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Let a,b,c,d be the four integers such that a+b+c+d = 4m+1, where m is a positive integer. Given m, which one of the following is true?
A) The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2+2m+1
B)The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1
Explain this question please?
A) The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2+2m+1
B)The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1
Explain this question please?
Verified Answer
Let a,b,c,d be the four integers such that a+b+c+d = 4m+1, where m is ...
For any expression of the form a+b+c+d, the minimum value of a^2 + b^2+c^2+ d^2 will be when a = b = c = d. Or in other words, the four numbers have to be as close to the average as much as is possible. As the sum is in the form of 4m+ 1, we can say that the four numbers would give the minimum value of the above expression if they are m, m, m and m+1.
So, we get the sum of their squares to be:
4m^2 + 2m + 1 and so, option b is correct.
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Most Upvoted Answer
Let a,b,c,d be the four integers such that a+b+c+d = 4m+1, where m is ...
Question Analysis
The question provides four integers, a, b, c, and d, and states that their product is equal to 4m + 1, where m is a positive integer. The task is to determine which statement is true about the maximum or minimum possible value of a^2 * b^2 * c^2 * d^2 based on the given information.
Solution
To solve this problem, let's consider the properties of the given integers and their product.
Properties of a, b, c, and d
Since a, b, c, and d are integers and their product is equal to 4m + 1, we can infer the following properties:
- The four integers can be positive, negative, or zero.
- The product of the four integers can be positive, negative, or zero.
Properties of a^2, b^2, c^2, and d^2
The square of any integer is always non-negative. Therefore, we can conclude the following properties:
- a^2, b^2, c^2, and d^2 are non-negative integers.
- The product a^2 * b^2 * c^2 * d^2 is also a non-negative integer.
Statement A: The maximum possible value of a^2 * b^2 * c^2 * d^2 is 4m^2 + 2m + 1
This statement claims that the maximum possible value of the product a^2 * b^2 * c^2 * d^2 is equal to 4m^2 + 2m + 1. Let's analyze this claim.
To maximize the value of a^2 * b^2 * c^2 * d^2, we should choose the largest possible values for a, b, c, and d. Since the integers can be positive, negative, or zero, we have several cases to consider:
1. All four integers are positive.
2. Three integers are positive, and one integer is zero.
3. Two integers are positive, and two integers are negative.
4. One integer is positive, and three integers are negative.
5. All four integers are negative.
6. Three integers are negative, and one integer is zero.
7. Two integers are negative, and two integers are positive.
8. One integer is negative, and three integers are positive.
9. Three integers are zero, and one integer is positive.
10. Two integers are zero, and two integers have the same sign.
11. One integer is zero, and three integers have the same sign.
12. All four integers are zero.
Considering all these cases, we can see that it is not possible to determine a single maximum value for a^2 * b^2 * c^2 * d^2 based solely on the given information. Therefore, statement A is not true.
Statement B: The minimum possible value of a^2 * b^2 * c^2 * d^2 is 4m^2 + 2m + 1
This statement claims that the minimum possible value of the product a^2 * b^2 * c^2 * d^2 is equal to 4m^2 +
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Let a,b,c,d be the four integers such that a+b+c+d = 4m+1, where m is a positive integer. Given m, which one of the following is true?A) The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2+2m+1B)The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1Explain this question please?
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Let a,b,c,d be the four integers such that a+b+c+d = 4m+1, where m is a positive integer. Given m, which one of the following is true?A) The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2+2m+1B)The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1Explain this question please? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Let a,b,c,d be the four integers such that a+b+c+d = 4m+1, where m is a positive integer. Given m, which one of the following is true?A) The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2+2m+1B)The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1Explain this question please? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let a,b,c,d be the four integers such that a+b+c+d = 4m+1, where m is a positive integer. Given m, which one of the following is true?A) The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2+2m+1B)The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 + 2m + 1Explain this question please?.
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