequality of arithmetic and geometric means. a + b ≥ 2√abSo, 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.



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Re: If the sum of squares of two numbers is 97, then which one of the foll
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25 Feb 2019, 01:17
I tried to solve it by evenodd concept however i failed. is it possible to solve these question by that concept?



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Re: If the sum of squares of two numbers is 97, then which one of the foll
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27 Feb 2019, 21:26
OhsostudiousMJ wrote: Can someone please explain me the answer? Let the two nos. be a and b Suppose we don't know the relation between square of two nos. and its product. Standard mathematics says that sum of a number and its reciprocal is always >=2 Thus a/b + b/a > =2 Let us multiply the two sides by ab The a^2 + b^2 >= 2ab Thus sum of squares of two nos. is always >= Twice its product. Thus 97 >= 2*ab In option A , Product = 64 Thus 2* Product = 128 > 97 This cant be true



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If the sum of squares of two numbers is 97, then which one of the foll
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Updated on: 28 Feb 2019, 15:12
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 AMGM inequality, yes, AMGM is out of GMAT scope, and yes, I basically just rederived the twounknown case of it. However, this doesn't mean that one needs to know AMGM, per se, to solve this problem, so I'm not entirely convinced that this problem is out of GMAT scope.)
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.



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Re: If the sum of squares of two numbers is 97, then which one of the foll
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28 Feb 2019, 02:47
AnthonyRitz wrote: 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 \(AB\) definitely cannot be \(64\), we're done.
(Yes, for those of you citing the AMGM inequality, it is out of GMAT scope, and I basically just rederived the twounknown case of it. However, this doesn't mean that one needs to know AMGM, per se, to solve this problem, so I'm not entirely convinced that this problem is out of GMAT scope.) The best explanation, thanks.



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Re: If the sum of squares of two numbers is 97, then which one of the foll
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28 Feb 2019, 04:01
raghavrf wrote: 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 Given a constant sum of two numbers, the product takes the maximum value when the numbers are equal. Since a^2 + b^2 is constant, a^2*b^2 takes the maximum value when a^2 = b^2 = 97/2 = 48.5 Maximum value of a^2b^2 = (48.5 * 48.5) = 48.5^2 So maximum value of ab = 48.5 Hence the product cannot be 64. Answer (A) For more on this, check: https://www.veritasprep.com/blog/2015/0 ... atpartv/
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Re: If the sum of squares of two numbers is 97, then which one of the foll
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