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If n is an integer, then which of the following statements is/are FALSE?
I. n3 – n is always even.
II. 8n3 +12n2 +6n +1 is always even.
III. √ (4n2 – 4n +1) is always odd.
- a)I only
- b)II only
- c)I and II only
- d)II and III only
- e)I, II and III
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If n is an integer, then which of the following statements is/are FALS...
Statement I: n^3 - n is always even
To determine the truth of this statement, we need to consider two cases: when n is even and when n is odd.
1. When n is even:
If n is even, it can be expressed as n = 2k, where k is an integer. Substituting this into the expression, we get:
n^3 - n = (2k)^3 - 2k = 8k^3 - 2k = 2(4k^3 - k)
Since 4k^3 - k is an integer, we can let it be represented by m, where m is an integer. Therefore, we have:
n^3 - n = 2m
This shows that when n is even, n^3 - n is divisible by 2 and therefore even.
2. When n is odd:
If n is odd, it can be expressed as n = 2k + 1, where k is an integer. Substituting this into the expression, we get:
n^3 - n = (2k + 1)^3 - (2k + 1) = 8k^3 + 12k^2 + 6k + 1 - 2k - 1 = 8k^3 + 12k^2 + 4k
In this case, the expression 8k^3 + 12k^2 + 4k is not divisible by 2, and therefore n^3 - n is not even when n is odd.
Since n^3 - n is not always even for all integers n, Statement I is FALSE.
Statement II: 8n^3 - 12n^2 + 6n - 1 is always even
To determine the truth of this statement, we need to consider two cases: when n is even and when n is odd.
1. When n is even:
Similar to the previous case, if n is even, it can be expressed as n = 2k, where k is an integer. Substituting this into the expression, we get:
8n^3 - 12n^2 + 6n - 1 = 8(2k)^3 - 12(2k)^2 + 6(2k) - 1 = 8(8k^3) - 12(4k^2) + 12k - 1 = 2(32k^3 - 6k^2 + 6k) - 1
Since 32k^3 - 6k^2 + 6k is an integer, we can let it be represented by m, where m is an integer. Therefore, we have:
8n^3 - 12n^2 + 6n - 1 = 2m - 1
This shows that when n is even, 8n^3 - 12n^2 + 6n - 1 is divisible by 2 and therefore even.
2. When n is odd:
Similar to the previous case, if n is odd, it can be expressed as n = 2k + 1, where k is an integer. Substituting this into the expression, we get:
8n^3 - 12n^2 + 6n -
To determine the truth of this statement, we need to consider two cases: when n is even and when n is odd.
1. When n is even:
If n is even, it can be expressed as n = 2k, where k is an integer. Substituting this into the expression, we get:
n^3 - n = (2k)^3 - 2k = 8k^3 - 2k = 2(4k^3 - k)
Since 4k^3 - k is an integer, we can let it be represented by m, where m is an integer. Therefore, we have:
n^3 - n = 2m
This shows that when n is even, n^3 - n is divisible by 2 and therefore even.
2. When n is odd:
If n is odd, it can be expressed as n = 2k + 1, where k is an integer. Substituting this into the expression, we get:
n^3 - n = (2k + 1)^3 - (2k + 1) = 8k^3 + 12k^2 + 6k + 1 - 2k - 1 = 8k^3 + 12k^2 + 4k
In this case, the expression 8k^3 + 12k^2 + 4k is not divisible by 2, and therefore n^3 - n is not even when n is odd.
Since n^3 - n is not always even for all integers n, Statement I is FALSE.
Statement II: 8n^3 - 12n^2 + 6n - 1 is always even
To determine the truth of this statement, we need to consider two cases: when n is even and when n is odd.
1. When n is even:
Similar to the previous case, if n is even, it can be expressed as n = 2k, where k is an integer. Substituting this into the expression, we get:
8n^3 - 12n^2 + 6n - 1 = 8(2k)^3 - 12(2k)^2 + 6(2k) - 1 = 8(8k^3) - 12(4k^2) + 12k - 1 = 2(32k^3 - 6k^2 + 6k) - 1
Since 32k^3 - 6k^2 + 6k is an integer, we can let it be represented by m, where m is an integer. Therefore, we have:
8n^3 - 12n^2 + 6n - 1 = 2m - 1
This shows that when n is even, 8n^3 - 12n^2 + 6n - 1 is divisible by 2 and therefore even.
2. When n is odd:
Similar to the previous case, if n is odd, it can be expressed as n = 2k + 1, where k is an integer. Substituting this into the expression, we get:
8n^3 - 12n^2 + 6n -
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If n is an integer, then which of the following statements is/are FALS...
√4n2-4n+1=√(2n-1)2=2n-1 for every value of n the expression becomes odd n3-n =n(n²-1)=n*(n-1)*(n+1) is always even for every n value
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If n is an integer, then which of the following statements is/are FALSE?I. n3 – n is always even.II. 8n3 +12n2 +6n +1 is always even.III. √ (4n2 – 4n +1) is always odd.a)I onlyb)II onlyc)I and II onlyd)II and III onlye)I, II and IIICorrect answer is option 'B'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If n is an integer, then which of the following statements is/are FALSE?I. n3 – n is always even.II. 8n3 +12n2 +6n +1 is always even.III. √ (4n2 – 4n +1) is always odd.a)I onlyb)II onlyc)I and II onlyd)II and III onlye)I, II and IIICorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If n is an integer, then which of the following statements is/are FALSE?I. n3 – n is always even.II. 8n3 +12n2 +6n +1 is always even.III. √ (4n2 – 4n +1) is always odd.a)I onlyb)II onlyc)I and II onlyd)II and III onlye)I, II and IIICorrect answer is option 'B'. Can you explain this answer?.
If n is an integer, then which of the following statements is/are FALSE?I. n3 – n is always even.II. 8n3 +12n2 +6n +1 is always even.III. √ (4n2 – 4n +1) is always odd.a)I onlyb)II onlyc)I and II onlyd)II and III onlye)I, II and IIICorrect answer is option 'B'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If n is an integer, then which of the following statements is/are FALSE?I. n3 – n is always even.II. 8n3 +12n2 +6n +1 is always even.III. √ (4n2 – 4n +1) is always odd.a)I onlyb)II onlyc)I and II onlyd)II and III onlye)I, II and IIICorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If n is an integer, then which of the following statements is/are FALSE?I. n3 – n is always even.II. 8n3 +12n2 +6n +1 is always even.III. √ (4n2 – 4n +1) is always odd.a)I onlyb)II onlyc)I and II onlyd)II and III onlye)I, II and IIICorrect answer is option 'B'. Can you explain this answer?.
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ample number of questions to practice If n is an integer, then which of the following statements is/are FALSE?I. n3 – n is always even.II. 8n3 +12n2 +6n +1 is always even.III. √ (4n2 – 4n +1) is always odd.a)I onlyb)II onlyc)I and II onlyd)II and III onlye)I, II and IIICorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice GMAT tests.
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