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If the sum of squares of two numbers is 97, then which one of the foll [#permalink]
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Question Stats: Hide Show timer StatisticsIf the sum of squares of two numbers is 97, then which one of the following cannot be their product?1. 642. 163. −324. 48 5. 24
Intern Joined: 29 Dec 2018 Posts: 21 GMAT 1: 720 Q50 V35
If the sum of squares of two numbers is 97, then which one of the foll [#permalink]
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√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 = 48.5A.
Re: If the sum of squares of two numbers is 97, then which one of the foll [#permalink] 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 [#permalink] Is the question like which one could be their product? Posted from my mobile device
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Re: If the sum of squares of two numbers is 97, then which one of the foll [#permalink] I tried to solve it by even-odd 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 [#permalink]
OhsostudiousMJ wrote: Can someone please explain me the answer? Let the two nos. be a and bSuppose 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 >=2Thusa/b + b/a > =2Let us multiply the two sides by abThe a^2 + b^2 >= 2abThus sum of squares of two nos. is always >= Twice its product.Thus97 >= 2*abIn option A , Product = 64Thus 2* Product = 128 > 97This cant be true
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If the sum of squares of two numbers is 97, then which one of the foll [#permalink] 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.) _________________
Originally posted by AnthonyRitz on 28 Feb 2019, 00:55.
Intern Joined: 29 Dec 2018 Posts: 21 GMAT 1: 720 Q50 V35
Re: If the sum of squares of two numbers is 97, then which one of the foll [#permalink]
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 AM-GM inequality, it is out of GMAT scope, and 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.) 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 [#permalink]
raghavrf wrote: If the sum of squares of two numbers is 97, then which one of the following cannot be their product?1. 642. 163. −324. 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 = 48.5Maximum value of a^2b^2 = (48.5 * 48.5) = 48.5^2So maximum value of ab = 48.5Hence the product cannot be 64.Answer (A)For more on this, check: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html?/2015/0 ... at-part-v/ _________________
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Re: If the sum of squares of two numbers is 97, then which one of the foll [#permalink] Hello from the GMAT Club BumpBot!Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________
Re: If the sum of squares of two numbers is 97, then which one of the foll [#permalink] 05 May 2020, 06:03 |
If the sum of squares of two numbers is 97, then which one of the following cannot be their product?
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