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While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers is
Correct answer is '40'. Can you explain this answer?
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Verified Answer
While multiplying three real numbers, Ashok took one of the numbers as...
We know that one of the 3 numbers is 37.
Let the product of the other 2 numbers be x.
It has been given that 73x-37x = 720
36x = 720
x = 20
Product of 2 real numbers is 20.
We have to find the minimum possible value of the sum of the squares of the 2 numbers.
Let x=a*b
It has been given that a*b=20
The least possible sum for a given product is obtained when the numbers are as close to each other as possible.
Therefore, when a=b, the value of a and b will be √20.
Sum of the squares of the 2 numbers = 20 + 20 = 40.
Therefore, 40 is the correct answer.
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Most Upvoted Answer
While multiplying three real numbers, Ashok took one of the numbers as...
Approach:
- Let the other two numbers be x and y.
- Initially, the product of the three numbers is xyz.
- After replacing one of the numbers with 73, the new product is (73)(xy) = 720 + xyz.
- Therefore, xyz - (73)(xy) = -720.
- We can factor this equation as follows: (x - 73)(y - 73)z = -720.
- Since x, y, and z are real numbers, we know that (x - 73)(y - 73) and z have the same sign.
- If (x - 73)(y - 73) and z are positive, then their product is positive, which means xyz - (73)(xy) > 0, which is a contradiction.
- Therefore, (x - 73)(y - 73) and z are negative.
- Since we want to minimize x^2 + y^2, we can assume without loss of generality that x <=>
- We can rewrite x^2 + y^2 as (x + y)^2 - 2xy.
- Let s = x + y and t = xy. Then we have s^2 - 2t = x^2 + y^2.
- We want to minimize s^2 - 2t subject to the constraint (x - 73)(y - 73)z = -720 and s = x + y.
- We can use Lagrange multipliers to solve this optimization problem.

Solution:
- Let L = s^2 - 2t + lambda[(x - 73)(y - 73)z + 720 - s].
- Taking partial derivatives with respect to s, t, x, y, and z, we get:
- ds/dt = -2, dt/ds = -1/2, dx/dt = -lambda(y - 73)z, dy/dt = -lambda(x - 73)z, dz/dt = -lambda(x - 73)(y - 73).
- Setting these equal to zero, we get:
- t = s/2,
- lambda = -2/s^2,
- (x - 73)(y - 73)z = -720/s + s,
- (x - 73)z = (y - 73)z,
- (y - 73)z = (x - 73)z.
- Solving these equations, we get:
- z = -720/[s(s - 146)],
- y = (73s - 720)/(s - 146),
- x = (73s + 720)/(s + 146).
- Since x <= y,="" we="">
- (73s + 720)/(s + 146) <= (73s="" -="" 720)/(s="" -="">
- 73s^2 - 720s - 105120 <= 73s^2="" +="" 720s="" -="">
- s^2 <=>
- Therefore, the minimum possible value of s^2 - 2t = x^2 + y^2 is (720/73) + 720/73 = 40.
- Hence
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Directions: Read the following information carefully and answer the given question.A minimum viable product (MVP) is a development technique in which a new product or website is developed with sufficient features to satisfy early adopters. The final, complete set of features is only designed and developed after considering feedback from the product's initial users. This technique helps them in making the final product much better. With the help of MVP concept, the research or the marketing team will come to know where the product is lacking and or what are its strengths or weaknesses.MVP has three distinct features. One is that it will have enough features for consumers to purchase the product (it becomes easier for the company to market it), the other is that it will have some sort of a feedback mechanism wherein users would be able to send their feedback about the product. And, lastly it should have enough future benefits for consumers who choose to adopt the product first (Google gave free upgrade of its OS to all Nexus users). The idea is to get feedback from the consumers which will in turn help in making the desired changes in the final product. MVP actually tests the usage scenario that is much more helpful for the company to make changes to the final product.The developing team of an MVP will not waste any time on anything beyond the bare minimum, and build every other feature over time as they assess the customers' wishes and preferences as they start using the product. The product may change even dramatically or even get abandoned as feedback from users may significantly diverge from the original project. However, the developing teams will not waste any resources (efforts, time, money, advertising) on a product that no one really wants, needs, or likes. The catch to this development technique is that it assumes that early adopters can see the vision or promise of the final product and provide the valuable feedback needed to guide developers forward.The whole focus is learning during product development by collecting information from customers that are validated already. Observing users as they utilize the product is much more reliable than trying to validate marketing hypotheses with questionnaires or forecasts. A common pitfall is to release a bare-bone product that is so much below the users' expectations that is not viable at all. In general, the MVP technique is particularly viable for technically-oriented products used by technical users who are often keen on providing practical insights on how to improve or update the product. The concept of MVP has, in fact, been widely adopted by many verticals in the informatics industry. For example, many video games today spend many months (sometimes years) as early access products and are slowly developed as the community grows. Some popular franchises, such as Farmville, fully leveraged the concept of MVP by implementing new gameplay features over time while progressively improving overall user experience.Q. Which of the following statements is the author most likely to disagree about?

Directions: Read the following information carefully and answer the given question.A minimum viable product (MVP) is a development technique in which a new product or website is developed with sufficient features to satisfy early adopters. The final, complete set of features is only designed and developed after considering feedback from the product's initial users. This technique helps them in making the final product much better. With the help of MVP concept, the research or the marketing team will come to know where the product is lacking and or what are its strengths or weaknesses.MVP has three distinct features. One is that it will have enough features for consumers to purchase the product (it becomes easier for the company to market it), the other is that it will have some sort of a feedback mechanism wherein users would be able to send their feedback about the product. And, lastly it should have enough future benefits for consumers who choose to adopt the product first (Google gave free upgrade of its OS to all Nexus users). The idea is to get feedback from the consumers which will in turn help in making the desired changes in the final product. MVP actually tests the usage scenario that is much more helpful for the company to make changes to the final product.The developing team of an MVP will not waste any time on anything beyond the bare minimum, and build every other feature over time as they assess the customers' wishes and preferences as they start using the product. The product may change even dramatically or even get abandoned as feedback from users may significantly diverge from the original project. However, the developing teams will not waste any resources (efforts, time, money, advertising) on a product that no one really wants, needs, or likes. The catch to this development technique is that it assumes that early adopters can see the vision or promise of the final product and provide the valuable feedback needed to guide developers forward.The whole focus is learning during product development by collecting information from customers that are validated already. Observing users as they utilize the product is much more reliable than trying to validate marketing hypotheses with questionnaires or forecasts. A common pitfall is to release a bare-bone product that is so much below the users' expectations that is not viable at all. In general, the MVP technique is particularly viable for technically-oriented products used by technical users who are often keen on providing practical insights on how to improve or update the product. The concept of MVP has, in fact, been widely adopted by many verticals in the informatics industry. For example, many video games today spend many months (sometimes years) as early access products and are slowly developed as the community grows. Some popular franchises, such as Farmville, fully leveraged the concept of MVP by implementing new gameplay features over time while progressively improving overall user experience.Q. Which of the following statements is true in context of the passage?

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While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers isCorrect answer is '40'. Can you explain this answer?
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While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers isCorrect answer is '40'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers isCorrect answer is '40'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for While multiplying three real numbers, Ashok took one of the numbers as 73 instead of 37. As a result, the product went up by 720. Then the minimum possible value of the sum of squares of the other two numbers isCorrect answer is '40'. Can you explain this answer?.
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