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Let a, b, c, d be four integers such that a + b + c + d = 4m + 1 where m is a positive integer. Given m, which one of the following is necessarily true?
- a)The minimum possible value of a2 + b2 + c2 + d2 is 4m2 – 2m + 1
- b)The minimum possible value of a2 + b2 + c2 + d2 is 4m2 + 2m + 1
- c)The maximum possible value of a2 + b2 + c2 + d2 is 4m2 – 2m +1
- d)The maximum possible value of a2 + b2 +c2 + d2 is 4m2 + 2m + 1
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Leta,b,c,dbe four integers such thata+b+c+d= 4m+ 1 wheremis a positive...
(a + b + + d)2 = (4m + 1)2
Thus, a2 + b2 + c2 + d2 + 2(ab + ac + ad + bc + bd + cd) = 16m2 + 8m + 1
a2 + b2 + c2 + d2 will have the minimum value if (ab + ac + ad + be + bd + cd) is the maximum.
This is possible if a=b=c=d= (m + 0.25) ... since a +b+c+d= 4m+ 1
In that case 2(ab + ac + ad + bc + bd + cd) = 12(m + 0.25)2 = 12m2 + 6m + 0.75
Thus, the minimum value of a2 + b2 + c2 + d2 = (16m2 + 8m + 1)— 2(ab + ac + ad + bc + bd + cd) = (16m2 + 8m + 1)— (12m2 + 6m + 0.75) = 4m2 + 2m + 0.25
Since it is an integer, the actual minimum value = 4m2 + 2m + 1
Most Upvoted Answer
Leta,b,c,dbe four integers such thata+b+c+d= 4m+ 1 wheremis a positive...
The minimum possible value of a² + b² + c² + d² is 4m².
To see why this is true, consider the case where all four integers are equal to m. In this case, we have a² + b² + c² + d² = m² + m² + m² + m² = 4m².
Now, consider any other combination of four integers that multiply to 4m + 1. We can write these integers as m + x, m + y, m + z, and m + w, where x, y, z, and w are positive or negative integers.
Expanding (m + x)² + (m + y)² + (m + z)² + (m + w)², we get m² + 2mx + x² + m² + 2my + y² + m² + 2mz + z² + m² + 2mw + w².
Simplifying this, we have 4m² + 2m(x + y + z + w) + (x² + y² + z² + w²).
Since x, y, z, and w can be positive or negative, their sum can be any integer. Therefore, we can make 2m(x + y + z + w) = 0, resulting in the minimum possible value of a² + b² + c² + d² being 4m² + 0 + (x² + y² + z² + w²) = 4m².
Therefore, the minimum possible value of a² + b² + c² + d² is 4m².
To see why this is true, consider the case where all four integers are equal to m. In this case, we have a² + b² + c² + d² = m² + m² + m² + m² = 4m².
Now, consider any other combination of four integers that multiply to 4m + 1. We can write these integers as m + x, m + y, m + z, and m + w, where x, y, z, and w are positive or negative integers.
Expanding (m + x)² + (m + y)² + (m + z)² + (m + w)², we get m² + 2mx + x² + m² + 2my + y² + m² + 2mz + z² + m² + 2mw + w².
Simplifying this, we have 4m² + 2m(x + y + z + w) + (x² + y² + z² + w²).
Since x, y, z, and w can be positive or negative, their sum can be any integer. Therefore, we can make 2m(x + y + z + w) = 0, resulting in the minimum possible value of a² + b² + c² + d² being 4m² + 0 + (x² + y² + z² + w²) = 4m².
Therefore, the minimum possible value of a² + b² + c² + d² is 4m².
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Leta,b,c,dbe four integers such thata+b+c+d= 4m+ 1 wheremis a positive integer. Givenm, which one of the following is necessarily true?a)The minimum possible value ofa2+b2+c2+d2is 4m2– 2m+ 1b)The minimum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1c)The maximum possible value ofa2+b2+c2+d2is 4m2– 2m+1d)The maximum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1Correct answer is option 'B'. Can you explain this answer?
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Leta,b,c,dbe four integers such thata+b+c+d= 4m+ 1 wheremis a positive integer. Givenm, which one of the following is necessarily true?a)The minimum possible value ofa2+b2+c2+d2is 4m2– 2m+ 1b)The minimum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1c)The maximum possible value ofa2+b2+c2+d2is 4m2– 2m+1d)The maximum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1Correct answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Leta,b,c,dbe four integers such thata+b+c+d= 4m+ 1 wheremis a positive integer. Givenm, which one of the following is necessarily true?a)The minimum possible value ofa2+b2+c2+d2is 4m2– 2m+ 1b)The minimum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1c)The maximum possible value ofa2+b2+c2+d2is 4m2– 2m+1d)The maximum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Leta,b,c,dbe four integers such thata+b+c+d= 4m+ 1 wheremis a positive integer. Givenm, which one of the following is necessarily true?a)The minimum possible value ofa2+b2+c2+d2is 4m2– 2m+ 1b)The minimum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1c)The maximum possible value ofa2+b2+c2+d2is 4m2– 2m+1d)The maximum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1Correct answer is option 'B'. Can you explain this answer?.
Leta,b,c,dbe four integers such thata+b+c+d= 4m+ 1 wheremis a positive integer. Givenm, which one of the following is necessarily true?a)The minimum possible value ofa2+b2+c2+d2is 4m2– 2m+ 1b)The minimum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1c)The maximum possible value ofa2+b2+c2+d2is 4m2– 2m+1d)The maximum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1Correct answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Leta,b,c,dbe four integers such thata+b+c+d= 4m+ 1 wheremis a positive integer. Givenm, which one of the following is necessarily true?a)The minimum possible value ofa2+b2+c2+d2is 4m2– 2m+ 1b)The minimum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1c)The maximum possible value ofa2+b2+c2+d2is 4m2– 2m+1d)The maximum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Leta,b,c,dbe four integers such thata+b+c+d= 4m+ 1 wheremis a positive integer. Givenm, which one of the following is necessarily true?a)The minimum possible value ofa2+b2+c2+d2is 4m2– 2m+ 1b)The minimum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1c)The maximum possible value ofa2+b2+c2+d2is 4m2– 2m+1d)The maximum possible value ofa2+b2+c2+d2is 4m2+ 2m+ 1Correct answer is option 'B'. Can you explain this answer?.
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